# The identities of the free product of a pair of two-element monoids

## TL;DR

This paper proves that the identities of the free product of any pair of two-element monoids cannot be finitely axiomatized, highlighting complexity in their algebraic structure.

## Contribution

It establishes that the identities of free products of two-element monoids have no finite basis, a novel result in algebraic theory.

## Key findings

- No finite basis for identities of free products of two-element monoids
- Identifies complexity in algebraic identities of simple monoids
- Extends understanding of free product structures

## Abstract

Up to isomorphism, there exist two non-isomorphic two-element monoids. We show that the identities of the free product of every pair of such monoids admit no finite basis.