Bisymmetric and quasitrivial operations: characterizations and enumerations

TL;DR

This paper characterizes and enumerates bisymmetric and quasitrivial binary operations on sets, including subclasses that are order-preserving, providing explicit class sizes for finite sets.

Contribution

It offers new characterizations and explicit enumeration formulas for bisymmetric, quasitrivial, and order-preserving binary operations on finite sets.

Findings

Characterizations of bisymmetric and quasitrivial operations

Explicit enumeration formulas for finite sets

Analysis of subclasses including order-preserving operations

Abstract

We investigate the class of bisymmetric and quasitrivial binary operations on a given set $X$ and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations. We also determine explicitly the sizes of these classes when the set $X$ is finite.