Equivalence of regular binoid expressions and regular expressions denoting binoid languages over free binoids

Abstract

A free binoid Σ ∗(∘,•) over a finite alphabet Σ is a free algebra generated by Σ with two independent associative operators, ∘ and •. It has also the same identity λ to both operations. Any element of Σ ∗(∘,•) is denoted uniquely by a sequence of symbols from the extended alphabet E(Σ) = Σ ∪ {∘,•,(,)} , and any subset of a free binoid is called a binoid language. The set of regular binoid expressions are introduced so that all languages denoted by regular binoid expressions are those which contain finite binoid languages, and closed under five operations, ∪,∘-concatenation, •-concatenation, ∘-closure and •-closure. It is shown that for any regular (monoid) expression denoting a binoid language R, there exists a regular binoid expression denoting R. This result together with the main result in a previous paper implies that the class of binoid languages denoted by binoid regular expressions is the same as the class of binoid languages denoted by regular expressions over free binoids.