Word operation closure and primitivity of languages

Abstract

Based on the general operation of words, called bw-operation, the notions of o-primitive words, o-closed languages, o-bases of languages and operation-left-quotient-closed languages are defined and investigated. These notions turn out to be generalizations of the classical notions of primitive words, plus-closed (star-closed) languages, minimal generating sets and deletion-closed languages. Properties of the set of all o-primitive words, the o-bases of non-empty languages, right o-residuals and operation-left-quotient closed languages are studied under the general concept of word operation. Properties of bi-catenation and related languages are discussed as examples and also by their own interests.