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13 pages, 386 KiB

Open AccessArticle

The Crossing Numbers of Join Products of Paths and Cycles with Four Graphs of Order Five

by Michal Staš

Mathematics 2021, 9(11), 1277; https://doi.org/10.3390/math9111277 - 2 Jun 2021

Cited by 9 | Viewed by 2247

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph [...] Read more.

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph

K_{1, 1, 3} \ e

obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph

K_{1, 1, 3}

. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

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9 pages, 263 KiB

Open AccessFeature PaperArticle

An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

by Javier Rodrigo, Susana Merchán, Danilo Magistrali and Mariló López

Mathematics 2021, 9(5), 525; https://doi.org/10.3390/math9050525 - 3 Mar 2021

Viewed by 1134

Abstract

In this paper, we improve the lower bound on the minimum number of

\leq k

-edges in sets of n points in general position in the plane when k is close to

\frac{n}{2}

. As a consequence, we improve the current [...] Read more.

In this paper, we improve the lower bound on the minimum number of

\leq k

-edges in sets of n points in general position in the plane when k is close to

\frac{n}{2}

. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph

K_{n}

for some values of n. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

13 pages, 235 KiB

Open AccessArticle

On BV-Algebras

by In Ho Hwang, Yong Lin Liu and Hee Sik Kim

Mathematics 2020, 8(10), 1779; https://doi.org/10.3390/math8101779 - 14 Oct 2020

Cited by 3 | Viewed by 2242

Abstract

In this paper, we introduce the notion of a

B V

-algebra, and we show that a

B V

-algebra is logically equivalent to several algebras, i.e.,

B M

-algebras,

B T

-algebras,

B O

-algebras and 0-commutative B-algebras. Moreover, we show [...] Read more.

In this paper, we introduce the notion of a

B V

-algebra, and we show that a

B V

-algebra is logically equivalent to several algebras, i.e.,

B M

-algebras,

B T

-algebras,

B O

-algebras and 0-commutative B-algebras. Moreover, we show that a

B V

-algebra with

(F)

is logically equivalent to several algebras, and we show some relationships between a

B V

-algebra with

(F)

and several related algebras. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

17 pages, 331 KiB

Open AccessArticle

On Minimal and Maximal Hyperidealsin n-ary Semihypergroups

by Jukkrit Daengsaen, Sorasak Leeratanavalee and Bijan Davvaz

Mathematics 2020, 8(10), 1656; https://doi.org/10.3390/math8101656 - 25 Sep 2020

Cited by 6 | Viewed by 1558

Abstract

The concept of j-hyperideals, for all positive integers

1 \leq j \leq n

and

n \geq 2

, in n-ary semihypergroups, is a generalization of the concept of left, lateral and right hyperideals in ternary semihypergroups. In this paper, we first [...] Read more.

The concept of j-hyperideals, for all positive integers

1 \leq j \leq n

and

n \geq 2

, in n-ary semihypergroups, is a generalization of the concept of left, lateral and right hyperideals in ternary semihypergroups. In this paper, we first introduce the concept of j-(0-)simple n-ary semihypergroups and discuss their related properties through terms of j-hyperideals. Furthermore, we characterize the minimality and maximality of j-hyperideals in n-ary semihypergroups and establish the relationships between the (0-)minimal, maximal j-hyperideals and the j-(0-)simple n-ary semihypergroups. Our main results are to extend and generalize the results on semihypergroups and ternary semihypergroups. Moreover, a related question raised by Petchkaew and Chinram is solved. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

11 pages, 275 KiB

Open AccessFeature PaperArticle

When Are Graded Rings Graded S-Noetherian Rings

by Dong Kyu Kim and Jung Wook Lim

Mathematics 2020, 8(9), 1532; https://doi.org/10.3390/math8091532 - 8 Sep 2020

Cited by 4 | Viewed by 2060

Abstract

Let

Γ

be a commutative monoid,

R = ⨁_{α \in Γ} R_{α}

a

Γ

-graded ring and S a multiplicative subset of

R_{0}

. We define R to be a graded S-Noetherian ring if every homogeneous ideal of R [...] Read more.

Let

Γ

be a commutative monoid,

R = ⨁_{α \in Γ} R_{α}

a

Γ

-graded ring and S a multiplicative subset of

R_{0}

. We define R to be a graded S-Noetherian ring if every homogeneous ideal of R is S-finite. In this paper, we characterize when the ring R is a graded S-Noetherian ring. As a special case, we also determine when the semigroup ring is a graded S-Noetherian ring. Finally, we give an example of a graded S-Noetherian ring which is not an S-Noetherian ring. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

14 pages, 277 KiB

Open AccessArticle

On Nonnil-S-Noetherian Rings

by Min Jae Kwon and Jung Wook Lim

Mathematics 2020, 8(9), 1428; https://doi.org/10.3390/math8091428 - 26 Aug 2020

Cited by 10 | Viewed by 2100

Abstract

Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, [...] Read more.

Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

10 pages, 299 KiB

Open AccessArticle

Endomorphism Spectra of Double Fan Graphs

by Mengdi Tong and Hailong Hou

Mathematics 2020, 8(6), 1009; https://doi.org/10.3390/math8061009 - 19 Jun 2020

Cited by 2 | Viewed by 1862

Abstract

There are six different classes of endomorphisms for a graph. The sets of these endomorphisms always form a chain under the inclusion of sets. For a more systematic treatment of different endomorphisms, Böttcher and Knauer proposed the concepts of the endomorphism type and [...] Read more.

There are six different classes of endomorphisms for a graph. The sets of these endomorphisms always form a chain under the inclusion of sets. For a more systematic treatment of different endomorphisms, Böttcher and Knauer proposed the concepts of the endomorphism type and the endomorphism spectrum of a graph in 1992. In this paper, we studied endomorphism types and endomorphism spectra of double fan graphs. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

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8 pages, 259 KiB

Open AccessArticle

Linear Operators That Preserve Two Genera of a Graph

by LeRoy B. Beasley, Kyung-Tae Kang and Seok-Zun Song

Mathematics 2020, 8(5), 676; https://doi.org/10.3390/math8050676 - 30 Apr 2020

Viewed by 1691

Abstract

If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus

g - 1

without edge crossings, then we say that the graph has genus g. We [...] Read more.

If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus

g - 1

without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus

g + j

to graphs of genus

g + j

for

j \leq g

and m sufficiently large. We show that such linear operators are necessarily vertex permutations. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

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10 pages, 241 KiB

Open AccessArticle

Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

by Hashem Bordbar, Young Bae Jun and Seok-Zun Song

Mathematics 2020, 8(4), 567; https://doi.org/10.3390/math8040567 - 11 Apr 2020

Cited by 3 | Viewed by 2221

Abstract

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras

φ : X \to Y

, we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping [...] Read more.

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras

φ : X \to Y

, we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping

c l_{Y} : I (Y) \to I (Y)

, we define a map

c l_{Y}^{\leftarrow}

on

I (X)

by

A \mapsto φ^{- 1} (φ {(A)}^{c l_{Y}}) .

We prove that, if “

c l_{Y}

” is a weak closure operation (respectively, semi-prime and meet) on

I (Y)

, then so is “

c l_{Y}^{\leftarrow}

” on

I (X)

. In addition, for mapping

c l_{X} : I (X) \to I (X)

, we define a map

c l_{X}^{\to}

on

I (Y)

as follows:

B \mapsto φ (φ^{- 1} {(B)}^{c l_{X}}) .

We show that, if “

c l_{X}

” is a weak closure operation (respectively, semi-prime and meet) on

I (X)

, then so is “

c l_{X}^{\to}

” on

I (Y)

. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

11 pages, 318 KiB

Open AccessArticle

Integral Domains in Which Every Nonzero w-Flat Ideal Is w-Invertible

by Hwankoo Kim and Jung Wook Lim

Mathematics 2020, 8(2), 247; https://doi.org/10.3390/math8020247 - 14 Feb 2020

Cited by 1 | Viewed by 2452

Abstract

Let D be an integral domain and w be the so-called w-operation on D. We define D to be a w-FF domain if every w-flat w-ideal of D is of w-finite type. This paper presents some properties [...] Read more.

Let D be an integral domain and w be the so-called w-operation on D. We define D to be a w-FF domain if every w-flat w-ideal of D is of w-finite type. This paper presents some properties of w-FF domains and related domains. Among other things, we study the w-FF property in the polynomial extension, the t-Nagata ring and the pullback construction. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

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8 pages, 315 KiB

Open AccessArticle

Unicyclic Graphs Whose Completely Regular Endomorphisms form a Monoid

by Rui Gu and Hailong Hou

Mathematics 2020, 8(2), 240; https://doi.org/10.3390/math8020240 - 13 Feb 2020

Cited by 3 | Viewed by 1787

Abstract

In this paper, completely regular endomorphisms of unicyclic graphs are explored. Let G be a unicyclic graph and let

c E n d (G)

be the set of all completely regular endomorphisms of G. The necessary and sufficient conditions under [...] Read more.

In this paper, completely regular endomorphisms of unicyclic graphs are explored. Let G be a unicyclic graph and let

c E n d (G)

be the set of all completely regular endomorphisms of G. The necessary and sufficient conditions under which

c E n d (G)

forms a monoid are given. It is shown that

c E n d (G)

forms a submonoid of

E n d (G)

if and only if G is an odd cycle or

G = G (n, m)

for some odd

n \geq 3

and integer

m \geq 1

. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

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