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10 pages, 278 KiB

Open AccessArticle

Generating Integrally Indecomposable Newton Polygons with Arbitrary Many Vertices

by Petar Ðapić, Ivan Pavkov, Siniša Crvenković and Ilija Tanackov

Mathematics 2022, 10(14), 2389; https://doi.org/10.3390/math10142389 - 07 Jul 2022

Viewed by 1066

In this paper we shall give another proof of a special case of Gao’s theorem for generating integrally indecomposable polygons in the sense of Minkowski. The approach of proving this theorem will enable us to give an effective algorithm for construction integrally indecomposable [...] Read more.

In this paper we shall give another proof of a special case of Gao’s theorem for generating integrally indecomposable polygons in the sense of Minkowski. The approach of proving this theorem will enable us to give an effective algorithm for construction integrally indecomposable convex integral polygons with arbitrary many vertices. In such a way, classes of absolute irreducible bivariate polynomials corresponding to those indecomposable Newton polygons are generated. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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19 pages, 1275 KiB

Open AccessArticle

Algebraic Perspective of Cubic Multi-Polar Structures on BCK/BCI-Algebras

by Anas Al-Masarwah and Halimah Alshehri

Mathematics 2022, 10(9), 1475; https://doi.org/10.3390/math10091475 - 28 Apr 2022

Cited by 6 | Viewed by 1296

Abstract

Cubic multipolar structure with finite degree (briefly, cubic k-polar (

C_{k} P

) structure) is a new hybrid extension of both k-polar fuzzy (

_{k} P F

) structure and cubic structure in which

C_{k} P

structure consists of [...] Read more.

Cubic multipolar structure with finite degree (briefly, cubic k-polar (

C_{k} P

) structure) is a new hybrid extension of both k-polar fuzzy (

_{k} P F

) structure and cubic structure in which

C_{k} P

structure consists of two parts; the first one is an interval-valued k-polar fuzzy (

I V_{k} P F

) structure acting as a membership grade extended from the interval

P [0, 1]

to

P {[0, 1]}^{k}

(i.e., from interval-valued of real numbers to the k-tuple interval-valued of real numbers), and the second one is a

_{k} P F

structure acting as a nonmembership grade extended from the interval

[0, 1]

to

{[0, 1]}^{k}

(i.e., from real numbers to the k-tuple of real numbers). This approach is based on generalized cubic algebraic structures using polarity concepts and therefore the novelty of a

C_{k} P

algebraic structure lies in its large range comparative to both

_{k} P F

algebraic structure and cubic algebraic structure. The aim of this manuscript is to apply the theory of

C_{k} P

structure on BCK/BCI-algebras. We originate the concepts of

C_{k} P

subalgebras and (closed)

C_{k} P

ideals. Moreover, some illustrative examples and dominant properties of these concepts are studied in detail. Characterizations of a

C_{k} P

subalgebra/ideal are given, and the correspondence between

C_{k} P

subalgebras and (closed)

C_{k} P

ideals are discussed. In this regard, we provide a condition for a

C_{k} P

subalgebra to be a

C_{k} P

ideal in a BCK-algebra. In a BCI-algebra, we provide conditions for a

C_{k} P

subalgebra to be a

C_{k} P

ideal, and conditions for a

C_{k} P

subalgebra to be a closed

C_{k} P

ideal. We prove that, in weakly BCK-algebra, every

C_{k} P

ideal is a closed

C_{k} P

ideal. Finally, we establish the

C_{k} P

extension property for a

C_{k} P

ideal. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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20 pages, 335 KiB

Open AccessArticle

Transposition Regular AG-Groupoids and Their Decomposition Theorems

by Yudan Du, Xiaohong Zhang and Xiaogang An

Mathematics 2022, 10(9), 1396; https://doi.org/10.3390/math10091396 - 22 Apr 2022

Cited by 3 | Viewed by 1169

Abstract

In this paper, we introduce transposition regularity into AG-groupoids, and a variety of transposition regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties and structures are discussed by their decomposition theorems: (1) L1/R1-transposition regular AG-groupoids are equivalent to each other, and they can be [...] Read more.

In this paper, we introduce transposition regularity into AG-groupoids, and a variety of transposition regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties and structures are discussed by their decomposition theorems: (1) L1/R1-transposition regular AG-groupoids are equivalent to each other, and they can be decomposed into the union of disjoint Abelian subgroups; (2) L1/R1-transposition regular AG-groupoids are LR-transposition regular AG-groupoids, and an example is given to illustrate that not every LR-transposition regular AG-groupoid is an L1/R1-transposition regular AG-groupoid; (3) an AG-groupoid is an L1/R1-transposition regular AG-groupoid if it is an LR-transposition regular AG-groupoid satisfying a certain condition; (4) strong L2/R3-transposition regular AG-groupoids are equivalent to each other, and they are union of disjoint Abelian subgroups; (5) strong L3/R2-transposition regular AG-groupoids are equivalent to each other and they can be decomposed into union of disjoint AG subgroups. Their relations are discussed. Finally, we introduce various transposition regular AG-groupoid semigroups and discuss the relationships among them and the commutative Clifford semigroup as well as the Abelian group. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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11 pages, 780 KiB

Open AccessArticle

Hamiltonian Cycles in Cayley Graphs of Gyrogroups

by Rasimate Maungchang, Charawi Detphumi, Prathomjit Khachorncharoenkul and Teerapong Suksumran

Mathematics 2022, 10(8), 1251; https://doi.org/10.3390/math10081251 - 11 Apr 2022

Cited by 1 | Viewed by 1467

Abstract

In this study, we investigate Hamiltonian cycles in the right-Cayley graphs of gyrogroups. More specifically, we give a gyrogroup version of the factor group lemma and show that some right-Cayley graphs of certain gyrogroups are Hamiltonian. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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16 pages, 311 KiB

Open AccessArticle

Normalizer Maps Modulo N

by Nazlı Yazıcı Gözütok

Mathematics 2022, 10(7), 1046; https://doi.org/10.3390/math10071046 - 24 Mar 2022

Cited by 1 | Viewed by 1196

Abstract

The present paper is devoted to studying the maps corresponding to the suborbital graphs for the normalizer

Γ_{B} (N)

of

Γ_{0} (N)

modulo N, where N denotes a positive integer. We reveal the complete structure of [...] Read more.

The present paper is devoted to studying the maps corresponding to the suborbital graphs for the normalizer

Γ_{B} (N)

of

Γ_{0} (N)

modulo N, where N denotes a positive integer. We reveal the complete structure of these maps, finding their vertices, edges, darts, and faces explicitly. The maps we investigated in the present paper were all regular maps of large genus except for some low values of N. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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16 pages, 494 KiB

Open AccessArticle

Left (Right) Regular and Transposition Regular Semigroups and Their Structures

by Xiaohong Zhang and Yudan Du

Mathematics 2022, 10(7), 1021; https://doi.org/10.3390/math10071021 - 22 Mar 2022

Cited by 8 | Viewed by 1984

Abstract

Regular semigroups and their structures are the most wonderful part of semigroup theory, and the contents are very rich. In order to explore more regular semigroups, this paper extends the relevant classical conclusions from a new perspective: by transforming the positions of the [...] Read more.

Regular semigroups and their structures are the most wonderful part of semigroup theory, and the contents are very rich. In order to explore more regular semigroups, this paper extends the relevant classical conclusions from a new perspective: by transforming the positions of the elements in the regularity conditions, some new regularity conditions (collectively referred to as transposition regularity) are obtained, and the concepts of various transposition regular semigroups are introduced (L1/L2/L3, R1/R2/R3-transposition regular semigroups, etc.). Their relations with completely regular semigroups and left (right) regular semigroups, proposed by Clifford and Preston, are analyzed. Their properties and structures are studied from the aspects of idempotents, local identity elements, local inverse elements, subsemigroups and so on. Their decomposition theorems are proved respectively, and some new necessary and sufficient conditions for semigroups to become completely regular semigroups are obtained. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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8 pages, 288 KiB

Open AccessArticle

The Basic Locally Primitive Graphs of Order Twice a Prime Square

by Yulong Ma and Bengong Lou

Mathematics 2022, 10(6), 985; https://doi.org/10.3390/math10060985 - 18 Mar 2022

Viewed by 1320

Abstract

A graph

Γ

is called G-basic if G is quasiprimitive or bi-quasiprimitive on the vertex set of

Γ

, where

G ⩽ Aut (Γ)

. It is known that locally primitive vertex-transitive graphs are normal covers of basic ones. In [...] Read more.

A graph

Γ

is called G-basic if G is quasiprimitive or bi-quasiprimitive on the vertex set of

Γ

, where

G ⩽ Aut (Γ)

. It is known that locally primitive vertex-transitive graphs are normal covers of basic ones. In this paper, a complete classification of the basic locally primitive vertex-transitive graph of order

2 p^{2}

is given, where p is an odd prime. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

30 pages, 4869 KiB

Open AccessArticle

On Cyclic Associative Semihypergroups and Neutrosophic Extended Triplet Cyclic Associative Semihypergroups

by Minghao Hu and Xiaohong Zhang

Mathematics 2022, 10(4), 535; https://doi.org/10.3390/math10040535 - 09 Feb 2022

Cited by 8 | Viewed by 1405

Abstract

This paper introduces a new concept called cyclic associative semihypergroup (CA-semihypergroup). The relationships among CA-semihypergroups, Semihypergroups and LA-semihypergroups are studied through some interesting examples. The relationships among various NET-CA-semihypergroups are also studied. The main properties of strong pure neutrosophic extended triplet CA-semihypergroups (SP-NET-CA-semihypergroups) [...] Read more.

This paper introduces a new concept called cyclic associative semihypergroup (CA-semihypergroup). The relationships among CA-semihypergroups, Semihypergroups and LA-semihypergroups are studied through some interesting examples. The relationships among various NET-CA-semihypergroups are also studied. The main properties of strong pure neutrosophic extended triplet CA-semihypergroups (SP-NET-CA-semihypergroups) are obtained. In particular, the algorithm of a generated CA-semihypergroup of order tm+n by two known CA-semihypergroups of order m and n is proven, and a CA-semihypergroup of order 19 is obtained by using a Python program. Moreover, it is proven that five different definitions, which can all be used as the definition of SP-NET-CA-Semihypergroup, are equivalent. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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11 pages, 298 KiB

Open AccessArticle

Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

by Ruixue Zhang, Fengming Dong and Meiqiao Zhang

Mathematics 2022, 10(2), 193; https://doi.org/10.3390/math10020193 - 09 Jan 2022

Viewed by 1163

Abstract

A mixed hypergraph

H

is a triple

(X, C, D)

, where X is a finite set and each of

C

and

D

is a family of subsets of X. For any positive integer

λ

, a proper [...] Read more.

A mixed hypergraph

H

is a triple

(X, C, D)

, where X is a finite set and each of

C

and

D

is a family of subsets of X. For any positive integer

λ

, a proper

λ

-coloring of

H

is an assignment of

λ

colors to vertices in

H

such that each member in

C

contains at least two vertices assigned the same color and each member in

D

contains at least two vertices assigned different colors. The chromatic polynomial of

H

is the graph-function counting the number of distinct proper

λ

-colorings of

H

whenever

λ

is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals

(- \infty, 0)

and

(0, 1)

, which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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7 pages, 337 KiB

Open AccessArticle

A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A₁₁₉

by Bo Ling, Wanting Li and Bengong Lou

Mathematics 2021, 9(22), 2935; https://doi.org/10.3390/math9222935 - 18 Nov 2021

Viewed by 1075

Abstract

A Cayley graph

Γ = Cay (G, S)

is said to be normal if the base group G is normal in

Aut Γ

. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 [...] Read more.

A Cayley graph

Γ = Cay (G, S)

is said to be normal if the base group G is normal in

Aut Γ

. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group

A_{119}

. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to

A_{120}

. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

13 pages, 287 KiB

Open AccessArticle

Commutative Ideals of BCI-Algebras Using MBJ-Neutrosophic Structures

by Seok-Zun Song, Mehmet Ali Öztürk and Young-Bae Jun

Mathematics 2021, 9(17), 2047; https://doi.org/10.3390/math9172047 - 25 Aug 2021

Cited by 3 | Viewed by 1351

Abstract

As a generalization of a neutrosophic set, the notion of MBJ-neutrosophic sets is introduced by Mohseni Takallo, Borzooei and Jun, and it is applied to BCK/BCI-algebras. In this article, MBJ-neutrosophic set is used to study commutative ideal in BCI-algebras. The concept of closed [...] Read more.

As a generalization of a neutrosophic set, the notion of MBJ-neutrosophic sets is introduced by Mohseni Takallo, Borzooei and Jun, and it is applied to BCK/BCI-algebras. In this article, MBJ-neutrosophic set is used to study commutative ideal in BCI-algebras. The concept of closed MBJ-neutrosophic ideal and commutative MBJ-neutrosophic ideal is introduced and their properties and relationships are studied. The conditions for an MBJ-neutrosophic ideal to be a commutative MBJ-neutrosophic ideal are given. The conditions for an MBJ-neutrosophic ideal to be a closed MBJ-neutrosophic ideal are provided. Characterization of a commutative MBJ-neutrosophic ideal is established. Finally, the extension property for a commutative MBJ-neutrosophic ideal is founded. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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