Expanded subalphabets in the theories of languages and semigroups†

Abstract

A new type of code, called an expanded subalphabet, is introduced. It is shown that the following four conditions on a subset L of the free monoid A ∗ over a finite or infinite alphabet A are equivalent: (1) L is the submonoid generated by an expanded subalphabet; (2) L is a retract; (3) L is the fixed language of an endomorphism; (4) L is the stationary language of an endomorphism. Expanded subalphabets are used as a tool for the investigation of fixed languages (=retracts). Special results for the case of a finite alphabet are given and the relationship with the theory of L-systems is indicated.