Quasitrivial semigroups: characterizations and enumerations

TL;DR

This paper explores quasitrivial semigroups, offering characterizations of specific subclasses, and explicitly enumerates their sizes on finite sets, leading to new integer sequences.

Contribution

It provides new characterizations and explicit enumerations of quasitrivial semigroups and their subclasses, including commutative and order-preserving cases.

Findings

Characterizations of quasitrivial, commutative, and order-preserving semigroups

Explicit enumeration formulas for finite set cases

Discovery of new integer sequences from enumerations

Abstract

We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order-preserving semigroups. We also determine explicitly the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences.