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Algebra and Discrete Mathematics
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Algebra and Discrete Mathematics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 November 2019) | Viewed by 20478

Special Issue Editor

Prof. Dr. Seok-Zun Song

E-Mail Website1 Website2
Guest Editor
Department of Mathematics, Jeju National University, Jeju 63243, Korea
Interests: linear operator; rank preserver; minimum permanents; BCK/BCI-algebras and related systems; fuzzy algebraic structures
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Algebra is well-known research subject for almost all mathematicians. It is important to develop computer science and cryptography. In this Special Issue, we hope to communicate new research topics and their applications. In mathematics, BCI/BCK-algebra is an algebraic structure that was introduced by Y. Imai, K. Iséki and S. Tanaka in 1966 to generalize the set difference in set theory, to describe fragments of the propositional calculus involving implications known as BCI and BCK logic. It is known that the class of BCK algebra is a proper subclass of the class of BCI algebra. We refer the reader to useful textbooks on BCI/BCK algebra (see Huang, Y. S. BCI-Algebra. Science Press: Beijing, China, 2006; Iorgulescu, A. Algebras of Logic as BCK Algebras. Editura ASE: Bucharest, Romania, 2008 and Meng J.; Jun, Y.B. BCK-Algebras. Kyung Moon Sa Co.: Seoul, Korea, 1994).

The aim of this Special Issue is to promote the exchange of ideas between researchers and to spread new trends in this area. It is focused on all aspects of algebra and BCK algebra and related algebraic systems from their foundations to applications in computer sciences, informatics and decision-making problems, etc. BCK algebra and related algebraic systems contain MV algebra, BL algebra, R0 algebra, MTL algebra, EQ algebra, lattice implication algebra, equality algebra, hoop algebra, etc.

Prof. Dr. Seok-Zun Song
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • algebra and its applications
  • linear algebra and its applications
  • discrete mathematics
  • BCK-algebras and related algebraic systems
  • (intuitionistic) fuzzy theory and applications
  • soft matrix theory and applications
  • (intuitionistic) fuzzy soft matrix theory and applications
  • neutrosophic soft matrix theory and applications
  • neutrosophic fuzzy matrix theory and applications
  • rough matrix theory and applications 
  • fuzzy soft rough matrix theory and applications

Published Papers (9 papers)

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Research

15 pages, 318 KiB  
Article
Composite Hurwitz Rings as PF-Rings and PP-Rings
by Dong Kyu Kim and Jung Wook Lim
Mathematics 2020, 8(1), 100; https://doi.org/10.3390/math8010100 - 7 Jan 2020
Cited by 1 | Viewed by 1808
Abstract
Let R ⊆ T be an extension of commutative rings with identity and H ( R , T ) (respectively, h ( R , T ) ) the composite Hurwitz series ring (respectively, composite Hurwitz polynomial ring). In this article, we study equivalent [...] Read more.
Let R ⊆ T be an extension of commutative rings with identity and H ( R , T ) (respectively, h ( R , T ) ) the composite Hurwitz series ring (respectively, composite Hurwitz polynomial ring). In this article, we study equivalent conditions for the rings H ( R , T ) and h ( R , T ) to be PF-rings and PP-rings. We also give some examples of PP-rings and PF-rings via the rings H ( R , T ) and h ( R , T ) . Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
8 pages, 252 KiB  
Article
Linear Maps that Preserve Any Two Term Ranks on Matrix Spaces over Anti-Negative Semirings
by Kyung Tae Kang, Seok-Zun Song and Young Bae Jun
Mathematics 2020, 8(1), 41; https://doi.org/10.3390/math8010041 - 1 Jan 2020
Viewed by 2665
Abstract
There are many characterizations of linear operators from various matrix spaces into themselves which preserve term rank. In this research, we characterize the linear maps which preserve any two term ranks between different matrix spaces over anti-negative semirings, which extends the previous results [...] Read more.
There are many characterizations of linear operators from various matrix spaces into themselves which preserve term rank. In this research, we characterize the linear maps which preserve any two term ranks between different matrix spaces over anti-negative semirings, which extends the previous results on characterizations of linear operators from some matrix spaces into themselves. That is, a linear map T from p × q matrix spaces into m × n matrix spaces preserves any two term ranks if and only if T preserves all term ranks if and only if T is a ( P , Q , B )-block map. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
13 pages, 260 KiB  
Article
Constructing Some Logical Algebras with Hoops
by M. Aaly Kologani, Seok-Zun Song, R. A. Borzooei and Young Bae Jun
Mathematics 2019, 7(12), 1243; https://doi.org/10.3390/math7121243 - 16 Dec 2019
Cited by 1 | Viewed by 3000
Abstract
In any logical algebraic structures, by using of different kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, in this paper by considering a hoop algebra or a hoop, that is introduced by Bosbach, the notion [...] Read more.
In any logical algebraic structures, by using of different kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, in this paper by considering a hoop algebra or a hoop, that is introduced by Bosbach, the notion of co-filter on hoops is introduced and related properties are investigated. Then by using of co-filter, a congruence relation on hoops is defined, and the associated quotient structure is studied. Thus Brouwerian semilattices, Heyting algebras, Wajsberg hoops, Hilbert algebras and BL-algebras are obtained. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
8 pages, 226 KiB  
Article
Some Implicativities for Groupoids and BCK-Algebras
by In Ho Hwang, Hee Sik Kim and Joseph Neggers
Mathematics 2019, 7(10), 973; https://doi.org/10.3390/math7100973 - 15 Oct 2019
Cited by 6 | Viewed by 1519
Abstract
In this paper, we generalize the notion of an implicativity discussed in B C K -algebras, and apply it to some groupoids and B C K -algebras. We obtain some relations among those axioms in the theory of groupoids. [...] Read more.
In this paper, we generalize the notion of an implicativity discussed in B C K -algebras, and apply it to some groupoids and B C K -algebras. We obtain some relations among those axioms in the theory of groupoids. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
18 pages, 303 KiB  
Article
Probability Functions on Posets
by Jae Hee Kim, Hee Sik Kim and Joseph Neggers
Mathematics 2019, 7(9), 785; https://doi.org/10.3390/math7090785 - 25 Aug 2019
Viewed by 1973
Abstract
In this paper, we define the notion of a probability function on a poset which is similar to the probability function discussed on d-algebras, and obtain three probability functions on posets. Moreover, we define a probability realizer of a poset, and we [...] Read more.
In this paper, we define the notion of a probability function on a poset which is similar to the probability function discussed on d-algebras, and obtain three probability functions on posets. Moreover, we define a probability realizer of a poset, and we provide some examples to describe its role for the standard probability function. We apply the notion of a probability function to the ordered plane and obtain three probability functions on it. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
20 pages, 351 KiB  
Article
Makgeolli Structures and Its Application in BCK/BCI-Algebras
by Sun Shin Ahn, Seok-Zun Song, Young Bae Jun and Hee Sik Kim
Mathematics 2019, 7(9), 784; https://doi.org/10.3390/math7090784 - 25 Aug 2019
Cited by 3 | Viewed by 1845
Abstract
A fuzzy set is an extension of an existing set using fuzzy logic. Soft set theory is a generalization of fuzzy set theory. Fuzzy and soft set theory are good mathematical tools for dealing with uncertainty in a parametric manner. The aim of [...] Read more.
A fuzzy set is an extension of an existing set using fuzzy logic. Soft set theory is a generalization of fuzzy set theory. Fuzzy and soft set theory are good mathematical tools for dealing with uncertainty in a parametric manner. The aim of this article is to introduce the concept of makgeolli structures using fuzzy and soft set theory and to apply it to BCK/BCI-algebras. The notion of makgeolli algebra and makgeolli ideal in BCK/BCI-algebras is defined, and several properties are investigated. It deals with the relationship between makgeolli algebra and makgeolli ideal, and several examples are given. Characterization of makgeolli algebra and makgeolli ideal are discussed, and a new makgeolli algebra from old one is established. A condition for makgeolli algebra to be makgeolli ideal in BCK-soft universe is considered, and we give example to show that makgeolli ideal is not makgeolli algebra in BCI-soft universe. Conditions for makgeolli ideal to be makgeolli algebra in BCI-soft universe are provided. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
9 pages, 232 KiB  
Article
Neutrosophic Quadruple BCI-Positive Implicative Ideals
by Young Bae Jun, Seok-Zun Song and Seon Jeong Kim
Mathematics 2019, 7(5), 385; https://doi.org/10.3390/math7050385 - 28 Apr 2019
Cited by 6 | Viewed by 1743
Abstract
By considering an entry (i.e., a number, an idea, an object, etc.) which is represented by a known part ( a ) and an unknown part ( b T , c I , d F ) where T , I , F have [...] Read more.
By considering an entry (i.e., a number, an idea, an object, etc.) which is represented by a known part ( a ) and an unknown part ( b T , c I , d F ) where T , I , F have their usual neutrosophic logic meanings and a , b , c , d are real or complex numbers, Smarandache introduced the concept of neutrosophic quadruple numbers. Using the concept of neutrosophic quadruple numbers based on a set, Jun et al. constructed neutrosophic quadruple BCK/BCI-algebras and implicative neutrosophic quadruple BCK-algebras. The notion of a neutrosophic quadruple BCI-positive implicative ideal is introduced, and several properties are dealt with in this article. We establish the relationship between neutrosophic quadruple ideal and neutrosophic quadruple BCI-positive implicative ideal. Given nonempty subsets I and J of a BCI-algebra, conditions for the neutrosophic quadruple ( I , J ) -set to be a neutrosophic quadruple BCI-positive implicative ideal are provided. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
6 pages, 223 KiB  
Article
Linear Operators That Preserve the Genus of a Graph
by LeRoy B. Beasley, Jeong Han Kim and Seok-Zun Song
Mathematics 2019, 7(4), 312; https://doi.org/10.3390/math7040312 - 28 Mar 2019
Cited by 2 | Viewed by 2308
Abstract
A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k − 1 . A mapping of the set of graphs on n vertices to itself [...] Read more.
A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k − 1 . A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k + 1 to graphs of genus k + 1 . We show that such linear operators are necessarily vertex permutations. Similar results with different restrictions on the genus k preserving operators give the same conclusion. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
12 pages, 672 KiB  
Article
Stanley Depth of Edge Ideals of Some Wheel-Related Graphs
by Jia-Bao Liu, Mobeen Munir, Raheel Farooki, Muhammad Imran Qureshi and Quratulien Muneer
Mathematics 2019, 7(2), 202; https://doi.org/10.3390/math7020202 - 21 Feb 2019
Cited by 3 | Viewed by 2550
Abstract
Stanley depth is a geometric invariant of the module and is related to an algebraic invariant called depth of the module. We compute Stanley depth of the quotient of edge ideals associated with some familiar families of wheel-related graphs. In particular, we establish [...] Read more.
Stanley depth is a geometric invariant of the module and is related to an algebraic invariant called depth of the module. We compute Stanley depth of the quotient of edge ideals associated with some familiar families of wheel-related graphs. In particular, we establish general closed formulas for Stanley depth of quotient of edge ideals associated with the m t h -power of a wheel graph, for m ≥ 3 , gear graphs and anti-web gear graphs. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
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