Free profinite R-trivial, locally idempotent and locally commutative semigroups

Abstract

This paper is concerned with the structure of implicit operations on R intersection with LJ1, the pseudovariety of all R-trivial, locally idempotent and locally commutative semigroups. We give a unique factorization statement, in terms of component projections and idempotent elements, for the implicit operations on R intersection with LJ1. As an application we give a combinatorial description of the languages that are both R-trivial and locally testable. A similar study is conducted for the pseudovariety DA intersection with LJ1 of locally idempotent and locally commutative semigroups in which each regular D-class is a rectangular band.