Transformation Semigroup Perspective on the Magma Monoid
arXiv:2607.13746 · doi:10.1007/s00233-026-10659-x
The paper studies the monoid of all binary operations (the magma monoid) using a transformation semigroup viewpoint, fully describing its principal left and right ideals, idempotent and regular elements, providing combinatorial counts, answering open questions, and correcting a previous error about its center.
Abstract
The monoid of all binary operations was first introduced by H. S. Kim and J. Neggers in 2008. Since then, different aspects and applications of this monoid were studied, while several questions about its semigroup-theoretic properties remain unanswered. We employ a transformation semigroup perspective to fully characterize principal left and right ideals, idempotent and regular elements of this monoid, as well as provide precise combinatorial enumerations of them. This approach gives a general framework for most of the existing results on ideals in the magma monoid. We also answer several open questions posed in the 2023 PhD dissertation of A. Rafieipour. Finally, we correct an error regarding the description of the center of the magma monoid from the 2011 paper of H. F. Fayomi.
To be published in Semigroup Forum