Generalized associative algebras

TL;DR

This paper explores generalized associative algebras parametrized by semigroups, describing their free structures, formal series, and Koszul properties, and investigates operadic morphisms and connections with other algebra types.

Contribution

It introduces new parametrized operads of associative algebras, analyzes their algebraic properties, and links them to existing algebraic frameworks.

Findings

Free algebras are explicitly described.

Operads are Koszul for finite parameter sets.

Connections with diassociative, dendriform, and post-Lie algebras are established.

Abstract

We study diverse parametrized versions of the operad of associative algebra, where the parameter are taken in an associative semigroup $\Omega$ (generalization of matching or family associative algebras) or in its cartesian square (two-parameters associative algebras). We give a description of the free algebras on these operads, study their formal series and prove that they are Koszul when the set of parameters is finite. We also study operadic morphisms between the operad of classical associative algebras and these objects, and links with other types of algebras (diassociative, dendriform, post-Lie...).