\(\Gamma \)-Semigroups: A Survey
01 Jan 2016-Vol. 174, pp 225-239
TL;DR: A survey of some works published by different authors on the concept of gamma-semigroups can be found in this article, where the authors present a survey of the generalization of semigroups.
Abstract: The concept of \(\Gamma \)-semigroup is a generalization of semigroup. Let S and \(\Gamma \) be two nonempty sets. S is called \(\Gamma \)-semigroup if there exists a mapping \(S\times \Gamma \times S\longrightarrow S\), written as \((a, \alpha , b)\longrightarrow a\alpha b\), satisfying the identity \( (a\alpha b)\beta c\) \(=\) \(a\alpha (b\beta c) \) for all \(a, b, c\in S\) and \( \alpha , \beta \in \Gamma \). This article is a survey of some works published by different authors on \(\Gamma \)-semigroups.
TL;DR: In this paper, a wide generalization of the Artin theorem using the concepts of hyperidentity and co-identity is proposed, referred to as $$g$$¯¯¯¯ -algebras.
Abstract: The following Artin theorem about alternative linear algebras defined on the commutative, associative ring with unity is well-known: in an alternative linear algebra, if $$(a,b,c)=0$$ , then the subalgebra generated by the elements $$a$$ , $$b$$ , and $$c$$ is associative. In this paper a wide generalization of this classical result is proposed using the concepts of hyperidentity and coidentity. The corresponding universal algebras are referred to as $$g$$ -algebras.
TL;DR: In this article, the authors studied both of these strategies in detail and showed that they are not equivalent in general (although they are in some cases) and proved several interesting properties.
Abstract: In a distributed system, one strategy for achieving mutual exclusion of groups of nodes without communication is to assign to each node a number of votes. Only a group with a majority of votes can execute the critical operations, and mutual exclusion is achieved because at any given time there is at most one such group. A second strategy, which appears to be similar to votes, is to define a priori a set of groups that intersect each other. Any group of nodes that finds itself in this set can perform the restricted operations. In this paper, both of these strategies are studied in detail and it is shown that they are not equivalent in general (although they are in some cases). In doing so, a number of other interesting properties are proved. These properties will be of use to a system designer who is selecting a vote assignment or a set of groups for a specific application.
TL;DR: In this article, the finest inverse semigroup congruence on an orthodox semigroup is shown to have a simple form and conversely, regular semigroups whose finest inverse congruences has this simple form are shown to be orthodox.
Abstract: For brevity the semigroups in the title are called orthodox semigroups. The finest inverse semigroup congruence on an orthodox semigroup is shown to have a simple form and conversely, regular semigroups whose finest inverse congruence has this simple form are shown to be orthodox. Next ideal extensions of orthodox semigroups by orthodox semigroups are shown to be also orthodox, whence a finite semigroup is orthodox if and only if each principal factor is orthodox and completely O-simple or simple. Finally it is determined which completely O-simple semigroups are orthodox.
TL;DR: In this paper, a structure theorem for generalized inverse semigroups is established and some theorems are given to clarify the mutual relations between several conditions on semiigroups, including permutation identities.
Abstract: This paper is concerned with a certain class of regular semigroups. It is well-known that a regular semigroup in which the set of idempotents satisfies commutativity xxx2 = x2xi is an inverse semigroup firstly introduced by V. V. Vagner, and the structure of inverse semigroups was clarified by A. E. Liber, W. D. Munn, G. B. Preston and V. V. Vagner, etc. By a generalized inverse semigroup is meant a regular semigroup in which the set of idempotents satisfies a permutation identity xxxz xn = xPlxP2 xPn (where (pl9 pZf , pn) is a nontrivial permutation of (1,2, —-,ri)). N. Kimura and the author proved in a previous paper that any band B satisfying a permutation identity satisfies normality x^XzX^ = XiXzXzX^. Such a B is called a normal band, and the structure of normal bands was completely determined. In this paper, first a structure theorem for generalized inverse semigroups is established. Next, as a special case, it is proved that a regular semigroup is isomorphic to the spined product (a special subdirect product) of a normal band and a commutative regular semigroup if and only if it satisfies a permutation identity. The problem of classifying all permutation identities on regular semigroups into equivalence classes is also solved. Finally, some theorems are given to clarify the mutual relations between several conditions on semigroups. In particular, it is proved that an inverse semigroup satisfying a permutation identity is necessarily commutative.