In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results
Y.B.Jun et al. [9] introduced the notion of Cubic sets and Cubic subgroups. In this paper we introduced the
notion of cubic BF- Algebra i.e., an interval-valued BF-Algebra and an anti fuzzy BF-Algebra. Intersection of two cubic
BF- Algebras is again a cubic BF-Algebra is also studied.
Pseudo Bipolar Fuzzy Cosets of Bipolar Fuzzy and Bipolar Anti-Fuzzy HX Subgroupsmathsjournal
ย
In this paper, we introduce the concept of pseudo bipolar fuzzy cosets, pseudo bipolar fuzzy double cosets of a bipolar fuzzy and bipolar anti-fuzzy subgroups. We also establish these concepts to bipolar fuzzy and bipolar anti-fuzzy HX subgroups of a HX group with suitable examples. Also we discuss some of their relative properties.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
ย
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
A Study on Intuitionistic Multi-Anti Fuzzy Subgroups mathsjournal
ย
For any intuitionistic multi-fuzzy set A = { < x , ยตA(x) , ฮฝA(x) > : xโX} of an universe set X, we study the set [A](ฮฑ, ฮฒ) called the (ฮฑ, ฮฒ)โlower cut of A. It is the crisp multi-set { xโX : ยตi(x) โค ฮฑi , ฮฝi(x) โฅ ฮฒi , โi } of X. In this paper, an attempt has been made to study some algebraic structure of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (ฮฑ, ฮฒ)โlower cut sets
Regularity of Generalized Derivations in P Semi Simple BCIK AlgebrasYogeshIJTSRD
ย
In this paper we study the regularity of inside or outside derivations in p semi simple BCIK โ algebra X and prove that let d X X be an inside derivation of X. If there exists a X such that d x a = 0, then d is regular for all x X. It is also show that if X is a BCIK algebra, then every inside or outside derivation of X is regular. Furthermore the concepts of ideal, ideal and invariant inside or outside derivation of X are introduced and their related properties are investigated. Finally we obtain the following result If d X X is an outside derivation of X, then d is regular if and only if every ideal of X is d invariant. S Rethina Kumar "Regularity of Generalized Derivations in P-Semi Simple BCIK-Algebras" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd39949.pdf Paper URL: https://www.ijtsrd.com/mathemetics/algebra/39949/regularity-of-generalized-derivations-in-psemi-simple-bcikalgebras/s-rethina-kumar
Y.B.Jun et al. [9] introduced the notion of Cubic sets and Cubic subgroups. In this paper we introduced the
notion of cubic BF- Algebra i.e., an interval-valued BF-Algebra and an anti fuzzy BF-Algebra. Intersection of two cubic
BF- Algebras is again a cubic BF-Algebra is also studied.
Pseudo Bipolar Fuzzy Cosets of Bipolar Fuzzy and Bipolar Anti-Fuzzy HX Subgroupsmathsjournal
ย
In this paper, we introduce the concept of pseudo bipolar fuzzy cosets, pseudo bipolar fuzzy double cosets of a bipolar fuzzy and bipolar anti-fuzzy subgroups. We also establish these concepts to bipolar fuzzy and bipolar anti-fuzzy HX subgroups of a HX group with suitable examples. Also we discuss some of their relative properties.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
ย
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
A Study on Intuitionistic Multi-Anti Fuzzy Subgroups mathsjournal
ย
For any intuitionistic multi-fuzzy set A = { < x , ยตA(x) , ฮฝA(x) > : xโX} of an universe set X, we study the set [A](ฮฑ, ฮฒ) called the (ฮฑ, ฮฒ)โlower cut of A. It is the crisp multi-set { xโX : ยตi(x) โค ฮฑi , ฮฝi(x) โฅ ฮฒi , โi } of X. In this paper, an attempt has been made to study some algebraic structure of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (ฮฑ, ฮฒ)โlower cut sets
Regularity of Generalized Derivations in P Semi Simple BCIK AlgebrasYogeshIJTSRD
ย
In this paper we study the regularity of inside or outside derivations in p semi simple BCIK โ algebra X and prove that let d X X be an inside derivation of X. If there exists a X such that d x a = 0, then d is regular for all x X. It is also show that if X is a BCIK algebra, then every inside or outside derivation of X is regular. Furthermore the concepts of ideal, ideal and invariant inside or outside derivation of X are introduced and their related properties are investigated. Finally we obtain the following result If d X X is an outside derivation of X, then d is regular if and only if every ideal of X is d invariant. S Rethina Kumar "Regularity of Generalized Derivations in P-Semi Simple BCIK-Algebras" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd39949.pdf Paper URL: https://www.ijtsrd.com/mathemetics/algebra/39949/regularity-of-generalized-derivations-in-psemi-simple-bcikalgebras/s-rethina-kumar
This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the
equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct
a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of
fuzzy congruence of group G (respectively ring R) is isomorphic to the lattice of fuzzy normal subgroup of G
(respectively fuzzy ideals of R).In Yuan Boond Wu wangrning investigated the relationship between the fuzzy
ideals and the fuzzy congruences on a distributive lattice and obtained that the lattice of fuzzy ideals is
isomorphic to the lattice of fuzzy congruences on a generalized Boolean algebra. Fuzzy group theory can be
used to describe, symmetries and permutation in nature and mathematics. The fuzzy group is one of the oldest
branches of abstract algebra. For example group can be used is classify to all of the forms chemical crystal can
take. Group can be used to count the number of non-equivalent objects and permutation or symmetries. For
example, the number of different is switching functions of n, variable when permutation of the input are
allowed. Beside crystallography and combinatory group have application of quantum mechanics.
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
ย
This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method
based on the C-means algorithm, using the defined partition, is presented in this paper, which will
be validated with the traditional iris clustering problem by measuring its petals.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
ย
In this article we assume that experts express their view points by way of approximation of Triangular fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A comparison has been made between approximation of triangular fuzzy systems and the corresponding fuzzy triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
ย
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
ย
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
In this paper, the notion a -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The
homomorphic images (pre images) ofa -anti fuzzy new-ideal under homomorphism of a PU-algebras has
been obtained. Some related result have been derived.
Intuitionistic Fuzzification of T-Ideals in Bci-Algebrasiosrjce
ย
The notions of intuitionistic fuzzy T-ideals in BCI-algebras are introduced.Conditions for an intuitionistic fuzzy ideal to be an intuitionistic fuzzy T-idealare provided. Using a collection of T-ideals, intuitionistic fuzzy T-ideals areestablished.
A Study on Intuitionistic Multi-Anti Fuzzy Subgroupsmathsjournal
ย
For any intuitionistic multi-fuzzy set A = { < x , ยตA(x) , ฮฝA(x) > : xโX} of an universe set X, we study the set [A](ฮฑ, ฮฒ) called the (ฮฑ, ฮฒ)โlower cut of A. It is the crisp multi-set { xโX : ยตi(x) โค ฮฑi ,ฮฝi(x) โฅ ฮฒi , โi } of X. In this paper, an attempt has been made to study some algebraic structure of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (ฮฑ, ฮฒ)โlower cut sets.
In this paper, the notion ฮฑ -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The homomorphic images (pre images) ofฮฑ -anti fuzzy new-ideal under homomorphism of a PU-algebras has been obtained. Some related result have been derived.
This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the
equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct
a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of
fuzzy congruence of group G (respectively ring R) is isomorphic to the lattice of fuzzy normal subgroup of G
(respectively fuzzy ideals of R).In Yuan Boond Wu wangrning investigated the relationship between the fuzzy
ideals and the fuzzy congruences on a distributive lattice and obtained that the lattice of fuzzy ideals is
isomorphic to the lattice of fuzzy congruences on a generalized Boolean algebra. Fuzzy group theory can be
used to describe, symmetries and permutation in nature and mathematics. The fuzzy group is one of the oldest
branches of abstract algebra. For example group can be used is classify to all of the forms chemical crystal can
take. Group can be used to count the number of non-equivalent objects and permutation or symmetries. For
example, the number of different is switching functions of n, variable when permutation of the input are
allowed. Beside crystallography and combinatory group have application of quantum mechanics.
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
ย
This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method
based on the C-means algorithm, using the defined partition, is presented in this paper, which will
be validated with the traditional iris clustering problem by measuring its petals.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
ย
In this article we assume that experts express their view points by way of approximation of Triangular fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A comparison has been made between approximation of triangular fuzzy systems and the corresponding fuzzy triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
ย
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
ย
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
In this paper, the notion a -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The
homomorphic images (pre images) ofa -anti fuzzy new-ideal under homomorphism of a PU-algebras has
been obtained. Some related result have been derived.
Intuitionistic Fuzzification of T-Ideals in Bci-Algebrasiosrjce
ย
The notions of intuitionistic fuzzy T-ideals in BCI-algebras are introduced.Conditions for an intuitionistic fuzzy ideal to be an intuitionistic fuzzy T-idealare provided. Using a collection of T-ideals, intuitionistic fuzzy T-ideals areestablished.
A Study on Intuitionistic Multi-Anti Fuzzy Subgroupsmathsjournal
ย
For any intuitionistic multi-fuzzy set A = { < x , ยตA(x) , ฮฝA(x) > : xโX} of an universe set X, we study the set [A](ฮฑ, ฮฒ) called the (ฮฑ, ฮฒ)โlower cut of A. It is the crisp multi-set { xโX : ยตi(x) โค ฮฑi ,ฮฝi(x) โฅ ฮฒi , โi } of X. In this paper, an attempt has been made to study some algebraic structure of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (ฮฑ, ฮฒ)โlower cut sets.
In this paper, the notion ฮฑ -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The homomorphic images (pre images) ofฮฑ -anti fuzzy new-ideal under homomorphism of a PU-algebras has been obtained. Some related result have been derived.
CUBIC STRUCTURES OF MEDIAL IDEAL ON BCI -ALGEBRAS Wireilla
ย
In this paper, we introduce the concept of cubic medial-ideal and investigate several properties. Also, we give relations between cubic medial-ideal and cubic BCI-ideal .The image and the pre-image of cubic medial-ideal under homomorphism of BCI-algebras are defined and how the image and the pre-image of cubic medial-ideal under homomorphism of BCI-algebras become cubic medial-ideal are studied. Moreover, the Cartesian product of cubic medial-ideal in Cartesian product BCI-algebras is given. 2010 AMS Classification. 06F35, 03G25, 08A72
CUBIC STRUCTURES OF MEDIAL IDEAL ON BCI -ALGEBRASijfls
ย
In this paper, we introduce the concept of cubic medial-ideal and investigate several properties. Also, we
give relations between cubic medial-ideal and cubic BCI-ideal .The image and the pre-image of cubic
medial-ideal under homomorphism of BCI-algebras are defined and how the image and the pre-image of
cubic medial-ideal under homomorphism of BCI-algebras become cubic medial-ideal are studied.
Moreover, the Cartesian product of cubic medial-ideal in Cartesian product BCI-algebras is given.
2010 AMS Classification. 06F35, 03G25, 08A72
Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Publishing House
ย
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Journals
ย
Abstract In this paper we proved fixed point theorem of four mapping on fuzzy metric space based on the concept of semi copatibility using implicit relation. These results generalize several corresponding relations in fuzzy metric space. All the results of this paper are new. Keywords: fuzzy metric space, compatibility, semi compatibility, implicit relation. 2000 AMS Mathematics Subject Classification: 47H10, 54H25
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ย
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result โNew generalized fuzzy metric space and fixed point theorem in fuzzy metric spaceโ. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
ย
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
In this paper, we introduce the concept of residual quotient of intuitionistic fuzzy subsets of ring and module and then define the notion of residual quotient intuitionistic fuzzy submodules , residual quotient intuitionistic fuzzy ideals. We study many properties of residual quotient relating to union, intersection, sum of intuitionistic fuzzy submodules (ideals). Using the concept of residual quotient, we investigate some important characterization of intuitionistic fuzzy annihilator of subsets of ring and module. We also study intuitionistic fuzzy prime submodules with the help of intuitionistic fuzzy annihilators. Many related properties are defined and discussed
Fixed Point Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
ย
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
In this paper, we introduce the concept of residual quotient of intuitionistic fuzzy subsets of ring and module and then define the notion of residual quotient intuitionistic fuzzy submodules , residual quotient intuitionistic fuzzy ideals. We study many properties of residual quotient relating to union, intersection, sum of intuitionistic fuzzy submodules (ideals). Using the concept of residual quotient, we investigate some important characterization of intuitionistic fuzzy annihilator of subsets of ring and module. We also study intuitionistic fuzzy prime submodules with the help of intuitionistic fuzzy annihilators. Many related properties are defined and discussed.
Pseudo Bipolar Fuzzy Cosets of Bipolar Fuzzy and Bipolar Anti-Fuzzy HX Subgroupsmathsjournal
ย
In this paper, we introduce the concept of pseudo bipolar fuzzy cosets, pseudo bipolar fuzzy double cosets of a bipolar fuzzy and bipolar anti-fuzzy subgroups. We also establish these concepts to bipolar fuzzy and bipolar anti-fuzzy HX subgroups of a HX group with suitable examples. Also we discuss some of their relative properties.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
ย
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
ย
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces.
Similar to FUZZY IDEALS AND FUZZY DOT IDEALS ON BH-ALGEBRAS (20)
Submission Deadline: 30th September 2022
Acceptance Notification: Within Three Daysโ time period
Online Publication: Within 24 Hrs. time Period
Expected Date of Dispatch of Printed Journal: 5th October 2022
MODELING AND ANALYSIS OF SURFACE ROUGHNESS AND WHITE LATER THICKNESS IN WIRE-...IAEME Publication
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White layer thickness (WLT) formed and surface roughness in wire electric discharge turning (WEDT) of tungsten carbide composite has been made to model through response surface methodology (RSM). A Taguchiโs standard Design of experiments involving five input variables with three levels has been employed to establish a mathematical model between input parameters and responses. Percentage of cobalt content, spindle speed, Pulse on-time, wire feed and pulse off-time were changed during the experimental tests based on the Taguchiโs orthogonal array L27 (3^13). Analysis of variance (ANOVA) revealed that the mathematical models obtained can adequately describe performance within the parameters of the factors considered. There was a good agreement between the experimental and predicted values in this study.
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The study explores the reasons for a transgender to become entrepreneurs. In this study transgender entrepreneur was taken as independent variable and reasons to become as dependent variable. Data were collected through a structured questionnaire containing a five point Likert Scale. The study examined the data of 30 transgender entrepreneurs in Salem Municipal Corporation of Tamil Nadu State, India. Simple Random sampling technique was used. Garrett Ranking Technique (Percentile Position, Mean Scores) was used as the analysis for the present study to identify the top 13 stimulus factors for establishment of trans entrepreneurial venture. Economic advancement of a nation is governed upon the upshot of a resolute entrepreneurial doings. The conception of entrepreneurship has stretched and materialized to the socially deflated uncharted sections of transgender community. Presently transgenders have smashed their stereotypes and are making recent headlines of achievements in various fields of our Indian society. The trans-community is gradually being observed in a new light and has been trying to achieve prospective growth in entrepreneurship. The findings of the research revealed that the optimistic changes are taking place to change affirmative societal outlook of the transgender for entrepreneurial ventureship. It also laid emphasis on other transgenders to renovate their traditional living. The paper also highlights that legislators, supervisory body should endorse an impartial canons and reforms in Tamil Nadu Transgender Welfare Board Association.
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Since ages gender difference is always a debatable theme whether caused by nature, evolution or environment. The birth of a transgender is dreadful not only for the child but also for their parents. The pain of living in the wrong physique and treated as second class victimized citizen is outrageous and fully harboured with vicious baseless negative scruples. For so long, social exclusion had perpetuated inequality and deprivation experiencing ingrained malign stigma and besieged victims of crime or violence across their life spans. They are pushed into the murky way of life with a source of eternal disgust, bereft sexual potency and perennial fear. Although they are highly visible but very little is known about them. The common public needs to comprehend the ravaged arrogance on these insensitive souls and assist in integrating them into the mainstream by offering equal opportunity, treat with humanity and respect their dignity. Entrepreneurship in the current age is endorsing the gender fairness movement. Unstable careers and economic inadequacy had inclined one of the gender variant people called Transgender to become entrepreneurs. These tiny budding entrepreneurs resulted in economic transition by means of employment, free from the clutches of stereotype jobs, raised standard of living and handful of financial empowerment. Besides all these inhibitions, they were able to witness a platform for skill set development that ignited them to enter into entrepreneurial domain. This paper epitomizes skill sets involved in trans-entrepreneurs of Thoothukudi Municipal Corporation of Tamil Nadu State and is a groundbreaking determination to sightsee various skills incorporated and the impact on entrepreneurship.
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Technology upgradation in banking sector took the economy to view that payment mode towards online transactions using mobile applications. This system enabled connectivity between banks, Merchant and user in a convenient mode. there are various applications used for online transactions such as Google pay, Paytm, freecharge, mobikiwi, oxygen, phonepe and so on and it also includes mobile banking applications. The study aimed at evaluating the predilection of the user in adopting digital transaction. The study is descriptive in nature. The researcher used random sample techniques to collect the data. The findings reveal that mobile applications differ with the quality of service rendered by Gpay and Phonepe. The researcher suggest the Phonepe application should focus on implementing the application should be user friendly interface and Gpay on motivating the users to feel the importance of request for money and modes of payments in the application.
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The prototype of a voice-based ATM for visually impaired using Arduino is to help people who are blind. This uses RFID cards which contain users fingerprint encrypted on it and interacts with the users through voice commands. ATM operates when sensor detects the presence of one person in the cabin. After scanning the RFID card, it will ask to select the mode like โnormal or blind. User can select the respective mode through voice input, if blind mode is selected the balance check or cash withdraw can be done through voice input. Normal mode procedure is same as the existing ATM.
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The main objective of this study is to analyze the impact of aspects of Organizational Culture on the Effectiveness of the Performance Management System (PMS) in the Health Care Organization at Thanjavur. Organizational Culture and PMS play a crucial role in present-day organizations in achieving their objectives. PMS needs employeesโ cooperation to achieve its intended objectives. Employees' cooperation depends upon the organizationโs culture. The present study uses exploratory research to examine the relationship between the Organization's culture and the Effectiveness of the Performance Management System. The study uses a Structured Questionnaire to collect the primary data. For this study, Thirty-six non-clinical employees were selected from twelve randomly selected Health Care organizations at Thanjavur. Thirty-two fully completed questionnaires were received.
Living in 21st century in itself reminds all of us the necessity of police and its administration. As more and more we are entering into the modern society and culture, the more we require the services of the so called โKhaki Worthyโ men i.e., the police personnel. Whether we talk of Indian police or the other nationโs police, they all have the same recognition as they have in India. But as already mentioned, their services and requirements are different after the like 26th November, 2008 incidents, where they without saving their own lives has sacrificed themselves without any hitch and without caring about their respective family members and wards. In other words, they are like our heroes and mentors who can guide us from the darkness of fear, militancy, corruption and other dark sides of life and so on. Now the question arises, if Gandhi would have been alive today, what would have been his reaction/opinion to the police and its functioning? Would he have some thing different in his mind now what he had been in his mind before the partition or would he be going to start some Satyagraha in the form of some improvement in the functioning of the police administration? Really these questions or rather night mares can come to any oneโs mind, when there is too much confusion is prevailing in our minds, when there is too much corruption in the society and when the polices working is also in the questioning because of one or the other case throughout the India. It is matter of great concern that we have to thing over our administration and our practical approach because the police personals are also like us, they are part and parcel of our society and among one of us, so why we all are pin pointing towards them.
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The goal of this study was to see how talent management affected employee retention in the selected IT organizations in Chennai. The fundamental issue was the difficulty to attract, hire, and retain talented personnel who perform well and the gap between supply and demand of talent acquisition and retaining them within the firms. The study's main goals were to determine the impact of talent management on employee retention in IT companies in Chennai, investigate talent management strategies that IT companies could use to improve talent acquisition, performance management, career planning and formulate retention strategies that the IT firms could use. The respondents were given a structured close-ended questionnaire with the 5 Point Likert Scale as part of the study's quantitative research design. The target population consisted of 289 IT professionals. The questionnaires were distributed and collected by the researcher directly. The Statistical Package for Social Sciences (SPSS) was used to collect and analyse the questionnaire responses. Hypotheses that were formulated for the various areas of the study were tested using a variety of statistical tests. The key findings of the study suggested that talent management had an impact on employee retention. The studies also found that there is a clear link between the implementation of talent management and retention measures. Management should provide enough training and development for employees, clarify job responsibilities, provide adequate remuneration packages, and recognise employees for exceptional performance.
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Globally, Millions of dollars were spent by the organizations for employing skilled Information Technology (IT) professionals. It is costly to replace unskilled employees with IT professionals possessing technical skills and competencies that aid in interconnecting the business processes. The organizationโs employment tactics were forced to alter by globalization along with technological innovations as they consistently diminish to remain lean, outsource to concentrate on core competencies along with restructuring/reallocate personnel to gather efficiency. As other jobs, organizations or professions have become reasonably more appropriate in a shifting employment landscape, the above alterations trigger both involuntary as well as voluntary turnover. The employee view on jobs is also afflicted by the COVID-19 pandemic along with the employee-driven labour market. So, having effective strategies is necessary to tackle the withdrawal rate of employees. By associating Emotional Intelligence (EI) along with Talent Management (TM) in the IT industry, the rise in attrition rate was analyzed in this study. Only 303 respondents were collected out of 350 participants to whom questionnaires were distributed. From the employees of IT organizations located in Bangalore (India), the data were congregated. A simple random sampling methodology was employed to congregate data as of the respondents. Generating the hypothesis along with testing is eventuated. The effect of EI and TM along with regression analysis between TM and EI was analyzed. The outcomes indicated that employee and Organizational Performance (OP) were elevated by effective EI along with TM.
INFLUENCE OF TALENT MANAGEMENT PRACTICES ON ORGANIZATIONAL PERFORMANCE A STUD...IAEME Publication
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By implementing talent management strategy, organizations would have the option to retain their skilled professionals while additionally working on their overall performance. It is the course of appropriately utilizing the ideal individuals, setting them up for future top positions, exploring and dealing with their performance, and holding them back from leaving the organization. It is employee performance that determines the success of every organization. The firm quickly obtains an upper hand over its rivals in the event that its employees having particular skills that cannot be duplicated by the competitors. Thus, firms are centred on creating successful talent management practices and processes to deal with the unique human resources. Firms are additionally endeavouring to keep their top/key staff since on the off chance that they leave; the whole store of information leaves the firm's hands. The study's objective was to determine the impact of talent management on organizational performance among the selected IT organizations in Chennai. The study recommends that talent management limitedly affects performance. On the off chance that this talent is appropriately management and implemented properly, organizations might benefit as much as possible from their maintained assets to support development and productivity, both monetarily and non-monetarily.
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Banking regulations act of India, 1949 defines banking as โacceptance of deposits for the purpose of lending or investment from the public, repayment on demand or otherwise and withdrawable through cheques, drafts order or otherwiseโ, the major participants of the Indian financial system are commercial banks, the financial institution encompassing term lending institutions. Investments institutions, specialized financial institution and the state level development banks, non banking financial companies (NBFC) and other market intermediaries such has the stock brokers and money lenders are among the oldest of the certain variants of NBFC and the oldest market participants. The asset quality of banks is one of the most important indicators of their financial health. The Indian banking sector has been facing severe problems of increasing Non- Performing Assets (NPAs). The NPAs growth directly and indirectly affects the quality of assets and profitability of banks. It also shows the efficiency of banks credit risk management and the recovery effectiveness. NPA do not generate any income, whereas, the bank is required to make provisions for such as assets that why is a double edge weapon. This paper outlines the concept of quality of bank loans of different types like Housing, Agriculture and MSME loans in state Haryana of selected public and private sector banks. This study is highlighting problems associated with the role of commercial bank in financing Small and Medium Scale Enterprises (SME). The overall objective of the research was to assess the effect of the financing provisions existing for the setting up and operations of MSMEs in the country and to generate recommendations for more robust financing mechanisms for successful operation of the MSMEs, in turn understanding the impact of MSME loans on financial institutions due to NPA. There are many research conducted on the topic of Non- Performing Assets (NPA) Management, concerning particular bank, comparative study of public and private banks etc. In this paper the researcher is considering the aggregate data of selected public sector and private sector banks and attempts to compare the NPA of Housing, Agriculture and MSME loans in state Haryana of public and private sector banks. The tools used in the study are average and Anova test and variance. The findings reveal that NPA is common problem for both public and private sector banks and is associated with all types of loans either that is housing loans, agriculture loans and loans to SMES. NPAs of both public and private sector banks show the increasing trend. In 2010-11 GNPA of public and private sector were at same level it was 2% but after 2010-11 it increased in many fold and at present there is GNPA in some more than 15%. It shows the dark area of Indian banking sector.
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An experiment conducted in this study found that BaSO4 changed Nylon 6's mechanical properties. By changing the weight ratios, BaSO4 was used to make Nylon 6. This Researcher looked into how hard Nylon-6/BaSO4 composites are and how well they wear. Experiments were done based on Taguchi design L9. Nylon-6/BaSO4 composites can be tested for their hardness number using a Rockwell hardness testing apparatus. On Nylon/BaSO4, the wear behavior was measured by a wear monitor, pinon-disc friction by varying reinforcement, sliding speed, and sliding distance, and the microstructure of the crack surfaces was observed by SEM. This study provides significant contributions to ultimate strength by increasing BaSO4 content up to 16% in the composites, and sliding speed contributes 72.45% to the wear rate
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The majority of the population in India lives in villages. The village is the back bone of the country. Village or rural industries play an important role in the national economy, particularly in the rural development. Developing the rural economy is one of the key indicators towards a countryโs success. Whether it be the need to look after the welfare of the farmers or invest in rural infrastructure, Governments have to ensure that rural development isnโt compromised. The economic development of our country largely depends on the progress of rural areas and the standard of living of rural masses. Village or rural industries play an important role in the national economy, particularly in the rural development. Rural entrepreneurship is based on stimulating local entrepreneurial talent and the subsequent growth of indigenous enterprises. It recognizes opportunity in the rural areas and accelerates a unique blend of resources either inside or outside of agriculture. Rural entrepreneurship brings an economic value to the rural sector by creating new methods of production, new markets, new products and generate employment opportunities thereby ensuring continuous rural development. Social Entrepreneurship has the direct and primary objective of serving the society along with the earning profits. So, social entrepreneurship is different from the economic entrepreneurship as its basic objective is not to earn profits but for providing innovative solutions to meet the society needs which are not taken care by majority of the entrepreneurs as they are in the business for profit making as a sole objective. So, the Social Entrepreneurs have the huge growth potential particularly in the developing countries like India where we have huge societal disparities in terms of the financial positions of the population. Still 22 percent of the Indian population is below the poverty line and also there is disparity among the rural & urban population in terms of families living under BPL. 25.7 percent of the rural population & 13.7 percent of the urban population is under BPL which clearly shows the disparity of the poor people in the rural and urban areas. The need to develop social entrepreneurship in agriculture is dictated by a large number of social problems. Such problems include low living standards, unemployment, and social tension. The reasons that led to the emergence of the practice of social entrepreneurship are the above factors. The research problem lays upon disclosing the importance of role of social entrepreneurship in rural development of India. The paper the tendencies of social entrepreneurship in India, to present successful examples of such business for providing recommendations how to improve situation in rural areas in terms of social entrepreneurship development. Indian government has made some steps towards development of social enterprises, social entrepreneurship, and social in- novation, but a lot remains to be improved.
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Distribution system is a critical link between the electric power distributor and the consumers. Most of the distribution networks commonly used by the electric utility is the radial distribution network. However in this type of network, it has technical issues such as enormous power losses which affect the quality of the supply. Nowadays, the introduction of Distributed Generation (DG) units in the system help improve and support the voltage profile of the network as well as the performance of the system components through power loss mitigation. In this study network reconfiguration was done using two meta-heuristic algorithms Particle Swarm Optimization and Gravitational Search Algorithm (PSO-GSA) to enhance power quality and voltage profile in the system when simultaneously applied with the DG units. Backward/Forward Sweep Method was used in the load flow analysis and simulated using the MATLAB program. Five cases were considered in the Reconfiguration based on the contribution of DG units. The proposed method was tested using IEEE 33 bus system. Based on the results, there was a voltage profile improvement in the system from 0.9038 p.u. to 0.9594 p.u.. The integration of DG in the network also reduced power losses from 210.98 kW to 69.3963 kW. Simulated results are drawn to show the performance of each case.
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Manufacturing industries have witnessed an outburst in productivity. For productivity improvement manufacturing industries are taking various initiatives by using lean tools and techniques. However, in different manufacturing industries, frugal approach is applied in product design and services as a tool for improvement. Frugal approach contributed to prove less is more and seems indirectly contributing to improve productivity. Hence, there is need to understand status of frugal approach application in manufacturing industries. All manufacturing industries are trying hard and putting continuous efforts for competitive existence. For productivity improvements, manufacturing industries are coming up with different effective and efficient solutions in manufacturing processes and operations. To overcome current challenges, manufacturing industries have started using frugal approach in product design and services. For this study, methodology adopted with both primary and secondary sources of data. For primary source interview and observation technique is used and for secondary source review has done based on available literatures in website, printed magazines, manual etc. An attempt has made for understanding application of frugal approach with the study of manufacturing industry project. Manufacturing industry selected for this project study is Mahindra and Mahindra Ltd. This paper will help researcher to find the connections between the two concepts productivity improvement and frugal approach. This paper will help to understand significance of frugal approach for productivity improvement in manufacturing industry. This will also help to understand current scenario of frugal approach in manufacturing industry. In manufacturing industries various process are involved to deliver the final product. In the process of converting input in to output through manufacturing process productivity plays very critical role. Hence this study will help to evolve status of frugal approach in productivity improvement programme. The notion of frugal can be viewed as an approach towards productivity improvement in manufacturing industries.
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Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
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Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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Forklift Classes Overview by Intella PartsIntella Parts
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Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
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NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
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Power plants release a large amount of water vapor into the
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plant. If a power plant could recover and reuse a portion of this
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requirement. One of the most practical way to recover water
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plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
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Here is a blog on the role of electrical and electronics engineers in IOT. Let's dig in!!!!
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Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
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This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
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When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Cosmetic shop management system project report.pdfKamal Acharya
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Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Fundamentals of Electric Drives and its applications.pptx
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FUZZY IDEALS AND FUZZY DOT IDEALS ON BH-ALGEBRAS
1. http://www.iaeme.com/IJARET/index.asp 350 editor@iaeme.com
International Journal of Advanced Research in Engineering and Technology (IJARET)
Volume 10, Issue 2, March - April 2019, pp. 350-361, Article ID: IJARET_10_02_034
Available online at http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=10&IType=02
ISSN Print: 0976-6480 and ISSN Online: 0976-6499
ยฉ IAEME Publication
FUZZY IDEALS AND FUZZY DOT IDEALS ON
BH-ALGEBRAS
K. Anitha
Research Scholar,
PG and Research Department of Mathematics,
Saiva Bhanu Kshatriya College,
Aruppukottai - 626 101, Tamil Nadu, India
Dr. N. Kandaraj
Associate Professor,
PG and Research Department of Mathematics,
Saiva Bhanu Kshatriya College,
Aruppukottai - 626 101, Tamil Nadu, India
ABSTRACT
In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results.
Keywords: BH-algebras, BH- Ideals, Fuzzy Dot BH-ideal.
Cite this Article: K. Anitha and Dr. N. Kandaraj, Fuzzy Ideals and Fuzzy Dot Ideals
on Bh-Algebras, International Journal of Advanced Research in Engineering and
Technology, 10(2), 2019, pp 350-361.
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=10&IType=2
Subject Classification: AMS (2000), 06F35, 03G25, 06D99, 03B47
1 INTRODUCTION
Y. Imai and K. Iseki [1, 2, and 3] introduced two classes of abstract algebras: BCK-algebras
and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class
of BCI-algebras. K. Iseki and S. Tanaka, [7] are introduced Ideal theory of BCK-algebras P.
Bhattacharya, N.P. Mukherjee and L.A. Zadeh [4] are introduced fuzzy relations and fuzzy
groups. The notion of BH-algebras is introduced by Y. B Jun, E. H. Roh and H. S. Kim[9]
Since then, several authors have studied BH-algebras. In particular, Q. Zhang, E. H. Roh and
Y. B. Jun [10] studied the fuzzy theory in BH-algebras. L.A. Zadeh [6] introduced notion of
fuzzy sets and A. Rosenfeld [8] introduced the notion of fuzzy group. O.G. Xi [5] introduced
the notion of fuzzy BCK-algebras. After that, Y.B. Jun and J. Meng [10] studied
Characterization of fuzzy sub algebras by their level sub algebras on BCK-algebras. J. Neggers
and H. S. Kim[11] introduced on d-algebras, M. Akram [12] introduced on fuzzy d-algebras In
this paper we classify the notion of Fuzzy Ideals on BH โ algebras and the notion of Fuzzy dot
2. K. Anitha and Dr. N. Kandaraj
http://www.iaeme.com/IJARET/index.asp 351 editor@iaeme.com
Ideals on BH โ algebras. And then we investigate several basic properties which are related to
fuzzy BH-ideals and fuzzy dot BH- ideals
2 PRELIMINARIES
In this section we cite the fundamental definitions that will be used in the sequel:
Definition 2.1 [1, 2, 3]
Let X be a nonempty set with a binary operation โ and a constant 0. Then (X, โ, 0) is called
a BCK-algebra if it satisfies the following conditions
1. ((๐ฅ โ ๐ฆ) โ (๐ฅ โ ๐ง)) โ (๐ง โ ๐ฆ) = 0
2.(๐ฅ โ (๐ฅ โ ๐ฆ)) โ ๐ฆ = 0
3.๐ฅ โ ๐ฅ = 0
4. ๐ฅ โ ๐ฆ = 0, ๐ฆ โ ๐ฅ = 0 โ ๐ฅ = ๐ฆ
5. 0 โ ๐ฅ = 0 ๐๐๐ ๐๐๐ ๐ฅ, ๐ฆ, ๐ง โ ๐
Definition 2.2 [1, 2, 3]
Let X be a BCK-algebra and I be a subset of X, then I is called an ideal of X if
(I1) 0โ ๐ผ
(I2) ๐ฆ ๐๐๐ ๐ฅ โ ๐ฆ โ ๐ผ โ ๐ฅ โ ๐ผ ๐๐๐ ๐๐๐ ๐ฅ, ๐ฆ โ ๐ผ
Definition 2.3 [9, 10]
A nonempty set X with a constant 0 and a binary operation โ is called a BH-algebra, if it
satisfies the following axioms
(BH1) ๐ฅ โ ๐ฅ = 0
(BH2) ๐ฅ โ 0 = 0
(BH3) ๐ฅ โ ๐ฆ = 0 ๐๐๐ ๐ฆ โ ๐ฅ = 0 โ ๐ฅ = ๐ฆ for all ๐ฅ, ๐ฆ โ ๐
Example 2.4
Let X= {0, 1, 2} be a set with the following cayley table
* 0 1 2
0 0 1 1
1 1 0 1
2 2 1 0
Then (X, โ, 0) is a BH-algebra
Definition 2.5[9, 10]
Let X be a BH-algebra and I be a subset of X, then I is called an ideal of X if
(BHI1) 0โ ๐ผ
(BHI2) ๐ฆ ๐๐๐ ๐ฅ โ ๐ฆ โ ๐ผ โ ๐ฅ โ ๐ผ
(BHI3) ๐ฅ โ ๐ผ and ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ ๐ผ ๐๐๐ ๐๐๐ ๐ฅ, ๐ฆ โ ๐ผ
A mapping ๐: ๐ โ ๐ of BH-algebras is called a homomorphism if ๐ (๐ฅ โ ๐ฆ ) = ๐(๐ฅ) โ
๐(๐ฆ) for all ๐ฅ, ๐ฆ ๐ โ ๐. Note that if ๐ โถ ๐ โ ๐ is homomorphism of BH-algebras, Then
๐ (0) = 0โ . We now review some fuzzy logic concepts. A fuzzy subset of a set X is a function
๐ โถ ๐ โ [0, 1]. For a fuzzy subset ฮผ of X and t โ [0, 1], define U (ฮผ; t) to be the set U (ฮผ; t) =
{x โ X | ฮผ(x) โฅ t}. For any fuzzy subsets ฮผ and ฮฝ of a set X, we define
(๐ โฉ ๐)(๐ฅ) = ๐๐๐{๐(๐ฅ), ๐(๐ฅ)} ๐๐๐ ๐๐๐ ๐ฅ โ ๐.
3. Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
http://www.iaeme.com/IJARET/index.asp 352 editor@iaeme.com
Let ๐ โถ ๐ โ ๐ be a function from a set X to a set Y and let ฮผ be a fuzzy subset of X. The
fuzzy subset v of Y defined by ๏ฝ)(yv {
otherwise
Yyyfifx
yfx
0
,)()(sup
1
)(1
๏๏ข๏น
๏ญ
๏ ๏ญ
๏ฆ๏ญ
is called the image of ยต under๐, denoted by ๐(ยต). If ๐ is a fuzzy subset o๐ ๐, the fuzzy
subset ฮผ of X given by ๐(๐ฅ) = ๐(๐(๐ฅ)) for all x โ X is called the Preimage of ฮฝ under f and is
denoted by ๐โ1
(๐ฃ) . A fuzzy subset ฮผ in X has the sup property if for any ๐ โ ๐ there exists
๐ฅ0โ T such that ๐(๐ฅ0) =
)(1
)(sup
yfx
z
๏ญ
๏
๏ญ . A fuzzy relation ฮผ on a set X is a fuzzy subset of ๐ ร ๐,
that is, a map ๐ โถ ๐ ร ๐ โ [0, 1].
Definition 2.6[4, 6, 8]
Let X be a nonempty set. A fuzzy (sub) set ๐ of the set X is a mapping ๐: ๐ โ [0,1
Definition 2.7[4, 6, 8]
Let ๐ be the fuzzy set of a set X. For a fixed sโ [0,1], the set ๐ ๐ = {๐ฅ โ ๐: ๐(๐ฅ) โฅ ๐ }is
called an upper level of ฮผ or level subset of ๐
Definition 2.8[5, 7]
A fuzzy set ๐ in X is called fuzzy BCK-ideal of X if it satisfies the following inequalities
1.๐(0) โฅ ๐(๐ฅ)
2. ๐(๐ฅ) โฅ min{๐(๐ฅ โ ๐ฆ), ๐(๐ฆ)}
Definition 2.9 [11]
Let X be a nonempty set with a binary operation โ and a constant 0. Then (X, โ, 0) is called
a d - algebra if it satisfies the following axioms.
1.๐ฅ โ ๐ฅ = 0
2. 0 โ ๐ฅ = 0
3. ๐ฅ โ ๐ฆ = 0, ๐ฆ โ ๐ฅ = 0 โ ๐ฅ = ๐ฆ for all ๐ฅ, ๐ฆ โ ๐
Definition 2.10 [12]
A fuzzy set ๐ in X is called fuzzy d-ideal of X if it satisfies the following inequalities
Fd1.๐(0) โฅ ๐(๐ฅ)
Fd2๐(๐ฅ) โฅ min{๐(๐ฅ โ ๐ฆ), ๐(๐ฆ)}
Fd3. ๐(๐ฅ โ ๐ฆ) โฅ min{๐(๐ฅ), ๐(๐ฆ)} For all ๐ฅ, ๐ฆ โ ๐
Definition 2.11[12] A fuzzy subset ฮผ of X is called a fuzzy dot d-ideal of X if it satisfies
The following conditions:
1. ๐(0) โฅ ๐(๐ฅ)
2. ๐(๐ฅ) โฅ ๐(๐ฅ โ ๐ฆ) . ๐(๐ฆ)
3. ๐(๐ฅ โ ๐ฆ) โฅ ๐(๐ฅ). ๐(๐ฆ)for all ๐ฅ, ๐ฆ โ ๐
3. FUZZY IDEALS ON BH-ALGEBRAS
Definition 3.1
A fuzzy set ๐ in X is called fuzzy BH-ideal of X if it satisfies the following inequalities
1.๐(0) โฅ ๐(๐ฅ)
2. ๐(๐ฅ) โฅ min{๐(๐ฅ โ ๐ฆ), ๐(๐ฆ)}
3. ๐(๐ฅ โ ๐ฆ) โฅ min{๐(๐ฅ), ๐(๐ฆ)} for all ๐ฅ, ๐ฆ โ ๐
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Example 3.2
Let X= {0, 1, 2, 3} be a set with the following cayley table
* 0 1 2 3
0 0 0 0 0
1 1 0 0 1
2 2 2 0 0
3 3 3 3 0
Then (X, โ ,0) is not BCK-algebra. Since {(1โ 3) โ (1 โ 2)} โ (2 โ 3) = 1 โ 0
We define fuzzy set ๐ in X by ๐(0) = 0.8 and ๐(๐ฅ) = 0.01 for all ๐ฅ โ 0 in X. then it is
easy to show that ๐ is a BH-ideal of X.
We can easily observe the following propositions
1. In a BH-algebra every fuzzy BH-ideal is a fuzzy BCK-ideal, and every fuzzy BCK-
ideal is a fuzzy BH -Sub algebra
2. Every fuzzy BH-ideal of a BH- algebra is a fuzzy BH-subalgebra.
Example 3.3
Let X = {0, 1, 2} be a set given by the following cayley table
* 0 1 2
0 0 0 0
1 1 0 1
2 2 2 0
Then (X, โ ,0) is a fuzzy BCK-algebra. We define fuzzy set ๐ in X by ๐(0) = 0.7, ๐(1) =
0.5, ๐(2) = 0.2
Then ๐ is a fuzzy BH-ideal of X.
Definition 3.4
Let ฮป ๐๐๐ ๐ be the fuzzy sets in a set X. The Cartesian product ฮป ร ๐: ๐ ร ๐ โ [0,1] is
defined by (ฮป ร ๐)(๐ฅ, ๐ฆ) = min{๐(๐ฅ), ๐(๐ฆ)} โ๐ฅ, ๐ฆ โ ๐.
Theorem 3.5
If ฮป ๐๐๐ ๐ be the fuzzy BH-ideals of a BH- algebra X, then ฮป ร ๐ is a fuzzy BH-ideals
of ๐ ร ๐
Proof
For any (๐ฅ, ๐ฆ) โ ๐ ร ๐, wehave
(ฮป ร ๐)(0,0) = min{๐(0), ๐(0)} โฅ min{ ๐(๐ฅ), ๐(๐ฆ)}
= (ฮป ร ๐)(๐ฅ, ๐ฆ)
That is (ฮป ร ๐)(0,0) = (ฮป ร ๐)(๐ฅ, ๐ฆ)
Let (๐ฅ1 , ๐ฅ2) ๐๐๐ (๐ฆ1, ๐ฆ2) โ ๐ ร ๐
Then, (ฮป ร ๐) (๐ฅ1 , ๐ฅ2) = ๐๐๐{๐(๐ฅ1), ๐(๐ฅ2)}
โฅ min{min{ ๐(๐ฅ1 โ ๐ฆ1), ๐(๐ฆ1)} , min{๐(๐ฅ2 โ ๐ฆ2), ๐(๐ฆ2)}
= min{min ๐(๐ฅ1 โ ๐ฆ1 ), ๐(๐ฅ2 โ ๐ฆ2), min{ ๐(๐ฆ1), ๐(๐ฆ2)}}
= min {(ฮป ร ๐) (๐ฅ1 โ ๐ฆ1, ๐ฅ2 โ ๐ฆ2)) , ( ฮป ร ๐) (๐ฆ1, ๐ฆ2)}
= min {((ฮป ร ๐) (๐ฅ1, ๐ฅ2) โ (๐ฆ1, ๐ฆ2)), ( ฮป ร ๐) (๐ฆ1, ๐ฆ2)}
That is ((ฮป ร ๐) (๐ฅ1, ๐ฅ2)) = min {(ฮป ร ๐) (๐ฅ1 , ๐ฅ2) โ (๐ฆ1, ๐ฆ2)), (ฮป ร ๐)(๐ฆ1, ๐ฆ2)}
And (ฮป ร ๐)( (๐ฅ1 , ๐ฅ2) โ (๐ฆ1, ๐ฆ2))
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= (ฮป ร ๐) (๐ฅ1 โ ๐ฆ1 , ๐ฅ2 โ ๐ฆ2)
= min{ ๐(๐ฅ1 โ ๐ฆ1 ), ๐(๐ฅ2 โ ๐ฆ2)}
โฅ min {min {ฮป (๐ฅ1), ฮป (๐ฆ1)}, min {๐(๐ฅ2), ๐(๐ฆ2)}}
= min {min (ฮป ( ๐ฅ1 ), ๐(๐ฅ2), min{( ฮป (๐ฆ1), ๐(๐ฆ2))}
= min {(ฮป ร ๐)(( ๐ฅ1 , ๐ฅ2), ( ฮป ร ๐)(๐ฆ1, ๐ฆ2))
That is (ฮป ร ๐)(( ๐ฅ1 , ๐ฅ2) โ (๐ฆ1, ๐ฆ2))
=min {(ฮป ร ๐)(( ๐ฅ1 , ๐ฅ2), ( ฮป ร ๐) (๐ฆ1, ๐ฆ2)}
Hence ฮป ร ๐ is a fuzzy BH-ideal of Xร ๐
Theorem 3.6
Let ฮป ๐๐๐ ๐ be fuzzy sets in a BH-algebra such that ๐ ร ๐ is a fuzzy BH-ideal of ๐ ร
๐.Then
i)Either ฮป(0)โฅ ฮป (x)๐๐ ๐(0) โฅ ๐(๐ฅ) โ๐ฅ โ ๐.
ii) If ฮป (0)โฅ ฮป (x) โ๐ฅ โ ๐, then either ๐(0) โฅ ฮป(๐ฅ) or ๐(0) โฅ ๐(๐ฅ)
iii) If ๐(0) โฅ ๐(๐ฅ) โ๐ฅ โ ๐, then either ฮป (0)โฅ ฮป (x) or ฮป (0)โฅ ยต(x)
Proof
We use reduction to absurdity
i) Assume ฮป(x)> ฮป (0) and ๐(๐ฅ) โฅ ๐(0) for some ๐ฅ, ๐ฆ โ ๐.
Then (ฮป ร ๐)(๐ฅ, ๐ฆ) = min{ฮป(x), ๐(๐ฆ)}
>min {{ฮป(0), ๐(0)}
= (ฮป ร ๐)(0,0)
(ฮป ร ๐)(๐ฅ, ๐ฆ) > (ฮป ร ๐)(0,0) โ๐ฅ, ๐ฆ โ ๐
Which is a contradiction to (ฮป ร ๐) is a fuzzy BH-ideal of Xร ๐
Therefore either ฮป (0)โฅ ฮป (x)๐๐ ๐(0) โฅ ๐(๐ฅ) โ๐ฅ โ ๐.
ii) Assume ๐(0) < ฮป(๐ฅ) and ๐(0) < ๐(๐ฆ) for some ๐ฅ, ๐ฆ โ ๐.
Then (ฮป ร ๐)(0,0) = min{ฮป(0), ๐(0)}= ๐(0)
And (ฮป ร ๐)(๐ฅ, ๐ฆ) = min{ฮป(x), ๐(๐ฆ)}> ๐(0)
= (ฮป ร ๐)(0,0)
This implies (ฮป ร ๐)(๐ฅ, ๐ฆ) > and ( ฮป ร ๐)(0,0)
Which is a contradiction to ฮป ร ๐ is a fuzzy BH-ideal of Xร ๐
Hence if ฮป (0)โฅ ฮป (x) โ๐ฅ โ ๐, ๐กโ๐๐
Either ๐(0) โฅ ฮป(๐ฅ) or ๐(0) โฅ ๐(๐ฅ) โ๐ฅ โ ๐
iii) Assume ฮป (0)< ฮป (x) or ฮป (0)< ยต(y) โ๐ฅ, ๐ฆ โ ๐
Then ((ฮป ร ๐)(0, 0) = min{ฮป(0), ๐(0)}= ๐(0)
And (ฮป ร ๐)(๐ฅ, ๐ฆ) = min{ฮป(x), ๐(๐ฆ)}> ๐(0)
= (ฮป ร ๐)(0,0)
This implies (ฮป ร ๐)(๐ฅ, ๐ฆ) >(ฮป ร ๐)(0,0)
Which is a contradiction to (ฮป ร ๐) is a fuzzy BH-ideal of ๐ ร ๐
Hence if ๐(0) โฅ ๐(๐ฅ) โ๐ฅ โ ๐ then either ฮป (0)โฅ ฮป (x) or ฮป (0)โฅ ยต(x)
This completes the proof
Theorem 3.7
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If ฮป ร ๐ is a fuzzy BH-idela of ๐ ร ๐ then ฮป ๐๐ ๐ is a fuzzy BH-ideal of X.
Proof
First we prove that ๐ is a fuzzy BH-ideal of X.
Given ฮป ร ๐ is a fuzzy BH-ideal of ๐ ร ๐, then by theorem 3.6(i), either ฮป (0)โฅ
ฮป (x)๐๐ ๐(0) โฅ ๐(๐ฅ) โ๐ฅ โ ๐.
Let ๐(0) โฅ ๐(๐ฅ)
By theorem 3.6(iii) then either ฮป (0)โฅ ฮป (x) or ฮป (0)โฅ ยต(x)
Now ๐(๐ฅ) = min{ฮป(0), ๐(๐ฅ)}
= (ฮป ร ๐)(0, ๐ฅ)
โฅ min{ ( (ฮป ร ๐)(0, ๐ฅ) โ (0, ๐ฆ)), (ฮป ร ๐)(0, ๐ฆ)}
= min{ (ฮป ร ๐)(0 โ 0), ๐ฅ โ ๐ฆ),(ฮป ร ๐)(0, ๐ฆ)}
= min{ (ฮป ร ๐)(0, ๐ฅ โ ๐ฆ),(ฮป ร ๐)(0, ๐ฆ)}
= min{ (ฮป ร ๐)(0 โ 0), ๐ฅ โ ๐ฆ),(ฮป ร ๐)(0, ๐ฆ)}
= min{ ๐(๐ฅ โ ๐ฆ), ( ๐)(๐ฆ)}
That is ๐(๐ฅ) โฅ min{ ๐(๐ฅ โ ๐ฆ), ๐(๐ฆ)}
๐(๐ฅ โ ๐ฆ) = min{ ฮป (0) , ๐(๐ฅ โ ๐ฆ)}
= (ฮป ร ๐)(0, ๐ฅ โ ๐ฆ)
= (ฮป ร ๐)(0 โ 0, ๐ฅ โ ๐ฆ)
= (ฮป ร ๐)(0, ๐ฅ) โ (0, ๐ฆ)
๐(๐ฅ โ ๐ฆ) โฅ min{ (ฮป ร ๐)(0, ๐ฅ), (ฮป ร ๐)(0, ๐ฆ)}
=min{๐(๐ฅ), ๐(๐ฆ)}
That is, ๐(๐ฅ โ ๐ฆ) โฅ min{๐(๐ฅ), ๐(๐ฆ)}
This proves that ๐ is a fuzzy BH-ideal of X.
Secondly to prove that ฮป is a Fuzzy BH-ideal of X.
Using theorem 4.6(i) and (ii) we get
This completes the proof.
Theorem 3.8
If ๐ is a fuzzy BH-idela of ๐, then ๐ ๐ก is a BH-idela of X for all ๐ก โ [0,1]
Proof
Let ๐ be a fuzzy BH-ideal of X,
Then By the definition of BH-ideal
๐(0) โฅ ๐(๐ฅ)
๐(๐ฅ) โฅ min{ ๐(๐ฅ โ ๐ฆ), (๐(๐ฆ)}
๐(๐ฅ โ ๐ฆ) โฅ min{ ๐(๐ฅ), (๐(๐ฆ)} โ๐ฅ, ๐ฆ โ ๐
To prove that ๐ ๐ก ๐๐ ๐ BH-ideal of x.
By the definition of level subset of ๐
๐ ๐ก = {๐ฅ/ ๐(๐ฅ) โฅ ๐ก}
Let ๐ฅ, ๐ฆ โ ๐ ๐ก and ๐ is a fuzzy BH-ideal of X.
Since ๐(0) โฅ ๐(๐ฅ) โฅ ๐ก implies 0โ ๐ ๐ก, for all ๐ก โ [0,1]
Let ๐ฅ, ๐ฆ โ ๐ ๐ก and ๐ฆ โ ๐ ๐ก
Therefore ๐(๐ฅ โ ๐ฆ) โฅ ๐ก and ๐(๐ฆ) โฅ ๐ก
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Now ๐(๐ฅ) โฅ min{ ๐(๐ฅ โ ๐ฆ), (๐(๐ฆ)}
โฅ min{๐ก, ๐ก} โฅ ๐ก
Hence (๐(๐ฅ) โฅ ๐ก
That is ๐ฅ โ ๐ ๐ก.
Let ๐ฅ โ ๐ ๐ก, ๐ฆ โ ๐
Choose y in X such that ๐(๐ฆ) โฅ ๐ก
Since ๐ฅ โ ๐ ๐ก implies ๐(๐ฅ) โฅ ๐ก
We know that ๐(๐ฅ โ ๐ฆ) โฅ min{ ๐(๐ฅ), (๐(๐ฆ)}
โฅ min{๐ก, ๐ก}
โฅ ๐ก
That is ๐(๐ฅ โ ๐ฆ) โฅ ๐ก implies ๐ฅ โ ๐ฆ โ ๐ ๐ก
Hence ๐ ๐ก is a BH-ideal of X.
Theorem 3.9
If X be a BH-algebra, โ๐ก โ [0,1] and ๐ ๐ก ๐๐ ๐ ๐ต๐ป โideal of X, then ๐ is a fuzzy BH-ideal of
X.
Proof
Since ๐ ๐ก ๐๐ ๐ ๐ต๐ป โideal of X
0โ ๐ ๐ก
ii) ๐ฅ โ ๐ฆ โ ๐ ๐ก And ๐ฆ โ ๐ ๐ก implies ๐ฅ โ ๐ ๐ก
iii)๐ฅ โ ๐ ๐ก, y โ ๐ implies ๐ฅ โ ๐ฆ โ ๐ ๐ก
To prove that ๐ ๐๐ ๐ ๐๐ข๐ง๐ง๐ฆ BH-ideal of X.
Let ๐ฅ, ๐ฆ โ ๐ ๐ก then ๐(๐ฅ) โฅ ๐ก and ๐(๐ฆ) โฅ ๐ก
Let ๐(๐ฅ) = ๐ก1 and ๐(๐ฆ) = ๐ก2
Without loss of generality let ๐ก1 โค ๐ก2
Then ๐ฅ โ ๐ ๐ก1
Now ๐ฅ โ ๐ ๐ก1
and y โ ๐ ๐๐๐๐๐๐๐ ๐ฅ โ ๐ฆ โ ๐ ๐ก1
That is ๐(๐ฅ โ ๐ฆ) โฅ ๐ก1
= min {๐ก1, ๐ก2}
= min { ๐(๐ฅ), ๐(๐ฆ)}
๐(๐ฅ โ ๐ฆ) โฅ min { ๐(๐ฅ), ๐(๐ฆ)}
๐๐) ๐ฟ๐๐ก ๐(0) = ๐(๐ฅ โ ๐ฅ) โฅ min { ๐(๐ฅ), ๐(๐ฆ)} โฅ ๐(๐ฅ) (by proof (i))
That is ๐(0) โฅ ๐(๐ฅ) for all x โ ๐
iii) Let ๐(๐ฅ) = ๐(๐ฅ โ ๐ฆ) โ (0 โ ๐ฆ))
โฅ min{ ๐(๐ฅ โ ๐ฆ), ๐(0 โ ๐ฆ)) (By (i))
โฅ min{ ๐(๐ฅ โ ๐ฆ), min{๐(0), ๐(๐ฆ)}
โฅ min{ ๐(๐ฅ โ ๐ฆ),{๐(๐ฆ)) (By (ii))
๐(๐ฅ) โฅ min{ ๐(๐ฅ โ ๐ฆ),{๐(๐ฆ))
Hence ๐ is a fuzzy BH-ideal of X.
Definition 3.10
A fuzzy set ๐ in X is said to be fuzzy BH- ๐ ideal if ๐(๐ฅ โ ๐ข โ ๐ฃ โ ๐ฆ) โฅ min { ๐(๐ฅ), ๐(๐ฆ)}
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Theorem 3.11
Every Fuzzy BH -ideal is a fuzzy BH- ๐- ideal
Proof
It is trivial
Remark
Converse of the above theorem is not true. That is every fuzzy BH-๐ -ideal is not true. That
is every fuzzy BH -ideal. Let us prove this by an example
Let X = {0, 1, 2} be a set given by the following cayley table
* 0 1 2
0 0 1 2
1 1 0 1
2 2 2 0
Then (X,โ ,0) is a BH-algebra. We define fuzzy set: Xโ [0,1] by ๐(0) = 0.8, ๐(๐ฅ) =
0.2 โ๐ฅ โ 0 clearly ๐ is a fuzzy BH-ideal of X But ๐ is not a BH-๐-ideal of X.
For Let ๐ฅ = 0 ๐ข = 1 ๐ฃ = 1 ๐ฆ = 1
๐(๐ฅ โ ๐ข โ ๐ฃ โ ๐ฆ = ๐(0 โ 1 โ 1 โ 1)= ๐(1) = 0.2
min{ ๐(๐ฅ), ๐(๐ฆ)} = min{ ๐(0), ๐(0)}
= ๐(0) = 0.8
๐(๐ฅ โ ๐ข โ ๐ฃ โ ๐ฆ) โคmin { ๐(๐ฅ), ๐(๐ฆ)}
Hence ๐ is not a fuzzy BH-๐-ideal of X.
And ๐(๐ฅ โ ๐ฆ)= ๐ (๐(๐) โ ๐(๐))
โฅ ๐ (๐(๐ โ ๐))
= ๐ ๐
(๐ โ ๐)
โฅ min{ ๐ ๐(๐), ๐ ๐
(๐)}
= min{๐(๐(๐), ๐(๐(๐))}
= min{๐(๐ฅ), ๐(๐ฆ)}
Hence ๐(๐ฅ โ ๐ฆ) โฅ min{๐(๐ฅ), ๐(๐ฆ)}
Hence ๐ is a fuzzy BH- ๐ ideal of Y.
4. FUZZY DOT BH-IDEALS OF BH-ALGEBRAS
Definition 4.1. A fuzzy subset ฮผ of X is called a fuzzy dot BH-ideal of X if it satisfies
The following conditions:
(FBH1). ๐(0) โฅ ๐(๐ฅ)
(FBH2). ๐(๐ฅ) โฅ ๐(๐ฅ โ ๐ฆ) . ๐(๐ฆ)
(FBH3). ๐(๐ฅ โ ๐ฆ) โฅ ๐(๐ฅ). ๐(๐ฆ)for all ๐ฅ, ๐ฆ โ ๐
Example 4.2. Let X = {0, 1, 2, 3} be a BH-algebra with Cayley table (Table 1) as follows:
(Table 1)
* 0 1 2 3
0 0 0 0 0
1 1 0 0 2
2 2 2 0 0
3 3 3 3 0
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Define ๐ โถ ๐ โ [0,1] by ฮผ (0) =0.9, ฮผ (a) = ฮผ (b) =0.6, ฮผ (c) =0.3 .It is easy to verify that ฮผ
is a Fuzzy dot BH-ideal of X.
Proposition 4.3. Every fuzzy BH-ideal is a fuzzy dot BH-ideal of a BH-algebra.
Remark. The converse of Proposition 4.3 is not true as shown in the following Example
4.2. Let ๐ = {0 ,1 ,2 ,3} be a BH-algebra with Cayley table (Table 1) as follows:
Example 4.4. Let ๐ = {0,1, 2} be a BH-algebra with Cayley table (Table 2) as follows:
(Table 2)
* 0 1 2
0 0 0 0
1 1 0 2
2 2 1 0
Define ฮผ: X โ [0, 1] by ฮผ (0) =0.8, ฮผ (1) = 0.5, ฮผ (2) =0.4. It is easy to verify that ฮผ is a
fuzzy Dot BH-ideal of X, but not a fuzzy d-ideal of X because
๐(๐ฅ) โค min{๐(๐ฅ โ ๐ฆ), ๐(๐ฆ)}
๐(1) = min{๐(1 โ 2), ๐(2)}
= ๐(2)
Proposition 4.5. Every fuzzy dot BH-ideal of a BH-algebra X is a fuzzy dot subalgebra of
X.
Remark. The converse of Proposition 4.5 is not true as shown in the following
Example:
Example 4.6. Let X be the BH-algebra in Example 4.4 and define ฮผ: X โ [0, 1] by
ฮผ (0)= ฮผ (1)=0.9, ฮผ (2)= 0.7. It is easy to verify that ฮผ is a fuzzy dot sub algebra of X, but
not a fuzzy dot BH-ideal of X because ฮผ (2)=0.7 โค 0.81=ฮผ (2 โ1)โ ฮผ (1) .
Proposition 4.7. If ฮผ and ฮฝ are fuzzy dot BH-ideals of a BH-algebra X, then so is ฮผ โฉ ฮฝ.
Proof. Let x, y โ X. Then
(๐ โฉ ๐)(0) = ๐๐๐ {๐ (0), ๐ (0)}
โฅ ๐๐๐ {๐(๐ฅ), ๐(๐ฅ)}
= (๐ โฉ ๐) (๐ฅ).
Also, (๐ โฉ ๐) (๐ฅ) = ๐๐๐ {๐(๐ฅ), ๐(๐ฅ)}
โฅ ๐๐๐ {๐(๐ฅ โ ๐ฆ) ยท ๐(๐ฆ), ๐(๐ฅ โ ๐ฆ) ยท ๐(๐ฆ)}
โฅ (๐๐๐ {๐(๐ฅ โ ๐ฆ), ๐(๐ฅ โ ๐ฆ)}) ยท (๐๐๐ {๐(๐ฆ), ๐(๐ฆ)})
= ((๐ โฉ ๐) (๐ฅ โ ๐ฆ)) ยท ((๐ โฉ ๐) (๐ฆ)).
๐ด๐๐, (๐ โฉ ๐) (๐ฅ โ ๐ฆ) = ๐๐๐ {๐(๐ฅ โ ๐ฆ), ๐(๐ฅ โ ๐ฆ)}
โฅ ๐๐๐ {๐(๐ฅ) ยท ๐(๐ฆ), ๐(๐ฅ) ยท ๐(๐ฆ)}
โฅ (๐๐๐ {๐(๐ฅ), ๐(๐ฅ)}) ยท (๐๐๐ {๐(๐ฆ), ๐(๐ฆ)})
= ((๐ โฉ ๐) (๐ฅ)) ยท ((๐ โฉ ๐) (๐ฆ)).
Hence ฮผ โฉ ฮฝ is a fuzzy dot BH-ideal of a d-algebra X.
Theorem 4.8. If each nonempty level subset U (ฮผ; t) of ฮผ is a fuzzy BH-ideal of X then
ฮผ is a fuzzy dot BH-ideal of X, where t โ[0, 1].
Definition 4.9
10. K. Anitha and Dr. N. Kandaraj
http://www.iaeme.com/IJARET/index.asp 359 editor@iaeme.com
Let ฯ be a fuzzy subset of X. The strongest fuzzy ฯ-relation on BH-algebra X is the fuzzy
subset ยตฯ of X รX given by ๐ ๐(x, y) = ฯ(x) ยท ฯ(y) for all x, y โ X. A fuzzy relation ฮผ on BH-
algebra X is called a Fuzzy ฯ-product relation if ยต(x, y) โฅ ฯ(x) ยท ฯ(y) for all x, y โ X. A
fuzzy relation ฮผ on BH-algebra is called a left fuzzy relation on ฯ if ยต(x, y) = ฯ(x) for all x, y
โ X.
Note that a left fuzzy relation on ฯ is a fuzzy ฯ-product relation.
Remark. The converse of Theorem 4.8 is not true as shown in the following example:
Example 4.10. Let X be the BH-algebra in Example 4.4 and define ฮผ: X โ [0, 1] by
ฮผ (0)=0.6, ฮผ (1)=0.7, ฮผ (2)= 0.8. We know that ฮผ is a fuzzy dot BH-ideal of X, but
U (ฮผ; 0.8) = {x โX | ฮผ(x) โฅ 0.8} = {2, 2} is not BH-ideal of X since 0โU (ฮผ; 0.8).
Theorem 4.11. Let ๐ โถ ๐ โ ๐ โฒ be an onto homomorphism of BH-algebras, ฮฝ be a fuzzy
Dot BH-ideal of Y. Then the Preimage ๐โ1
(๐ฃ) of ฮฝ under f is a fuzzy dot BH-ideal of X.
Proof. Let xโ X,
๐โ1(๐ฃ)(0) = ๐ฃ(๐(0)) = ๐ฃ(0โฒ
)
โฅ ๐ฃ(๐(๐ฅ)) = ๐โ1
(๐ฃ)(๐ฅ)
For any x, yโ X, we have
๐โ1
(๐ฃ)(๐ฅ) = ฮฝ (f (x)) โฅ ฮฝ (f (x)* f(y)) ยท ฮฝ (f (y))
= ฮฝ (f (x* y)) ยท ฮฝ (f (y)) = ๐โ1
(๐ฃ)(๐ฅ โ ๐ฆ). ๐โ1(๐ฃ)(๐ฆ)
Also,
๐โ1
(๐ฃ)(๐ฅ โ ๐ฆ)= ฮฝ (f (x * y)) = ฮฝ (f (x) * f (y))
โฅ ฮฝ (f (x)) ยท ฮฝ(f (y)) = ๐โ1
(๐ฃ)(๐ฅ). ๐โ1
(๐ฃ)(๐ฆ)
Thus ๐โ1
(๐ฃ)is a fuzzy dot BH-ideal of X.
Theorem: 4.12 An onto homomorphic image of a fuzzy dot BH-ideal with the sup
Property is a fuzzy dot BH-ideal.
Theorem 4.13. If ฮป and ฮผ are fuzzy dot BH-ideal of a BH-algebra X, then ฮป ร ฮผ is a fuzzy
Dot BH-ideal of X ร X.
Proof.
Let ๐ฅ, ๐ฆ โ ๐
ฮป ร ฮผ (0,0)=ฮป(0), ฮผ(0)
โฅ ฮป(x).ฮผ(y)= (ฮป ร ฮผ) (x, y)
For any ๐ฅ, ๐ฅโฒ
, ๐ฆ, ๐ฆโฒ โ ๐ wehave
(ฮป ร ยต) (x, y) = ฮป(x). ยต(y)
โฅ ฮป(x โ xโฒ). ฮป(xโฒ))ยต(y โ yโฒ). ยต(yโฒ)
= ฮป(x โ xโฒ). ยต(y โ yโฒ)). ฮป(xโฒ). ยต(yโฒ)
= (ฮปร ยต)(x, y) โ (xโฒ
, yโฒ
)). (ฮปร ยต)(xโฒ
, yโฒ
)
Also (ฮปร ยต)(x, y) โ (xโฒ
, yโฒ
)) = (ฮปร ยต)(x โ xโฒ) โ (y โ yโฒ
)).
= ฮป(x โ xโฒ). ยต(y โ yโฒ)
โฅ ฮป(x). ฮป(xโฒ
)). ( ยต(y). ยต(yโฒ))
(ฮป(x). ยต(y)). (ฮป(xโฒ). ยต(yโฒ))
= (ฮปร ยต)(x, y). (ฮป ร ยต)(xโฒ, yโฒ)
Hence ฮป ร ฮผ is a fuzzy dot BH-ideal of X ร X.
11. Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
http://www.iaeme.com/IJARET/index.asp 360 editor@iaeme.com
Theorem 4.14. Let ฯ be a fuzzy subset of a BH-algebra X and ๐ ๐be the strongest fuzzy
ฯ -relation on BH-algebra X. Then ฯ is a fuzzy dot BH-ideal of X if and only if ๐ ๐ is a
Fuzzy dot BH-ideal of X ร X.
Proof. Assume that ฯ is a fuzzy dot BH-ideal of X. For any x, y โ X we have
๐ ๐ (0, 0)= ฯ (0) ยท ฯ (0) โฅ ฯ(x) ยท ฯ(y) = ๐ ๐ (x, y).
Let x, x โฒ, y, y โฒโ X. Then
๐ ๐ ((๐ฅ, ๐ฅ โฒ) โ (๐ฆ, ๐ฆ โฒ)) ยท ๐ ๐ (๐ฆ, ๐ฆ โฒ)
= ๐ ๐ (๐ฅ โ ๐ฆ, ๐ฅ โฒ โ ๐ฆ โฒ) ยท ๐ ๐ (๐ฆ, ๐ฆ โฒ)
= (๐(๐ฅ โ ๐ฆ) ยท ๐(๐ฅ โฒ โ ๐ฆ โฒ)) ยท (๐(๐ฆ) ยท ๐(๐ฆ โฒ))
= (๐(๐ฅ โ ๐ฆ) ยท ๐(๐ฆ)) ยท (๐(๐ฅ โฒ โ ๐ฆ โฒ) ยท ๐(๐ฆ โฒ))
โค ๐ (๐ฅ) ยท ๐(๐ฅ โฒ ) = ๐ ๐ (๐ฅ, ๐ฅ โฒ ). ๐ด๐๐,
๐ ๐ (x, x โฒ )ยท ๐ ๐ (y, y โฒ) = (ฯ(x) ยท ฯ(x โฒ)) ยท (ฯ(y) ยท ฯ(y โฒ))
= (ฯ(x) ยท ฯ(y)) ยท (ฯ(x โฒ) ยท ฯ(y โฒ))
โค ๐ (๐ฅ โ ๐ฆ) ยท ๐( ๐ฅ โฒ โ ๐ฆ โฒ )
= ๐ ๐ (๐ฅ โ ๐ฆ, ๐ฅ โฒ โ ๐ฆ โฒ) = ๐ ๐ ((๐ฅ, ๐ฅ โฒ) โ (๐ฆ, ๐ฆ โฒ)).
Thus ๐ ๐ is a fuzzy dot BH-ideal of X ร X.
Conversely suppose that ๐ ๐ is a fuzzy dot BH-ideal of X ร X. From (FBH1) we get
(ฯ(0) )2
= ฯ (0) ยท ฯ(0)= ๐ (0, 0)
And so ฯ (0) โฅ ฯ(x) for all x โ X. Also we have
( ฯ(x) )2
= ยตฯ ((x, x )
โฅ ยตฯ ((x, x)*(y, y)). ยตฯ (y, y)
=ยตฯ ((xโy ), (x โ y)). ยตฯ (y, y)
=((ฯ(x โ y) ยท ฯ(y)) )2
Which implies that ๐(๐ฅ) โฅ ๐(๐ฅ โ y) ยท ฯ(y) for all x, y โ X.
Also we have
(๐(๐ฅ โ ๐ฆ)2
=๐ ๐ ((xโy ), ๐ฅ โ ๐ฆ)
=๐ ๐ ((๐ฅ, ๐ฅ) โ (๐ฆ, ๐ฆ)) โฅ ยตฯ(๐ฅ, ๐ฅ). ยตฯ(๐ฆ, ๐ฆ)
= (ฯ(๐ฅ) ยท ฯ( ๐ฆ ))2
So ๐(๐ฅ โ ๐ฆ) โฅ ๐(๐ฅ) ยท ๐(๐ฆ) for all x, y โX.
Therefore ฯ is a fuzzy dot BH-ideal of X.
Proposition 4.15. Let ฮผ be a left fuzzy relation on a fuzzy subset ฯ of a BH-algebra X. If
ฮผ is a fuzzy dot BH-ideal of X ร X, then ฯ is a fuzzy dot BH-ideal of a BH-algebra X.
Proof.
Suppose that a left fuzzy relation ฮผ on ฯ is a fuzzy dot BH-ideal of X ร X.
Then
ฯ (0) = ฮผ (0, z ) โz โ X
By putting z=0
ฯ (0) = ฮผ (0,0) โฅ ฮผ (x , y )=ฯ (x ),
For all xโ X.
For any x, x โฒ, y, y โฒโ X
12. K. Anitha and Dr. N. Kandaraj
http://www.iaeme.com/IJARET/index.asp 361 editor@iaeme.com
ฯ(x) = ยต(x, y) โฅ ยต((x, y) โ (xโฒ
, yโฒ)). ยต(xโฒ, yโฒ)
= ยต((x โ xโฒ), (y โ yโฒ)). ยต(y, yโฒ))
= ฯ(x โ xโฒ). ฯ(x โฒ)
Also
ฯ(x โ xโฒ).= ยต(x โ xโฒ
, y โ yโฒ) = ยต((x, y) โ (xโฒ
, yโฒ
))
โฅ ยต(x, y). ยต(xโฒ
, yโฒ
)
=ฯ(x). ฯ(๐ฅโฒ)
Thus ฯ is a fuzzy dot BH-ideal of a BH-algebra X.
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