Independence of algebras with edge term

TL;DR

This paper characterizes when two finite algebras with an edge term generate independent varieties, providing necessary and sufficient conditions and a polynomial time decision algorithm for finitely generated cases.

Contribution

It establishes criteria for independence of varieties generated by finite algebras with an edge term and introduces an efficient decision algorithm.

Findings

Necessary and sufficient conditions for independence with edge terms

Polynomial time algorithm for finitely generated varieties

Extension of results from Mal'cev to edge terms

Abstract

Two varieties $V, W$ of the same type are independent if there is a binary term $t(x,y)$ such that $V \models t(x,y) \approx x$ and $W \models t(x,y) \approx y$. In this note, we give necessary and sufficient conditions for two finite algebras with a Mal'cev term (or, more generally, with an edge term) to generate independent varieties. In particular we show that the independence of finitely generated varieties with edge term can be decided by a polynomial time algorithm.