Search Results (25)

This paper establishes a graph-theoretic framework for idealization semigroups arising from Bruck–Reilly extensions. Building on a recent study by Wazzan and Ozalan, we introduce five graph families—${\Gamma }_E$, ${\Gamma }_0$, ${\Gamma }_{\mathrm{Cay}}$, ${\Gamma }_{\mathcal{K}}$, and $\Gamma \left(G_k\right)$—each encoding a distinct algebraic facet of $S{B}_i(\bowtie )B$. We prove explicit correspondences linking combinatorial invariants to algebraic structure: diameter captures generating efficiency and semilattice height; girth signals short relations; chromatic number bounds idempotent cardinalities and $\mathcal{D}$-class counts; clique number measures maximal commuting subsets; and Laplacian spectra encode ideal size and Schützenberger groups. Our central result demonstrates that Green’s relations are combinatorially recoverable from graph pairs. For commutative $S{B}_i(\bowtie )B$, $({\Gamma }_E,{\Gamma }_{\mathcal{K}})$ uniquely determines $\mathcal{J}$-order, $\mathcal{D}$-classes, and $\mathcal{H}$-classes via neighborhood inclusions, bipartite components, and automorphism orbits, yielding the first algorithmic reconstruction of ideal-theoretic structure from graph data. The framework is implemented in SageMath as a reproducible open-source toolkit validated on concrete examples. This work synthesizes algebraic graph theory, semigroup theory, and computational mathematics into a unified algebraic-combinatorial dictionary, providing both new analytical tools and a methodological template for studying algebraic constructions via graph invariants. Full article

The concept of symmetry is fundamental to the study of algebra; it serves as the basis for a branch of group theory that is essential to abstract algebra. A semigroup is a structure that builds upon the concept of a group, similarly extending the idea of symmetry found within groups. In this study, we specifically focus on semigroups. The main objective of this research is to apply the notion of m-polar picture fuzzy sets (m-PPFSs), with m being a natural number, in investigations into semigroups, as this concept generalizes m-polar fuzzy sets (m-PFSs) and picture fuzzy sets (PFSs). This research introduces the concepts of m-polar picture fuzzy left ideals (m-PPFLs), m-polar picture fuzzy right ideals (m-PPFRs), m-polar picture fuzzy ideals (m-PPFIs), m-polar picture fuzzy bi-ideals (m-PPFBs), and m-polar picture fuzzy generalized bi-ideals (m-PPFGBs) in semigroups. This study examines the relationships between these concepts, showing that every m-PPFL (m-PPFR) in the semigroups is also an m-PPFB, and that every m-PPFB in the semigroups is an m-PPFGB. However, the opposite is not true. Additionally, we provide the characteristics of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs in semigroups. We further discuss the connections between the m-PPFLs (m-PPFIs) and the m-PPFBs within the framework of regular semigroups, and most importantly, we show that, if the semigroup is regular, then the m-PPFBs and m-PPFGBs are equal. Finally, we utilize the properties of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs within semigroups to explore the classifications of regular semigroups. Full article

Let $\mathbb{S}$ be a numerical monoid, while a ${\mathcal{P}}_{\mathrm{fin}}\left(\mathbb{S}\right)$-monoid S is a monoid generated by a finite number of finite non-empty subsets of $\mathbb{S}$. That is, S is a non-cancellative commutative monoid obtained from the sumset of finite non-negative integer sets. This work provides an algorithm for computing the ideals associated with some ${\mathcal{P}}_{\mathrm{fin}}\left(\mathbb{S}\right)$-monoids. These are the key to studying some factorization properties of ${\mathcal{P}}_{\mathrm{fin}}\left(\mathbb{S}\right)$-monoids and some additive properties of sumsets. This approach links computational commutative algebra with additive number theory. Full article

This paper aims to introduce a novel idea of possibility multi-fuzzy soft ordered semigroups for ideals and interior ideals. Various results, formulated as theorems based on these concepts, are presented and further validated with suitable examples. This paper also explores the broad applicability of possibility multi-fuzzy soft ordered semigroups in solving modern decision-making problems. Furthermore, this paper explores various classes of ordered semigroups, such as simple, regular, and intra-regular, using this innovative method. Based on these concepts, some important conclusions are drawn with supporting examples. Moreover, it defines the possibility of multi-fuzzy soft ideals for semiprime ordered semigroups. Full article

Let S and $\mathcal{C}$ be affine semigroups in ${\mathbb{N}}^d$ such that $S\subseteq \mathcal{C}$. We provide a characterization for the set $\mathcal{C}\setminus S$ to be finite, together with a procedure and computational tools to check whether such a set is finite and, if so, compute its elements. As a consequence of this result, we provide a characterization for an ideal I of an affine semigroup S so that $S\setminus I$ is a finite set. If so, we provide some procedures to compute the set $S\setminus I$. Full article

The concept of possibility fuzzy soft sets is a step in a new direction towards a soft set approach that can be used to solve decision-making issues. In this piece of research, an innovative and comprehensive conceptual framework for possibility multi-fuzzy soft ordered semigroups by making use of the notions that are associated with possibility multi-fuzzy soft sets as well as ordered semigroups is introduced. Possibility multi-fuzzy soft ordered semigroups mark a newly developed theoretical avenue, and the central aim of this paper is to investigate it. The focus lies on investigating this newly developed theoretical direction, with practical examples drawn from decision-making and diagnosis practices to enhance understanding and appeal to researchers’ interests. We strictly build the notions of possibility multi-fuzzy soft left (right) ideals, as well as l-idealistic and r-idealistic possibility multi-fuzzy soft ordered semigroups. Furthermore, various algebraic operations, such as union, intersection, as well as AND and OR operations are derived, while also providing a comprehensive discussion of their properties. To clarify these innovative ideas, the theoretical constructs are further reinforced with a set of demonstrative examples in order to guarantee deep and improved comprehension of the proposed framework. Full article

In this paper, we discuss the hypothesis that an ordered $\Gamma$-semigroup can be constructed on the M-left(right)-tri-basis. In order to generalize the left(right)-tri-basis using $\Gamma$-semigroups and ordered semigroups, we examined M-tri-ideals from a purely algebraic standpoint. We also present the form of the M-tri-ideal generator. We investigated the M-left(right)-tri-ideal using the ordered $\Gamma$-semigroup. In order to obtain their properties, we used M-left(right)-tri-basis. It was possible to generate a M-left(right)-tri-basis from elements and their subsets. Throughout this paper, we will present an interesting example of order ${⪯}_{mlt}\left({⪯}_{mrt}\right)$, which is not a partial order of $\mathcal{S}$. Additionally, we introduce the notion of quasi-order. As an example, we demonstrate the relationship between M-left(right)-tri-basis and partial order. Full article

The N-soft sets are newly defined structures with many applications in the real world. We aim for combining the semigroup theory and N-soft sets to provide a comprehensive account of the hybrid framework of N-soft Semigroups. In this paper, we define the $\gamma$-inclusive set, int N-soft subsemigroups, int N-soft left [right] ideals of S, int N-soft product and int N-soft characteristic function, $\theta$-Generalized int N-soft subsemigroups and $\theta$-Generalized int N-soft left [right] ideals of S. We also discuss some examples and theorems based on the restricted (extended) union, restricted (extended) intersection, and $\gamma$-inclusive set. Full article

Ordered semigroups are understood through their subsets. The aim of this article is to study ordered semigroups through some new substructures. In this regard, quasi-filters and $(m,n)$-quasi-filters of ordered semigroups are introduced as new types of filters. Some properties of the new concepts are investigated, different examples are constructed, and the relations between quasi-filters and quasi-ideals as well as between $(m,n)$-quasi-filters and $(m,n)$-quasi-ideals are discussed. Full article

First, semigroup structure is constructed by providing binary operations for the crossing cubic set structure. The concept of commutative crossing cubic ideal is introduced by applying crossing cubic set structure to commutative ideal in BCK-algebra, and several properties are investigated. The relationship between crossing cubic ideal and commutative crossing cubic ideal is discussed. An example to show that crossing cubic ideal is not commutative crossing cubic ideal is given, and then the conditions in which crossing cubic ideal can be commutative crossing cubic ideal are explored. Characterizations of commutative crossing cubic ideal are discussed, and the relationship between commutative crossing cubic ideal and crossing cubic level set is considered. An extension property of commutative crossing cubic ideal is established, and the translation of commutative crossing cubic ideal is studied. Conditions for the translation of crossing cubic set structure to be commutative crossing cubic ideal are provided, and its characterization is processed. Full article

Let $a,b$ and $n>1$ be three positive integers such that a and ${\sum }_{j=0}^{n-1}{b}^j$ are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of $\mathbb{N}$ generated by $\left\{{\sum }_{j=0}^{n-1}{b}^j\right\}\cup \{{\sum }_{j=0}^{n-1}{b}^j+a{\sum }_{j=0}^{i-2}{b}^j\mid i=2,\dots ,n\}$ is determinantal. Moreover, we prove that for $n>3$, the ideal I has a unique minimal system of generators if and only if $a<b-1$. Full article

The central objective of the proposed work in this research is to introduce the innovative concept of an m-polar fuzzy set (m-PFS) in semigroups, that is, the expansion of bipolar fuzzy set (BFS). Our main focus in this study is the generalization of some important results of BFSs to the results of m-PFSs. This paper provides some important results related to m-polar fuzzy subsemigroups (m-PFSSs), m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs), m-polar fuzzy quasi-ideals (m-PFQIs) and m-polar fuzzy interior ideals (m-PFIIs) in semigroups. This research paper shows that every m-PFBI of semigroups is the m-PFGBI of semigroups, but the converse may not be true. Furthermore this paper deals with several important properties of m-PFIs and characterizes regular and intra-regular semigroups by the properties of m-PFIs and m-PFBIs. Full article

Let S and T be two numerical semigroups. We say that T is an $\mathrm{I}(S)$-semigroup if $T\setminus \{0\}$ is an ideal of S. Given k a positive integer, we denote by $\Delta (k)$ the symmetric numerical semigroup generated by $\{2,2k+1\}$. In this paper we present a formula which calculates the number of $\mathrm{I}(S)$-semigroups with genus $\mathrm{g}(\Delta (k))+h$ for some nonnegative integer h and which we will denote by $\mathrm{i}(\Delta (k),h)$. As a consequence, we obtain that the sequence ${\{\mathrm{i}(\Delta (k),h)\}}_{h\in \mathbb{N}}$ is never decreasing. Besides, it becomes stationary from a certain term. Full article

We study the concept of i-ideal of an ordered n-ary semigroup and give a construction of the i-ideal of an ordered n-ary semigroup generated by its nonempty subset. Moreover, we study the notions of prime, weakly prime, semiprime and weakly semiprime ideals of an ordered n-ary semigroup. Full article

Let $\mathsf{\Gamma }$ be a commutative monoid, $R={⨁}_{\alpha \in \mathsf{\Gamma }}{R}_{\alpha }$ a $\mathsf{\Gamma }$-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