Sets with two associative operations

arXiv: Group Theory, 2017

In this paper we introduce novel views of monoids and groups. More specifically, for a given set $S$, let $S^{S\times S}$ be the set of binary operations on $S$. We equip $S^{S\times S}$ with canonical binary operations induced by the elements of $S$. Let $S^{S\times S}_{mn}$ (respectively, $S^{S\times S}_{gr}$) be the set of binary operations that make $S$ monoids (respectively, groups). Then we have the following "duality": for each $z\in S^{S\times S}_{mn}$ a certain subset of $S^{S\times S}$, denoted by $S^*_z$, is a monoid with a canonical binary operation and is isomorphic to $(S,z)$. If $z\in S^{S\times S}_{gr}$, then $S^{S\times S}_{gr}$ can be partitioned into copies of $S^*_z$. We also give a new characterization of group binary operations which distinguishes them from the other binary operations. These results give us new insights into monoids and groups, and will provide new tools and directions in studying these objects.

2020

We study a semigroup which represents a semigroup of sets of Boolean functions on a finite set using the concept of transformations with restricted range. For this semigroup, we determine the algebraic structure. In particular, we characterize the (left, right, and two-sided) ideals and the Green’s relations. Moreover, for each of the Green’s relations, we provide the greatest included congruence. MSC: 20M14; 20M18; 20M20

Bulletin mathématiques de la Société des sciences mathématiques de Roumanie

The notion of a semi-commutativity set for word mappings was defined in as an abstraction of a problem in cryptography. The notion is of special interest in case the mappings are morphisms. Then rather surprising constructions become possible. We investigate such constructions, paying special attention to exceptional values of inverse mappings. Some of our results bear a close relation to certain important issues in the theory of formal languages.

Journal of Computer and System Sciences, 1967

Information and Computation, 1987

Journal of Algebra, 2017

Tridendriform algebras are a type of associative algebras, introduced independently by F. Chapoton and by J.-L. Loday and the third author, in order to describe operads related to the Stasheff polytopes. The vector space ST spanned by the faces of permutohedra has a natural structure of tridendriform bialgebra, we prove that it is free as a tridendriform algebra and exhibit a basis. Our result implies that the subspace of primitive elements of the coalgebra ST, equipped with the coboundary map of permutohedra, is a free cacti algebra.

Journal of Mathematics and Computer Science, 2015

In this paper, suitable operations are defined on the class of partitions of a language which give rise to certain monoids and semigroups. In particular, certain algebraic structures of a language defined over a string are described.

2006

Throughout this paper, a semigroup S will denote a torsion free grading monoid, and it is a non-zero semigroup with 0. The operation is written additively. The aim of this paper is to study semigroup version of an integral domain ([1],[3],[4] and [티).

2022

The purpose of this article is to introduce \textit{the geometry of operations}. Let $\psi$ be a bijection from a nonempty set $G$ into another set $G^{'}$, we prove that for any fixed operation on $G$, there exists a \textit{unique} operation on $G^{'}$, where $\psi$ is an \textit{isomorphism}. This finding allow us to show that we can make the set of natural numbers a \textit{commutative field}. Therefore we establish that the symmetric group $\mathcal{S}_{G}$ of a given nonempty set $G$ \textit{acts} on the set of all operations on $G$. Then we generalize the notions of $t$-norms, $t$-conorms and we give the equivalent of the $t$-conorm for \textit{the triangle function}.