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Algebraic Structures and Applications
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
Let S be a nonempty set. A binary operation on S is any function from S × S to S. We shall denote an arbitrary operation on S by “⋆.” Thus ⋆ : S × S → S sending (x, y) → x ⋆ y assigns to each ordered pair (x, y) of elements of S an element x ⋆ y in S. Binary operation ⋆ on S is said to be associative if (x ⋆ y) ⋆ z = x ⋆ (y ⋆ z) for all x, y, z ∈ S. (S, ⋆) is called a semigroup if S is a nonempty set and ⋆ defines an associative binary operation on S.
Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
A semigroup is an associative binary system. An element e of a binary system (S, · ) is an identity element of S if a · e = e · a = a for all a ∈ S.
Introduction
Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002
John N. Mordeson, Davender S. Malik
Let (X,*) be a semigroup and let S be a nonempty subset of X. Then (S,*) is said to be a subsemigroup of (X,*) if (S,*) is a mathematical system. (Here for (S,*), we mean * restricted to S×S.) Let (X,*) be a monoid and let (S,*) be a subsemigroup of (X,*). If e is the identity of (X,*) and e∈S, then (S,*) is called a submonoid of (X,*). We often write X for (X,*) when the operation * is understood. Let a∈X. We define a1=a and if an is defined for n∈N, we define an+1=an*a.
A new characterization of fuzzy ideals of semigroups and its applications
Published in Automatika, 2021
Chunhua Li, Baogen Xu, Huawei Huang
Recently, considerable attention has been paid to study of fuzzy semigroups such as fuzzy ordered semigroups, fuzzy regular semigroups, and so on. As a generalized regular semigroup, the class of abundant semigroups encompasses a wide variety of semigroups, from regular semigroups and abundant monoids first arose in connection with the theory of S-systems. It is natural in this paper to consider these semigroups which satisfies abundance. Here we develop an approach to fuzzy semigroups, in the framework of fuzzy star ideals. In particular such as an approach based on the use of and is strong and completable. By fuzziness of certain ideal of a semigroup, it is possible to deal with fuzzy semigroup structure and to establish the relationships between fuzzy star ideals and semigroup abundance and semigroup regularity for an arbitrary semigroup. This is also the highlight of this paper. As a concrete application, we advance sufficient and necessary conditions for an arbitrary semigroup to be regular and inverse, respectively. Fuzzy star ideals introduced in this paper could improve applications of the theory of regular semigroups, and might be important for further study of fuzzy semigroup theory.
Global attractors for semigroup actions on uniformizable spaces
Published in Dynamical Systems, 2020
Josiney A. Souza, Richard W. M. Alves
In this paper we extend the notion of global attractor to the setting of semigroup actions on uniformizable spaces. This setting covers a wide scope of mathematical analysis, since almost every topological space studied in this branch of mathematics is at least uniformizable (completely regular): metric and pseudometric spaces, locally compact regular spaces, manifolds, topological groups, Lie groups, paracompact regular spaces, normal regular spaces, etc. Besides, semigroup actions include semigroups of operators, multi-time dynamical systems, polisystems, control systems, Lie group actions, Ellis actions, etc.. Then studying global attractors in this general setting contributes to the branch of mathematical analysis and topological dynamics.