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
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
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
For any intuitionistic multi-fuzzy set A = { < x , µA(x) , νA(x) > : x∈X} of an universe set X, we study the set [A](α, β) called the (α, β)–lower cut of A. It is the crisp multi-set { x∈X : µi(x) ≤ αi , νi(x) ≥ βi , ∀i } 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 (α, β)–lower cut sets
Regularity of Generalized Derivations in P Semi Simple BCIK AlgebrasYogeshIJTSRD
In this paper we study the regularity of inside or outside derivations in p semi simple BCIK – 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
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
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
For any intuitionistic multi-fuzzy set A = { < x , µA(x) , νA(x) > : x∈X} of an universe set X, we study the set [A](α, β) called the (α, β)–lower cut of A. It is the crisp multi-set { x∈X : µi(x) ≤ αi , νi(x) ≥ βi , ∀i } 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 (α, β)–lower cut sets
Regularity of Generalized Derivations in P Semi Simple BCIK AlgebrasYogeshIJTSRD
In this paper we study the regularity of inside or outside derivations in p semi simple BCIK – 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
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
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
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
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
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
For any intuitionistic multi-fuzzy set A = { < x , µA(x) , νA(x) > : x∈X} of an universe set X, we study the set [A](α, β) called the (α, β)–lower cut of A. It is the crisp multi-set { x∈X : µi(x) ≤ αi ,νi(x) ≥ βi , ∀i } 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 (α, β)–lower cut sets.
In this paper, the notion α -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The homomorphic images (pre images) ofα -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
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
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
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
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
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
For any intuitionistic multi-fuzzy set A = { < x , µA(x) , νA(x) > : x∈X} of an universe set X, we study the set [A](α, β) called the (α, β)–lower cut of A. It is the crisp multi-set { x∈X : µi(x) ≤ αi ,νi(x) ≥ βi , ∀i } 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 (α, β)–lower cut sets.
In this paper, the notion α -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The homomorphic images (pre images) ofα -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
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
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
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
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
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 “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. 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
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
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
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
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
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 Days’ 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
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 Taguchi’s 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 Taguchi’s 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.
A STUDY ON THE REASONS FOR TRANSGENDER TO BECOME ENTREPRENEURSIAEME Publication
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.
BROAD UNEXPOSED SKILLS OF TRANSGENDER ENTREPRENEURSIAEME Publication
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.
DETERMINANTS AFFECTING THE USER'S INTENTION TO USE MOBILE BANKING APPLICATIONSIAEME Publication
The banking and financial services industries are experiencing increased technology penetration. Among them, the banking industry has made technological advancements to better serve the general populace. The economy focused on transforming the banking sector's system into a cashless, paperless, and faceless one. The researcher wants to evaluate the user's intention for utilising a mobile banking application. The study also examines the variables affecting the user's behaviour intention when selecting specific applications for financial transactions. The researcher employed a well-structured questionnaire and a descriptive study methodology to gather the respondents' primary data utilising the snowball sampling technique. The study includes variables like performance expectations, effort expectations, social impact, enabling circumstances, and perceived risk. Each of the aforementioned variables has a major impact on how users utilise mobile banking applications. The outcome will assist the service provider in comprehending the user's history with mobile banking applications.
ANALYSE THE USER PREDILECTION ON GPAY AND PHONEPE FOR DIGITAL TRANSACTIONSIAEME Publication
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.
VOICE BASED ATM FOR VISUALLY IMPAIRED USING ARDUINOIAEME Publication
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 –normal 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.
IMPACT OF EMOTIONAL INTELLIGENCE ON HUMAN RESOURCE MANAGEMENT PRACTICES AMONG...IAEME Publication
There is increasing acceptability of emotional intelligence as a major factor in personality assessment and effective human resource management. Emotional intelligence as the ability to build capacity, empathize, co-operate, motivate and develop others cannot be divorced from both effective performance and human resource management systems. The human person is crucial in defining organizational leadership and fortunes in terms of challenges and opportunities and walking across both multinational and bilateral relationships. The growing complexity of the business world requires a great deal of self-confidence, integrity, communication, conflict and diversity management to keep the global enterprise within the paths of productivity and sustainability. Using the exploratory research design and 255 participants the result of this original study indicates strong positive correlation between emotional intelligence and effective human resource management. The paper offers suggestions on further studies between emotional intelligence and human capital development and recommends for conflict management as an integral part of effective human resource management.
VISUALISING AGING PARENTS & THEIR CLOSE CARERS LIFE JOURNEY IN AGING ECONOMYIAEME Publication
Our life journey, in general, is closely defined by the way we understand the meaning of why we coexist and deal with its challenges. As we develop the "inspiration economy", we could say that nearly all of the challenges we have faced are opportunities that help us to discover the rest of our journey. In this note paper, we explore how being faced with the opportunity of being a close carer for an aging parent with dementia brought intangible discoveries that changed our insight of the meaning of the rest of our life journey.
A STUDY ON THE IMPACT OF ORGANIZATIONAL CULTURE ON THE EFFECTIVENESS OF PERFO...IAEME Publication
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 employees’ cooperation to achieve its intended objectives. Employees' cooperation depends upon the organization’s 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 ‘Khaki Worthy’ men i.e., the police personnel. Whether we talk of Indian police or the other nation’s 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 one’s 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.
A STUDY ON TALENT MANAGEMENT AND ITS IMPACT ON EMPLOYEE RETENTION IN SELECTED...IAEME Publication
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.
ATTRITION IN THE IT INDUSTRY DURING COVID-19 PANDEMIC: LINKING EMOTIONAL INTE...IAEME Publication
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 organization’s 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
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.
A STUDY OF VARIOUS TYPES OF LOANS OF SELECTED PUBLIC AND PRIVATE SECTOR BANKS...IAEME Publication
Banking regulations act of India, 1949 defines banking as “acceptance of deposits for the purpose of lending or investment from the public, repayment on demand or otherwise and withdrawable through cheques, drafts order or otherwise”, 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.
EXPERIMENTAL STUDY OF MECHANICAL AND TRIBOLOGICAL RELATION OF NYLON/BaSO4 POL...IAEME Publication
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
ROLE OF SOCIAL ENTREPRENEURSHIP IN RURAL DEVELOPMENT OF INDIA - PROBLEMS AND ...IAEME Publication
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 country’s 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 isn’t 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.
OPTIMAL RECONFIGURATION OF POWER DISTRIBUTION RADIAL NETWORK USING HYBRID MET...IAEME Publication
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.
APPLICATION OF FRUGAL APPROACH FOR PRODUCTIVITY IMPROVEMENT - A CASE STUDY OF...IAEME Publication
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.
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENTIAEME Publication
In this paper, we investigated a queuing model of fuzzy environment-based a multiple channel queuing model (M/M/C) ( /FCFS) and study its performance under realistic conditions. It applies a nonagonal fuzzy number to analyse the relevant performance of a multiple channel queuing model (M/M/C) ( /FCFS). Based on the sub interval average ranking method for nonagonal fuzzy number, we convert fuzzy number to crisp one. Numerical results reveal that the efficiency of this method. Intuitively, the fuzzy environment adapts well to a multiple channel queuing models (M/M/C) ( /FCFS) are very well.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
The Internet of Things (IoT) is a revolutionary concept that connects everyday objects and devices to the internet, enabling them to communicate, collect, and exchange data. Imagine a world where your refrigerator notifies you when you’re running low on groceries, or streetlights adjust their brightness based on traffic patterns – that’s the power of IoT. In essence, IoT transforms ordinary objects into smart, interconnected devices, creating a network of endless possibilities.
Here is a blog on the role of electrical and electronics engineers in IOT. Let's dig in!!!!
For more such content visit: https://nttftrg.com/
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Forklift Classes Overview by Intella PartsIntella Parts
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.
For more technical information, visit our website https://intellaparts.com
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
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
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
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.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
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2. K. Anitha and Dr. N. Kandaraj
http://www.iaeme.com/IJARET/index.asp 351 editor@iaeme.com
Ideals on BH – 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 ∗ and a constant 0. Then (X, ∗, 0) is called
a BCK-algebra if it satisfies the following conditions
1. ((𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑧)) ∗ (𝑧 ∗ 𝑦) = 0
2.(𝑥 ∗ (𝑥 ∗ 𝑦)) ∗ 𝑦 = 0
3.𝑥 ∗ 𝑥 = 0
4. 𝑥 ∗ 𝑦 = 0, 𝑦 ∗ 𝑥 = 0 ⇒ 𝑥 = 𝑦
5. 0 ∗ 𝑥 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦, 𝑧 ∈ 𝑋
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) 0∈ 𝐼
(I2) 𝑦 𝑎𝑛𝑑 𝑥 ∗ 𝑦 ∈ 𝐼 ⇒ 𝑥 ∈ 𝐼 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝐼
Definition 2.3 [9, 10]
A nonempty set X with a constant 0 and a binary operation ∗ is called a BH-algebra, if it
satisfies the following axioms
(BH1) 𝑥 ∗ 𝑥 = 0
(BH2) 𝑥 ∗ 0 = 0
(BH3) 𝑥 ∗ 𝑦 = 0 𝑎𝑛𝑑 𝑦 ∗ 𝑥 = 0 ⇒ 𝑥 = 𝑦 for all 𝑥, 𝑦 ∈ 𝑋
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, ∗, 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) 0∈ 𝐼
(BHI2) 𝑦 𝑎𝑛𝑑 𝑥 ∗ 𝑦 ∈ 𝐼 ⇒ 𝑥 ∈ 𝐼
(BHI3) 𝑥 ∈ 𝐼 and 𝑦 ∈ 𝑋 ⇒ 𝑥 ∗ 𝑦 ∈ 𝐼 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝐼
A mapping 𝑓: 𝑋 → 𝑌 of BH-algebras is called a homomorphism if 𝑓 (𝑥 ∗ 𝑦 ) = 𝑓(𝑥) ∗
𝑓(𝑦) for all 𝑥, 𝑦 𝑋 ∈ 𝑋. Note that if 𝑓 ∶ 𝑋 → 𝑌 is homomorphism of BH-algebras, Then
𝑓 (0) = 0’ . We now review some fuzzy logic concepts. A fuzzy subset of a set X is a function
𝜇 ∶ 𝑋 → [0, 1]. For a fuzzy subset μ of X and t ∈ [0, 1], define U (μ; t) to be the set U (μ; t) =
{x ∈ X | μ(x) ≥ t}. For any fuzzy subsets μ and ν of a set X, we define
(𝜇 ∩ 𝜈)(𝑥) = 𝑚𝑖𝑛{𝜇(𝑥), 𝜈(𝑥)} 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋.
3. Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
http://www.iaeme.com/IJARET/index.asp 352 editor@iaeme.com
Let 𝑓 ∶ 𝑋 → 𝑌 be a function from a set X to a set Y and let μ be a fuzzy subset of X. The
fuzzy subset v of Y defined by )(yv {
otherwise
Yyyfifx
yfx
0
,)()(sup
1
)(1
is called the image of µ under𝑓, denoted by 𝑓(µ). If 𝜈 is a fuzzy subset o𝑓 𝑌, the fuzzy
subset μ of X given by 𝜇(𝑥) = 𝜈(𝑓(𝑥)) for all x ∈ X is called the Preimage of ν under f and is
denoted by 𝑓−1
(𝑣) . A fuzzy subset μ in X has the sup property if for any 𝑇 ⊆ 𝑋 there exists
𝑥0∈ T such that 𝜇(𝑥0) =
)(1
)(sup
yfx
z
. A fuzzy relation μ on a set X is a fuzzy subset of 𝑋 × 𝑋,
that is, a map 𝜇 ∶ 𝑋 × 𝑋 → [0, 1].
Definition 2.6[4, 6, 8]
Let X be a nonempty set. A fuzzy (sub) set 𝜇 of the set X is a mapping 𝜇: 𝑋 → [0,1
Definition 2.7[4, 6, 8]
Let 𝜇 be the fuzzy set of a set X. For a fixed s∈ [0,1], the set 𝜇 𝑠 = {𝑥 ∈ 𝑋: 𝜇(𝑥) ≥ 𝑠}is
called an upper level of μ or level subset of 𝜇
Definition 2.8[5, 7]
A fuzzy set 𝜇 in X is called fuzzy BCK-ideal of X if it satisfies the following inequalities
1.𝜇(0) ≥ 𝜇(𝑥)
2. 𝜇(𝑥) ≥ min{𝜇(𝑥 ∗ 𝑦), 𝜇(𝑦)}
Definition 2.9 [11]
Let X be a nonempty set with a binary operation ∗ and a constant 0. Then (X, ∗, 0) is called
a d - algebra if it satisfies the following axioms.
1.𝑥 ∗ 𝑥 = 0
2. 0 ∗ 𝑥 = 0
3. 𝑥 ∗ 𝑦 = 0, 𝑦 ∗ 𝑥 = 0 ⇒ 𝑥 = 𝑦 for all 𝑥, 𝑦 ∈ 𝑋
Definition 2.10 [12]
A fuzzy set 𝜇 in X is called fuzzy d-ideal of X if it satisfies the following inequalities
Fd1.𝜇(0) ≥ 𝜇(𝑥)
Fd2𝜇(𝑥) ≥ min{𝜇(𝑥 ∗ 𝑦), 𝜇(𝑦)}
Fd3. 𝜇(𝑥 ∗ 𝑦) ≥ min{𝜇(𝑥), 𝜇(𝑦)} For all 𝑥, 𝑦 ∈ 𝑋
Definition 2.11[12] A fuzzy subset μ of X is called a fuzzy dot d-ideal of X if it satisfies
The following conditions:
1. 𝜇(0) ≥ 𝜇(𝑥)
2. 𝜇(𝑥) ≥ 𝜇(𝑥 ∗ 𝑦) . 𝜇(𝑦)
3. 𝜇(𝑥 ∗ 𝑦) ≥ 𝜇(𝑥). 𝜇(𝑦)for all 𝑥, 𝑦 ∈ 𝑋
3. FUZZY IDEALS ON BH-ALGEBRAS
Definition 3.1
A fuzzy set 𝜇 in X is called fuzzy BH-ideal of X if it satisfies the following inequalities
1.𝜇(0) ≥ 𝜇(𝑥)
2. 𝜇(𝑥) ≥ min{𝜇(𝑥 ∗ 𝑦), 𝜇(𝑦)}
3. 𝜇(𝑥 ∗ 𝑦) ≥ min{𝜇(𝑥), 𝜇(𝑦)} for all 𝑥, 𝑦 ∈ 𝑋
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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, ∗ ,0) is not BCK-algebra. Since {(1∗ 3) ∗ (1 ∗ 2)} ∗ (2 ∗ 3) = 1 ≠ 0
We define fuzzy set 𝜇 in X by 𝜇(0) = 0.8 and 𝜇(𝑥) = 0.01 for all 𝑥 ≠ 0 in X. then it is
easy to show that 𝜇 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, ∗ ,0) is a fuzzy BCK-algebra. We define fuzzy set 𝜇 in X by 𝜇(0) = 0.7, 𝜇(1) =
0.5, 𝜇(2) = 0.2
Then 𝜇 is a fuzzy BH-ideal of X.
Definition 3.4
Let λ 𝑎𝑛𝑑 𝜇 be the fuzzy sets in a set X. The Cartesian product λ × 𝜇: 𝑋 × 𝑋 → [0,1] is
defined by (λ × 𝜇)(𝑥, 𝑦) = min{𝜆(𝑥), 𝜇(𝑦)} ∀𝑥, 𝑦 ∈ 𝑋.
Theorem 3.5
If λ 𝑎𝑛𝑑 𝜇 be the fuzzy BH-ideals of a BH- algebra X, then λ × 𝜇 is a fuzzy BH-ideals
of 𝑋 × 𝑋
Proof
For any (𝑥, 𝑦) ∈ 𝑋 × 𝑋, wehave
(λ × 𝜇)(0,0) = min{𝜆(0), 𝜇(0)} ≥ min{ 𝜆(𝑥), 𝜇(𝑦)}
= (λ × 𝜇)(𝑥, 𝑦)
That is (λ × 𝜇)(0,0) = (λ × 𝜇)(𝑥, 𝑦)
Let (𝑥1 , 𝑥2) 𝑎𝑛𝑑 (𝑦1, 𝑦2) ∈ 𝑋 × 𝑋
Then, (λ × 𝜇) (𝑥1 , 𝑥2) = 𝑚𝑖𝑛{𝜆(𝑥1), 𝜇(𝑥2)}
≥ min{min{ 𝜆(𝑥1 ∗ 𝑦1), 𝜆(𝑦1)} , min{𝜇(𝑥2 ∗ 𝑦2), 𝜇(𝑦2)}
= min{min 𝜆(𝑥1 ∗ 𝑦1 ), 𝜇(𝑥2 ∗ 𝑦2), min{ 𝜆(𝑦1), 𝜇(𝑦2)}}
= min {(λ × 𝜇) (𝑥1 ∗ 𝑦1, 𝑥2 ∗ 𝑦2)) , ( λ × 𝜇) (𝑦1, 𝑦2)}
= min {((λ × 𝜇) (𝑥1, 𝑥2) ∗ (𝑦1, 𝑦2)), ( λ × 𝜇) (𝑦1, 𝑦2)}
That is ((λ × 𝜇) (𝑥1, 𝑥2)) = min {(λ × 𝜇) (𝑥1 , 𝑥2) ∗ (𝑦1, 𝑦2)), (λ × 𝜇)(𝑦1, 𝑦2)}
And (λ × 𝜇)( (𝑥1 , 𝑥2) ∗ (𝑦1, 𝑦2))
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= (λ × 𝜇) (𝑥1 ∗ 𝑦1 , 𝑥2 ∗ 𝑦2)
= min{ 𝜆(𝑥1 ∗ 𝑦1 ), 𝜇(𝑥2 ∗ 𝑦2)}
≥ min {min {λ (𝑥1), λ (𝑦1)}, min {𝜇(𝑥2), 𝜇(𝑦2)}}
= min {min (λ ( 𝑥1 ), 𝜇(𝑥2), min{( λ (𝑦1), 𝜇(𝑦2))}
= min {(λ × 𝜇)(( 𝑥1 , 𝑥2), ( λ × 𝜇)(𝑦1, 𝑦2))
That is (λ × 𝜇)(( 𝑥1 , 𝑥2) ∗ (𝑦1, 𝑦2))
=min {(λ × 𝜇)(( 𝑥1 , 𝑥2), ( λ × 𝜇) (𝑦1, 𝑦2)}
Hence λ × 𝜇 is a fuzzy BH-ideal of X× 𝑋
Theorem 3.6
Let λ 𝑎𝑛𝑑 𝜇 be fuzzy sets in a BH-algebra such that 𝜆 × 𝜇 is a fuzzy BH-ideal of 𝑋 ×
𝑋.Then
i)Either λ(0)≥ λ (x)𝑜𝑟 𝜇(0) ≥ 𝜇(𝑥) ∀𝑥 ∈ 𝑋.
ii) If λ (0)≥ λ (x) ∀𝑥 ∈ 𝑋, then either 𝜇(0) ≥ λ(𝑥) or 𝜇(0) ≥ 𝜇(𝑥)
iii) If 𝜇(0) ≥ 𝜇(𝑥) ∀𝑥 ∈ 𝑋, then either λ (0)≥ λ (x) or λ (0)≥ µ(x)
Proof
We use reduction to absurdity
i) Assume λ(x)> λ (0) and 𝜇(𝑥) ≥ 𝜇(0) for some 𝑥, 𝑦 ∈ 𝑋.
Then (λ × 𝜇)(𝑥, 𝑦) = min{λ(x), 𝜇(𝑦)}
>min {{λ(0), 𝜇(0)}
= (λ × 𝜇)(0,0)
(λ × 𝜇)(𝑥, 𝑦) > (λ × 𝜇)(0,0) ∀𝑥, 𝑦 ∈ 𝑋
Which is a contradiction to (λ × 𝜇) is a fuzzy BH-ideal of X× 𝑋
Therefore either λ (0)≥ λ (x)𝑜𝑟 𝜇(0) ≥ 𝜇(𝑥) ∀𝑥 ∈ 𝑋.
ii) Assume 𝜇(0) < λ(𝑥) and 𝜇(0) < 𝜇(𝑦) for some 𝑥, 𝑦 ∈ 𝑋.
Then (λ × 𝜇)(0,0) = min{λ(0), 𝜇(0)}= 𝜇(0)
And (λ × 𝜇)(𝑥, 𝑦) = min{λ(x), 𝜇(𝑦)}> 𝜇(0)
= (λ × 𝜇)(0,0)
This implies (λ × 𝜇)(𝑥, 𝑦) > and ( λ × 𝜇)(0,0)
Which is a contradiction to λ × 𝜇 is a fuzzy BH-ideal of X× 𝑋
Hence if λ (0)≥ λ (x) ∀𝑥 ∈ 𝑋, 𝑡ℎ𝑒𝑛
Either 𝜇(0) ≥ λ(𝑥) or 𝜇(0) ≥ 𝜇(𝑥) ∀𝑥 ∈ 𝑋
iii) Assume λ (0)< λ (x) or λ (0)< µ(y) ∀𝑥, 𝑦 ∈ 𝑋
Then ((λ × 𝜇)(0, 0) = min{λ(0), 𝜇(0)}= 𝜆(0)
And (λ × 𝜇)(𝑥, 𝑦) = min{λ(x), 𝜇(𝑦)}> 𝜆(0)
= (λ × 𝜇)(0,0)
This implies (λ × 𝜇)(𝑥, 𝑦) >(λ × 𝜇)(0,0)
Which is a contradiction to (λ × 𝜇) is a fuzzy BH-ideal of 𝑋 × 𝑋
Hence if 𝜇(0) ≥ 𝜇(𝑥) ∀𝑥 ∈ 𝑋 then either λ (0)≥ λ (x) or λ (0)≥ µ(x)
This completes the proof
Theorem 3.7
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If λ × 𝜇 is a fuzzy BH-idela of 𝑋 × 𝑋 then λ 𝑜𝑟 𝜇 is a fuzzy BH-ideal of X.
Proof
First we prove that 𝜇 is a fuzzy BH-ideal of X.
Given λ × 𝜇 is a fuzzy BH-ideal of 𝑋 × 𝑋, then by theorem 3.6(i), either λ (0)≥
λ (x)𝑜𝑟 𝜇(0) ≥ 𝜇(𝑥) ∀𝑥 ∈ 𝑋.
Let 𝜇(0) ≥ 𝜇(𝑥)
By theorem 3.6(iii) then either λ (0)≥ λ (x) or λ (0)≥ µ(x)
Now 𝜇(𝑥) = min{λ(0), 𝜇(𝑥)}
= (λ × 𝜇)(0, 𝑥)
≥ min{ ( (λ × 𝜇)(0, 𝑥) ∗ (0, 𝑦)), (λ × 𝜇)(0, 𝑦)}
= min{ (λ × 𝜇)(0 ∗ 0), 𝑥 ∗ 𝑦),(λ × 𝜇)(0, 𝑦)}
= min{ (λ × 𝜇)(0, 𝑥 ∗ 𝑦),(λ × 𝜇)(0, 𝑦)}
= min{ (λ × 𝜇)(0 ∗ 0), 𝑥 ∗ 𝑦),(λ × 𝜇)(0, 𝑦)}
= min{ 𝜇(𝑥 ∗ 𝑦), ( 𝜇)(𝑦)}
That is 𝜇(𝑥) ≥ min{ 𝜇(𝑥 ∗ 𝑦), 𝜇(𝑦)}
𝜇(𝑥 ∗ 𝑦) = min{ λ (0) , 𝜇(𝑥 ∗ 𝑦)}
= (λ × 𝜇)(0, 𝑥 ∗ 𝑦)
= (λ × 𝜇)(0 ∗ 0, 𝑥 ∗ 𝑦)
= (λ × 𝜇)(0, 𝑥) ∗ (0, 𝑦)
𝜇(𝑥 ∗ 𝑦) ≥ min{ (λ × 𝜇)(0, 𝑥), (λ × 𝜇)(0, 𝑦)}
=min{𝜇(𝑥), 𝜇(𝑦)}
That is, 𝜇(𝑥 ∗ 𝑦) ≥ min{𝜇(𝑥), 𝜇(𝑦)}
This proves that 𝜇 is a fuzzy BH-ideal of X.
Secondly to prove that λ is a Fuzzy BH-ideal of X.
Using theorem 4.6(i) and (ii) we get
This completes the proof.
Theorem 3.8
If 𝜇 is a fuzzy BH-idela of 𝑋, then 𝜇 𝑡 is a BH-idela of X for all 𝑡 ∈ [0,1]
Proof
Let 𝜇 be a fuzzy BH-ideal of X,
Then By the definition of BH-ideal
𝜇(0) ≥ 𝜇(𝑥)
𝜇(𝑥) ≥ min{ 𝜇(𝑥 ∗ 𝑦), (𝜇(𝑦)}
𝜇(𝑥 ∗ 𝑦) ≥ min{ 𝜇(𝑥), (𝜇(𝑦)} ∀𝑥, 𝑦 ∈ 𝑋
To prove that 𝜇 𝑡 𝑖𝑠 𝑎 BH-ideal of x.
By the definition of level subset of 𝜇
𝜇 𝑡 = {𝑥/ 𝜇(𝑥) ≥ 𝑡}
Let 𝑥, 𝑦 ∈ 𝜇 𝑡 and 𝜇 is a fuzzy BH-ideal of X.
Since 𝜇(0) ≥ 𝜇(𝑥) ≥ 𝑡 implies 0∈ 𝜇 𝑡, for all 𝑡 ∈ [0,1]
Let 𝑥, 𝑦 ∈ 𝜇 𝑡 and 𝑦 ∈ 𝜇 𝑡
Therefore 𝜇(𝑥 ∗ 𝑦) ≥ 𝑡 and 𝜇(𝑦) ≥ 𝑡
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Now 𝜇(𝑥) ≥ min{ 𝜇(𝑥 ∗ 𝑦), (𝜇(𝑦)}
≥ min{𝑡, 𝑡} ≥ 𝑡
Hence (𝜇(𝑥) ≥ 𝑡
That is 𝑥 ∈ 𝜇 𝑡.
Let 𝑥 ∈ 𝜇 𝑡, 𝑦 ∈ 𝑋
Choose y in X such that 𝜇(𝑦) ≥ 𝑡
Since 𝑥 ∈ 𝜇 𝑡 implies 𝜇(𝑥) ≥ 𝑡
We know that 𝜇(𝑥 ∗ 𝑦) ≥ min{ 𝜇(𝑥), (𝜇(𝑦)}
≥ min{𝑡, 𝑡}
≥ 𝑡
That is 𝜇(𝑥 ∗ 𝑦) ≥ 𝑡 implies 𝑥 ∗ 𝑦 ∈ 𝜇 𝑡
Hence 𝜇 𝑡 is a BH-ideal of X.
Theorem 3.9
If X be a BH-algebra, ∀𝑡 ∈ [0,1] and 𝜇 𝑡 𝑖𝑠 𝑎 𝐵𝐻 −ideal of X, then 𝜇 is a fuzzy BH-ideal of
X.
Proof
Since 𝜇 𝑡 𝑖𝑠 𝑎 𝐵𝐻 −ideal of X
0∈ 𝜇 𝑡
ii) 𝑥 ∗ 𝑦 ∈ 𝜇 𝑡 And 𝑦 ∈ 𝜇 𝑡 implies 𝑥 ∈ 𝜇 𝑡
iii)𝑥 ∈ 𝜇 𝑡, y ∈ 𝑋 implies 𝑥 ∗ 𝑦 ∈ 𝜇 𝑡
To prove that 𝜇 𝑖𝑠 𝑎 𝑓𝑢𝑧𝑧𝑦 BH-ideal of X.
Let 𝑥, 𝑦 ∈ 𝜇 𝑡 then 𝜇(𝑥) ≥ 𝑡 and 𝜇(𝑦) ≥ 𝑡
Let 𝜇(𝑥) = 𝑡1 and 𝜇(𝑦) = 𝑡2
Without loss of generality let 𝑡1 ≤ 𝑡2
Then 𝑥 ∈ 𝜇 𝑡1
Now 𝑥 ∈ 𝜇 𝑡1
and y ∈ 𝑋 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑥 ∗ 𝑦 ∈ 𝜇 𝑡1
That is 𝜇(𝑥 ∗ 𝑦) ≥ 𝑡1
= min {𝑡1, 𝑡2}
= min { 𝜇(𝑥), 𝜇(𝑦)}
𝜇(𝑥 ∗ 𝑦) ≥ min { 𝜇(𝑥), 𝜇(𝑦)}
𝑖𝑖) 𝐿𝑒𝑡 𝜇(0) = 𝜇(𝑥 ∗ 𝑥) ≥ min { 𝜇(𝑥), 𝜇(𝑦)} ≥ 𝜇(𝑥) (by proof (i))
That is 𝜇(0) ≥ 𝜇(𝑥) for all x ∈ 𝑋
iii) Let 𝜇(𝑥) = 𝜇(𝑥 ∗ 𝑦) ∗ (0 ∗ 𝑦))
≥ min{ 𝜇(𝑥 ∗ 𝑦), 𝜇(0 ∗ 𝑦)) (By (i))
≥ min{ 𝜇(𝑥 ∗ 𝑦), min{𝜇(0), 𝜇(𝑦)}
≥ min{ 𝜇(𝑥 ∗ 𝑦),{𝜇(𝑦)) (By (ii))
𝜇(𝑥) ≥ min{ 𝜇(𝑥 ∗ 𝑦),{𝜇(𝑦))
Hence 𝜇 is a fuzzy BH-ideal of X.
Definition 3.10
A fuzzy set 𝜇 in X is said to be fuzzy BH- 𝞆 ideal if 𝜇(𝑥 ∗ 𝑢 ∗ 𝑣 ∗ 𝑦) ≥ min { 𝜇(𝑥), 𝜇(𝑦)}
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Theorem 3.11
Every Fuzzy BH -ideal is a fuzzy BH- 𝞆- ideal
Proof
It is trivial
Remark
Converse of the above theorem is not true. That is every fuzzy BH-𝞆 -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,∗ ,0) is a BH-algebra. We define fuzzy set: X→ [0,1] by 𝜇(0) = 0.8, 𝜇(𝑥) =
0.2 ∀𝑥 ≠ 0 clearly 𝜇 is a fuzzy BH-ideal of X But 𝜇 is not a BH-𝞆-ideal of X.
For Let 𝑥 = 0 𝑢 = 1 𝑣 = 1 𝑦 = 1
𝜇(𝑥 ∗ 𝑢 ∗ 𝑣 ∗ 𝑦 = 𝜇(0 ∗ 1 ∗ 1 ∗ 1)= 𝜇(1) = 0.2
min{ 𝜇(𝑥), 𝜇(𝑦)} = min{ 𝜇(0), 𝜇(0)}
= 𝜇(0) = 0.8
𝜇(𝑥 ∗ 𝑢 ∗ 𝑣 ∗ 𝑦) ≤min { 𝜇(𝑥), 𝜇(𝑦)}
Hence 𝜇 is not a fuzzy BH-𝞆-ideal of X.
And 𝜇(𝑥 ∗ 𝑦)= 𝜇 (𝑓(𝑎) ∗ 𝑓(𝑏))
≥ 𝜇 (𝑓(𝑎 ∗ 𝑏))
= 𝜇 𝑓
(𝑎 ∗ 𝑏)
≥ min{ 𝜇 𝑓(𝑎), 𝜇 𝑓
(𝑏)}
= min{𝜇(𝑓(𝑎), 𝜇(𝑓(𝑏))}
= min{𝜇(𝑥), 𝜇(𝑦)}
Hence 𝜇(𝑥 ∗ 𝑦) ≥ min{𝜇(𝑥), 𝜇(𝑦)}
Hence 𝜇 is a fuzzy BH- 𝞆 ideal of Y.
4. FUZZY DOT BH-IDEALS OF BH-ALGEBRAS
Definition 4.1. A fuzzy subset μ of X is called a fuzzy dot BH-ideal of X if it satisfies
The following conditions:
(FBH1). 𝜇(0) ≥ 𝜇(𝑥)
(FBH2). 𝜇(𝑥) ≥ 𝜇(𝑥 ∗ 𝑦) . 𝜇(𝑦)
(FBH3). 𝜇(𝑥 ∗ 𝑦) ≥ 𝜇(𝑥). 𝜇(𝑦)for all 𝑥, 𝑦 ∈ 𝑋
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 𝜇 ∶ 𝑋 → [0,1] by μ (0) =0.9, μ (a) = μ (b) =0.6, μ (c) =0.3 .It is easy to verify that μ
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 𝑋 = {0 ,1 ,2 ,3} be a BH-algebra with Cayley table (Table 1) as follows:
Example 4.4. Let 𝑋 = {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 μ: X → [0, 1] by μ (0) =0.8, μ (1) = 0.5, μ (2) =0.4. It is easy to verify that μ is a
fuzzy Dot BH-ideal of X, but not a fuzzy d-ideal of X because
𝜇(𝑥) ≤ min{𝜇(𝑥 ∗ 𝑦), 𝜇(𝑦)}
𝜇(1) = min{𝜇(1 ∗ 2), 𝜇(2)}
= 𝜇(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 μ: X → [0, 1] by
μ (0)= μ (1)=0.9, μ (2)= 0.7. It is easy to verify that μ is a fuzzy dot sub algebra of X, but
not a fuzzy dot BH-ideal of X because μ (2)=0.7 ≤ 0.81=μ (2 ∗1)⋅ μ (1) .
Proposition 4.7. If μ and ν are fuzzy dot BH-ideals of a BH-algebra X, then so is μ ∩ ν.
Proof. Let x, y ∈ X. Then
(𝜇 ∩ 𝜈)(0) = 𝑚𝑖𝑛 {𝜇 (0), 𝜈 (0)}
≥ 𝑚𝑖𝑛 {𝜇(𝑥), 𝜈(𝑥)}
= (𝜇 ∩ 𝜈) (𝑥).
Also, (𝜇 ∩ 𝜈) (𝑥) = 𝑚𝑖𝑛 {𝜇(𝑥), 𝜈(𝑥)}
≥ 𝑚𝑖𝑛 {𝜇(𝑥 ∗ 𝑦) · 𝜇(𝑦), 𝜈(𝑥 ∗ 𝑦) · 𝜈(𝑦)}
≥ (𝑚𝑖𝑛 {𝜇(𝑥 ∗ 𝑦), 𝜈(𝑥 ∗ 𝑦)}) · (𝑚𝑖𝑛 {𝜇(𝑦), 𝜈(𝑦)})
= ((𝜇 ∩ 𝜈) (𝑥 ∗ 𝑦)) · ((𝜇 ∩ 𝜈) (𝑦)).
𝐴𝑛𝑑, (𝜇 ∩ 𝜈) (𝑥 ∗ 𝑦) = 𝑚𝑖𝑛 {𝜇(𝑥 ∗ 𝑦), 𝜈(𝑥 ∗ 𝑦)}
≥ 𝑚𝑖𝑛 {𝜇(𝑥) · 𝜇(𝑦), 𝜈(𝑥) · 𝜈(𝑦)}
≥ (𝑚𝑖𝑛 {𝜇(𝑥), 𝜈(𝑥)}) · (𝑚𝑖𝑛 {𝜇(𝑦), 𝜈(𝑦)})
= ((𝜇 ∩ 𝜈) (𝑥)) · ((𝜇 ∩ 𝜈) (𝑦)).
Hence μ ∩ ν is a fuzzy dot BH-ideal of a d-algebra X.
Theorem 4.8. If each nonempty level subset U (μ; t) of μ is a fuzzy BH-ideal of X then
μ is a fuzzy dot BH-ideal of X, where t ∈[0, 1].
Definition 4.9
10. K. Anitha and Dr. N. Kandaraj
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Let σ be a fuzzy subset of X. The strongest fuzzy σ-relation on BH-algebra X is the fuzzy
subset µσ of X ×X given by 𝜇 𝜎(x, y) = σ(x) · σ(y) for all x, y ∈ X. A fuzzy relation μ on BH-
algebra X is called a Fuzzy σ-product relation if µ(x, y) ≥ σ(x) · σ(y) for all x, y ∈ X. A
fuzzy relation μ on BH-algebra is called a left fuzzy relation on σ if µ(x, y) = σ(x) for all x, y
∈ X.
Note that a left fuzzy relation on σ is a fuzzy σ-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 μ: X → [0, 1] by
μ (0)=0.6, μ (1)=0.7, μ (2)= 0.8. We know that μ is a fuzzy dot BH-ideal of X, but
U (μ; 0.8) = {x ∈X | μ(x) ≥ 0.8} = {2, 2} is not BH-ideal of X since 0∉U (μ; 0.8).
Theorem 4.11. Let 𝑓 ∶ 𝑋 → 𝑋 ′ be an onto homomorphism of BH-algebras, ν be a fuzzy
Dot BH-ideal of Y. Then the Preimage 𝑓−1
(𝑣) of ν under f is a fuzzy dot BH-ideal of X.
Proof. Let x∈ X,
𝑓−1(𝑣)(0) = 𝑣(𝑓(0)) = 𝑣(0′
)
≥ 𝑣(𝑓(𝑥)) = 𝑓−1
(𝑣)(𝑥)
For any x, y∈ X, we have
𝑓−1
(𝑣)(𝑥) = ν (f (x)) ≥ ν (f (x)* f(y)) · ν (f (y))
= ν (f (x* y)) · ν (f (y)) = 𝑓−1
(𝑣)(𝑥 ∗ 𝑦). 𝑓−1(𝑣)(𝑦)
Also,
𝑓−1
(𝑣)(𝑥 ∗ 𝑦)= ν (f (x * y)) = ν (f (x) * f (y))
≥ ν (f (x)) · ν(f (y)) = 𝑓−1
(𝑣)(𝑥). 𝑓−1
(𝑣)(𝑦)
Thus 𝑓−1
(𝑣)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 λ and μ are fuzzy dot BH-ideal of a BH-algebra X, then λ × μ is a fuzzy
Dot BH-ideal of X × X.
Proof.
Let 𝑥, 𝑦 ∈ 𝑋
λ × μ (0,0)=λ(0), μ(0)
≥ λ(x).μ(y)= (λ × μ) (x, y)
For any 𝑥, 𝑥′
, 𝑦, 𝑦′ ∈ 𝑋 wehave
(λ × µ) (x, y) = λ(x). µ(y)
≥ λ(x ∗ x′). λ(x′))µ(y ∗ y′). µ(y′)
= λ(x ∗ x′). µ(y ∗ y′)). λ(x′). µ(y′)
= (λ× µ)(x, y) ∗ (x′
, y′
)). (λ× µ)(x′
, y′
)
Also (λ× µ)(x, y) ∗ (x′
, y′
)) = (λ× µ)(x ∗ x′) ∗ (y ∗ y′
)).
= λ(x ∗ x′). µ(y ∗ y′)
≥ λ(x). λ(x′
)). ( µ(y). µ(y′))
(λ(x). µ(y)). (λ(x′). µ(y′))
= (λ× µ)(x, y). (λ × µ)(x′, y′)
Hence λ × μ is a fuzzy dot BH-ideal of X × X.
11. Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
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Theorem 4.14. Let σ be a fuzzy subset of a BH-algebra X and 𝜇 𝜎be the strongest fuzzy
σ -relation on BH-algebra X. Then σ is a fuzzy dot BH-ideal of X if and only if 𝜇 𝜎 is a
Fuzzy dot BH-ideal of X × X.
Proof. Assume that σ is a fuzzy dot BH-ideal of X. For any x, y ∈ X we have
𝜇 𝜎 (0, 0)= σ (0) · σ (0) ≥ σ(x) · σ(y) = 𝜇 𝜎 (x, y).
Let x, x ′, y, y ′∈ X. Then
𝜇 𝜎 ((𝑥, 𝑥 ′) ∗ (𝑦, 𝑦 ′)) · 𝜇 𝜎 (𝑦, 𝑦 ′)
= 𝜇 𝜎 (𝑥 ∗ 𝑦, 𝑥 ′ ∗ 𝑦 ′) · 𝜇 𝜎 (𝑦, 𝑦 ′)
= (𝜎(𝑥 ∗ 𝑦) · 𝜎(𝑥 ′ ∗ 𝑦 ′)) · (𝜎(𝑦) · 𝜎(𝑦 ′))
= (𝜎(𝑥 ∗ 𝑦) · 𝜎(𝑦)) · (𝜎(𝑥 ′ ∗ 𝑦 ′) · 𝜎(𝑦 ′))
≤ 𝜎 (𝑥) · 𝜎(𝑥 ′ ) = 𝜇 𝜎 (𝑥, 𝑥 ′ ). 𝐴𝑛𝑑,
𝜇 𝜎 (x, x ′ )· 𝜇 𝜎 (y, y ′) = (σ(x) · σ(x ′)) · (σ(y) · σ(y ′))
= (σ(x) · σ(y)) · (σ(x ′) · σ(y ′))
≤ 𝜎 (𝑥 ∗ 𝑦) · 𝜎( 𝑥 ′ ∗ 𝑦 ′ )
= 𝜇 𝜎 (𝑥 ∗ 𝑦, 𝑥 ′ ∗ 𝑦 ′) = 𝜇 𝜎 ((𝑥, 𝑥 ′) ∗ (𝑦, 𝑦 ′)).
Thus 𝜇 𝜎 is a fuzzy dot BH-ideal of X × X.
Conversely suppose that 𝜇 𝜎 is a fuzzy dot BH-ideal of X × X. From (FBH1) we get
(σ(0) )2
= σ (0) · σ(0)= 𝑐 (0, 0)
And so σ (0) ≥ σ(x) for all x ∈ X. Also we have
( σ(x) )2
= µσ ((x, x )
≥ µσ ((x, x)*(y, y)). µσ (y, y)
=µσ ((x∗y ), (x ∗ y)). µσ (y, y)
=((σ(x ∗ y) · σ(y)) )2
Which implies that 𝜎(𝑥) ≥ 𝜎(𝑥 ∗ y) · σ(y) for all x, y ∈ X.
Also we have
(𝜎(𝑥 ∗ 𝑦)2
=𝜇 𝜎 ((x∗y ), 𝑥 ∗ 𝑦)
=𝜇 𝜎 ((𝑥, 𝑥) ∗ (𝑦, 𝑦)) ≥ µσ(𝑥, 𝑥). µσ(𝑦, 𝑦)
= (σ(𝑥) · σ( 𝑦 ))2
So 𝜎(𝑥 ∗ 𝑦) ≥ 𝜎(𝑥) · 𝜎(𝑦) for all x, y ∈X.
Therefore σ is a fuzzy dot BH-ideal of X.
Proposition 4.15. Let μ be a left fuzzy relation on a fuzzy subset σ of a BH-algebra X. If
μ is a fuzzy dot BH-ideal of X × X, then σ is a fuzzy dot BH-ideal of a BH-algebra X.
Proof.
Suppose that a left fuzzy relation μ on σ is a fuzzy dot BH-ideal of X × X.
Then
σ (0) = μ (0, z ) ∀z ∈ X
By putting z=0
σ (0) = μ (0,0) ≥ μ (x , y )=σ (x ),
For all x∈ X.
For any x, x ′, y, y ′∈ X
12. K. Anitha and Dr. N. Kandaraj
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σ(x) = µ(x, y) ≥ µ((x, y) ∗ (x′
, y′)). µ(x′, y′)
= µ((x ∗ x′), (y ∗ y′)). µ(y, y′))
= σ(x ∗ x′). σ(x ′)
Also
σ(x ∗ x′).= µ(x ∗ x′
, y ∗ y′) = µ((x, y) ∗ (x′
, y′
))
≥ µ(x, y). µ(x′
, y′
)
=σ(x). σ(𝑥′)
Thus σ is a fuzzy dot BH-ideal of a BH-algebra X.
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