On extended associative semigroups
Loïc Foissy
TL;DR
This work develops the theory of extended associative semigroups ($\mathrm{EAS}$) and their commutative diassociative variants, introducing linear versions ($\ell\mathrm{EAS}$) and connecting them to bialgebra and Hopf-algebra theory. It provides a complete classification of $\mathrm{EAS}$ on a 2-element set, and proves a Structure Theorem (Theorem 3.14) showing that finite non-degenerate $\mathrm{CEDS}$ decompose as a product involving abelian-group and group-based $\mathrm{EAS}$ factors, plus an extra $\mathrm{EAS}$ component. The paper then develops linearizations, studies special vectors, left units, and left counits in low dimensions, and constructs two functors from (co)algebraic structures to $\boldsymbol{\ell\mathrm{EAS}}$, enabling translations from bialgebras and Hopf algebras to $\ell\mathrm{EAS}$ and back. Applications to non-degenerate finite CEDS and Hopf algebras of groups illustrate how EAS data encode semidirect products and provide ways to reconstruct bialgebra structures from $\ell\mathrm{EAS}$ data, linking operadic generalizations to classical algebraic frameworks.
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.