The Join of the Varieties of R-trivial and L-trivial Monoids via Combinatorics on Words

TL;DR

This paper provides a new proof for the effective characterization of the join of R-trivial and L-trivial monoids, using a single identity of omega-terms with three variables, advancing finite semigroup theory.

Contribution

It offers a novel proof of the known characterization, simplifying the understanding of the join of R-trivial and L-trivial monoids in finite semigroup theory.

Findings

New proof of Almeida and Azevedo's characterization

Single identity of omega-terms with three variables

Decidability of the join established

Abstract

The join of two varieties is the smallest variety containing both. In finite semigroup theory, the varieties of R-trivial and L-trivial monoids are two of the most prominent classes of finite monoids. Their join is known to be decidable due to a result of Almeida and Azevedo. In this paper, we give a new proof for Almeida and Azevedo's effective characterization of the join of R-trivial and L-trivial monoids. This characterization is a single identity of omega-terms using three variables.