On extended associative semigroups

TL;DR

This paper explores the algebraic structure of extended associative semigroups (EAS) and their commutative subclass (CEDS), providing classifications, examples, and connections to bialgebras and Hopf algebras.

Contribution

It introduces linear EAS, classifies finite CEDS, and establishes links between these structures and well-known algebraic objects like bialgebras.

Findings

Classified EAS of size two.

Constructed examples from semigroups and groups.

Linked linear EAS to braid equations and Hopf algebras.

Abstract

We study extended associative semigroups (briefly, EAS), an algebraic structure used to define generalizations of the operad of associative algebras, and the subclass of commutative extended diassociative semigroups (briefly, CEDS), which are used to define generalizations of the operad of pre-Lie algebras. We give families of examples based on semigroups or on groups, as well as a classification of EAS of cardinality two. We then define linear extended associative semigroups as linear maps satisfying a variation of the braid equation. We explore links between linear EAS and bialgebras and Hopf algebras. We also study the structure of nondegenerate finite CEDS and show that they are obtained by semidirect and direct products involving two groups.