Characterizations of biselective operations

TL;DR

This paper characterizes $(i,j)$-selective binary operations on sets, exploring their properties and subclasses with algebraic features like associativity and bisymmetry.

Contribution

It provides a comprehensive characterization of $(i,j)$-selective operations and examines subclasses with additional algebraic properties.

Findings

Characterizations of $(i,j)$-selective operations

Identification of subclasses with associativity or bisymmetry

Structural insights into the behavior of these operations

Abstract

Let $X$ be a nonempty set and let $i,j \in \{1,2,3,4\}$. We say that a binary operation $F:X^2\to X$ is $(i,j)$-selective if $$ F(F(x_1,x_2),F(x_3,x_4))~=~F(x_i,x_j), $$ for all $x_1,x_2,x_3,x_4\in X$. In this paper we provide characterizations of the class of $(i,j)$-selective operations. We also investigate some subclasses by adding algebraic properties such as associativity or bisymmetry.