This document discusses the category of U-complexes and proves that it is an abelian category. It begins by reviewing the concepts of a chain U-complex, chain (U,U)-map, and isomorphism of U-complexes. It then defines the category of U-complexes and proves it satisfies the properties of an abelian category, namely that it has kernels, cokernels, products, coproducts, and every morphism has an image and coimage that form a short exact sequence. This establishes the category of U-complexes as a generalization of the category of complexes that retains an important algebraic structure.
Professor Timoteo Carletti presented a seminar titled "A journey in the zoo of Turing patterns: the topology does matter as part of the SMART Seminar Series on 8th March 2018.
More information: http://www.uoweis.co/event/a-journey-in-the-zoo-of-turing-patterns-the-topology-does-matter/
Keep updated with future events: http://www.uoweis.co/events/category/smart-infrastructure-facility/
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
Professor Timoteo Carletti presented a seminar titled "A journey in the zoo of Turing patterns: the topology does matter as part of the SMART Seminar Series on 8th March 2018.
More information: http://www.uoweis.co/event/a-journey-in-the-zoo-of-turing-patterns-the-topology-does-matter/
Keep updated with future events: http://www.uoweis.co/events/category/smart-infrastructure-facility/
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
In this paper, natural inner product structure for the space of fuzzy n−tuples is introduced. Also we have
introduced the ortho vector, stochastic fuzzy vectors, ortho- stochastic fuzzy vectors, ortho-stochastic fuzzy
matrices and the concept of orthogonal complement of fuzzy vector subspace of a fuzzy vector space.
Talk accompanying the paper:
Lihua You and Richard Southern and Jian J. Zhang, Motion in Games (Lecture Notes in Computer Science), 2009/06/01, 5884(1):207-218, doi:10.1007/978-3-642-10347-6_19
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 2.
More info at http://summerschool.ssa.org.ua
Lattice rules are one of the two main classes of methods for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) integration. In this tutorial, we recall the definition and summarize the key properties of lattice rules. We discuss what classes of functions these rules are good to integrate, and how their parameters can be chosen in terms of variance bounds for these classes of functions. We consider integration lattices in the real space as well as in a polynomial space over the finite field F2. We provide various numerical examples of how these rules perform compared with standard Monte Carlo. Some examples involve high-dimensional integrals, others involve Markov chains. We also discuss software design for RQMC and what software is available.
Existence, Uniqueness and Stability Solution of Differential Equations with B...iosrjce
In this work, we investigate the existence ,uniqueness and stability solution of non-linear
differential equations with boundary conditions by using both method Picard approximation and
Banach fixed point theorem which were introduced by [6] .These investigations lead us to improving
and extending the above method. Also we expand the results obtained by [1] to change the non-linear
differential equations with initial condition to non-linear differential equations with boundary
conditions
Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)Yandex
We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.
Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one $k$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).
We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.
References
[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.
1: 97–115, 1988.
In case of spatial data, the OLS estimates are inconsistent. An alternative procedure is to use the MLE method. Here I discussed the parameter estimation procedure in a spatial regressive-autoregressive model using Ord's eigenvalue method. I wrote R code to implement this method in real world data.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
In this paper, natural inner product structure for the space of fuzzy n−tuples is introduced. Also we have
introduced the ortho vector, stochastic fuzzy vectors, ortho- stochastic fuzzy vectors, ortho-stochastic fuzzy
matrices and the concept of orthogonal complement of fuzzy vector subspace of a fuzzy vector space.
Talk accompanying the paper:
Lihua You and Richard Southern and Jian J. Zhang, Motion in Games (Lecture Notes in Computer Science), 2009/06/01, 5884(1):207-218, doi:10.1007/978-3-642-10347-6_19
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 2.
More info at http://summerschool.ssa.org.ua
Lattice rules are one of the two main classes of methods for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) integration. In this tutorial, we recall the definition and summarize the key properties of lattice rules. We discuss what classes of functions these rules are good to integrate, and how their parameters can be chosen in terms of variance bounds for these classes of functions. We consider integration lattices in the real space as well as in a polynomial space over the finite field F2. We provide various numerical examples of how these rules perform compared with standard Monte Carlo. Some examples involve high-dimensional integrals, others involve Markov chains. We also discuss software design for RQMC and what software is available.
Existence, Uniqueness and Stability Solution of Differential Equations with B...iosrjce
In this work, we investigate the existence ,uniqueness and stability solution of non-linear
differential equations with boundary conditions by using both method Picard approximation and
Banach fixed point theorem which were introduced by [6] .These investigations lead us to improving
and extending the above method. Also we expand the results obtained by [1] to change the non-linear
differential equations with initial condition to non-linear differential equations with boundary
conditions
Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)Yandex
We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.
Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one $k$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).
We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.
References
[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.
1: 97–115, 1988.
In case of spatial data, the OLS estimates are inconsistent. An alternative procedure is to use the MLE method. Here I discussed the parameter estimation procedure in a spatial regressive-autoregressive model using Ord's eigenvalue method. I wrote R code to implement this method in real world data.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
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.
On Some Continuous and Irresolute Maps In Ideal Topological Spacesiosrjce
In this paper we introduce some continuous and irresolute maps called
δ
ˆ
-continuity,
δ
ˆ
-irresolute,
δ
ˆ
s-continuity and
δ
ˆ
s-irresolute maps in ideal topological spaces and study some of their properties.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
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.
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, 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.
μ-πrα Closed Sets in Bigeneralized Topological SpacesIJERA Editor
The aim of the paper is to introduce the concept of μ(m,n)-πrα closed sets in bigeneralized topological spaces and study some of their properties. We also introduce the notion of μ(m,n)-πrα continuous function and μ(m,n)-πrα T1/2 spaces on bigeneralized topological spaces and investigate some of their properties. Mathematics subject classification: 54A05, 54A10
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
Continuous functions play a dominant role in analysis and homotopy theory. They
have applications to image processing, signal processing, information, statistics,
engineering and technology. Recently topologists studied the continuous like functions
between two different topological structures. For example, semi continuity between a
topological structure, α-continuity between a topology and an α-topology.
Nithyanantha Jothi and Thangavelu introduced the concept of binary topology in
2011. Recently the authors extended the notion of binary topology to n-ary topology
where n˃1 an integer. In this paper continuous like functions are defined between a
topological and an n-ary topological structures and their basic properties are
studied.
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
1. International Journal of Mathematical Analysis
Vol. 10, 2016, no. 17, 849 - 853
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijma.2016.6682
Abelian Property of
the Category of U-Complexes
Gustina Elfiyanti
Mathematics Department
Faculty of Sciences and Technology, UIN Jakarta
Faculty of Natural Sciences, Institut Teknologi Bandung, Indonesia
Intan Muchtadi - Alamsyah
Mathematics Department
Faculty of Natural Sciences, Institut Teknologi Bandung, Indonesia
Dellavitha Nasution
Mathematics Department
Faculty of Natural Sciences, Institut Teknologi Bandung, Indonesia
Utih Amartiwi
Mathematics Department
Faculty of Sciences and Technology, UIN Jakarta, Indonesia
Copyright c 2015 Gustina Elfiyanti, Intan Muchtadi-Alamsyah, Dellavitha Nasution
and Utih Amartiwi. This article is distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
Abstract
The notion of a chain U−complex and chain (U, U )−map were in-
troduced by Davvaz and Shabbani as a generalization of a chain complex
and a chain map respectively. In this paper we continue their research
by proposing a category of U−complexes as a generalization of the ca-
tegory of complexes. We show that the category of U−complexes is an
abelian category.
2. 850 G. Elfiyanti, I. Muchtadi-Alamsyah, D. Nasution and U. Amartiwi
Mathematics Subject Classification: 55U15, 18E10
Keywords: chain U−complex, chain (U, U )−map, category of U−complexes,
abelian category
1 Introduction
A sequence of R−modules and R−homomorphism
· · · → Xn+1
dn+1
−→ Xn
dn
−→ Xn−1
dn−1
−→ Xn−2 −→ · · · (1)
is called exact sequence if Im(dn+1) = d−1
n (0). It is a natural question what if
0 is replaced by Un−1, a submodul of Xn−1. Davvaz and Parnian [1] modified
the definition of exact sequence of modules which is called U−exact sequences
and generalized some results from existing ones to the modified case. Their
research was motivated by the exact sequence of hypergroups which generally
has no zero element, introduced by Freni and Elderberry in [2].
Davvaz and Shabbani continued working on this topic and proposed the
concept of U−complex as a generalization of complex [3]. They defined the
concepts of chain U−complex, U−homology, chain (U, U )−map, chain (U, U )−
homotopy and U−functor and used the concepts to find a generalization of se-
veral results in homological algebra.
This paper aims to apply the previous results to examine the concepts
category of U−complexes. We show that the category of U−complexes is an
abelian category.
2 Chain of U-Complexes
In this section we review some results introduced by Davvaz and Shabbani.
Definition 2.1 Given a family X = (X, UX
, dX
) = (Xn, UX
n , dX
n )n∈Z where
Xn, Un are R−modules and each of Xn consists Un and dn : Xn → Xn−1. A
chain
(X, UX
, dX
) : · · · → Xn+1
dX
n+1
−→ Xn
dX
n
−→ Xn−1
dX
n−1
−→ Xn−2 −→ · · ·
is called UX
−complex if for all n ∈ Z we have:
1. dX
n dX
n+1(Xn+1) ⊆ UX
n−1 and
2. Im dX
n ⊇ UX
n−1
The definition implisitly say that a chain complex is a chain 0−complex.
3. Abelian property of the category of U−complexes 851
Definition 2.2 Let (X, UX
, dX
) be a UX
−complex and (Y, UY
, dY
) be a
UY
−complex. The sequence f = (fn : Xn → Yn)n∈Z is called chain (UX
, UY
)−map
if following diagram is commutative and fn(UX
n ) ⊆ UY
n for each n ∈ Z.
· · · −→ Xn+1
dX
n+1
−→ Xn
dX
n
−→ Xn−1 −→ · · ·
↓fn+1 ↓fn ↓fn−1
· · · −→ Yn+1
dY
n+1
−→ Yn
dY
n
−→ Yn−1 −→ · · ·
Proposition 2.3 Let (X, UX
, dX
) be a UX
−complex such that dX
n dX
n+1(Xn+1) =
UX
n−1 and (Y, UX
, dY
) is a chain UY
−complex. If f = (fn : Xn → Yn)n∈Z is a
chain map then it is also a chain (UX
, UY
)−map.
Definition 2.4 Let Let (X, UX
, dX
) and (Y, UX
, dY
) be a chain UX
−complex
and UY
−complex respectively. A chain (UX
, UY
)−map f = (fn)n∈Z is an iso-
morphism if fn is R−modules isomorphism for all n ∈ Z and f−1
= (f−1
n )n∈Z
is a chain (UY
, UX
)−map.
If there exists an isomorphism from (X, UX
, dX
) to (Y, UX
, dY
) we say that
(X, UX
, dX
) isomorphic to (Y, UX
, dY
). The isomorphism of chain U−complexes
is an equivalence relation.
Proposition 2.5 If chain UX
−complex and UY
−complex are isomorphic
then UX
n UY
n for all n ∈ Z.
3 The Category of U-Complexes
In this section we introduce the concept of a category of U−complexes and
study its property. Let A be an abelian category R−Mod.
Definition 3.1 The category of U−complexes C(A, U) is a category whose
objects are chain U−complexes in A, the morphisms are chain (U, U )−map
and the composition operation is the usual composition function.
Theorem 3.2 The category of U-complexes C(A, U) is an abelian category
Proof
A1 Let (X, UX
, dX
) and (Y, UX
, dY
) be a chain UX
−complex and UY
−complex
respectively. Assume that f = (fn)n∈Z and g = (gn)n∈Z are two chain
(UX
, UY
)−maps. By defining f +g = (fn +gn)n∈Z it is easy to prove that
HomC(A,U) (X, Y ) is an abelian group and the composition of morphisms
HomC(A,U) (Y, Z) × HomC(A,U) (X, Y ) → HomC(A,U) (X, Z)
is bilinier over integer.
4. 852 G. Elfiyanti, I. Muchtadi-Alamsyah, D. Nasution and U. Amartiwi
A2 The zero object in C(A, U) is the chain of 0−complex which all modules
are zero.
A3 A coproduct of two objects X = (X, UX
, dX
n ) and Y = (Y, UY
, dY
n ) is and
object
X ⊕ Y = X ⊕ Y, UX⊕Y
, dX⊕Y
= Xn ⊕ Yn, UX⊕Y
n , dX⊕Y
n n∈Z
where
UX⊕Y
n =
UX
n
UY
n
and dX⊕Y
n =
dX
n 0
0 dY
n
together with chain UX
, UX⊕Y
−map ιX and chain UY
, UX⊕Y
−map
ιY satisfying the universal property: for every objects Z in C (A, U),
chain UX
, UZ
−map fX and chain UY
, UZ
−map fY there is a unique
chain UX⊕Y
, UZ
−map f making following diagram commutative.
Zn
(fX )n fn ↑ (fY )n
Xn
(ιX )n
−→ Xn ⊕ Yn
(ιY )n
←− Yn
(2)
A4 Let f = (fn : Xn → Yn)n∈Z be a chain UX
, UY
−map, then each fn is a
morphism in A. We show the existence of a cokernel and leave the dual.
Since A is an abelian category, each fn has a cokernel Cn = Yn/Im (fn) in
A together with a morphism cn : Yn → Cn such that cnfn = 0 satisfying
the universal property of cokernel, i.e. there is a unique morphism dC
n :
Cn → Cn−1 such that cn−1dC
n = dY
n cn. Let f = (fn : Xn → Yn)n∈Z be
a chain UX
, UY
−map, then each fn is a morphism in A. Hence the
following diagram is commutative.
X · · · → Xn+1
dX
n+1
→ Xn
dX
n
→ Xn−1 · · · →
↓fn+1 ↓fn ↓fn−1
Y · · · → Yn+1
dY
n+1
→ Yn
dY
n
→ Yn−1 · · · →
↓cn+1 ↓cn ↓cn−1
C · · · → Cn+1
dC
n+1
→ Cn
dC
n
→ Cn−1 · · · →
(3)
By choosing UC
n = UY
n / Im (fn) , it is easy to check that C = Cn, dC
n , UC
n
is a chain UC
−complex and satisfying the universal property of cokernel
for f.
A5 Let U
Im(f)
n = fn UX
n and U
coIm(f)
n = UX
n /ker(fn) then coIm(f) and
Im(f) are objects in C (A, U) . Consider the natural morphism coIm(f) →
coIm(f), since A is abelian then for every n the natural morphism
5. Abelian property of the category of U−complexes 853
gn : coIm(fn) → Im(fn) is an isomophism, hence the invers g−1
n :
coIm(fn) → Im(fn) is also isomorphism and g−1
= (g−1
n )n∈Z is a chain
map. As U
Im(f)
n = fn UX
n we have g−1
n U
Im(f)
n = UX
n , hence g−1
is
a chain UY
, UX
−map. By the Definition 2.4, g is an isomorphism of
U−complexes.
Acknowledgements. This research is supported by ITB and UIN Jakarta
Research Grant.
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Received: November 1, 2015; Published: June 20, 2016