Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 092, 34 pages      arXiv:2105.01326      https://doi.org/10.3842/SIGMA.2025.092

On Extended Associative Semigroups

Loïc Foissy
Université Littoral Côte d'Opale, UR 2597 LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, 62100 Calais, France

Received June 12, 2025, in final form October 15, 2025; Published online October 26, 2025

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 non-degenerate finite CEDS and show that they are obtained by semi-direct and direct products involving two groups.

Key words: semigroups; diassociative semigroups; braid equation.

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