d-minimal languages Original Research Article

A word v is said to be a proper d-factor of a word u if v ≠ u and u = vx = yv for some words x,y. The family of words which have i distinct proper d-factors is denoted by D(i). According to the number of distinct proper d-factors of words, the free semigroup X+ generated by X can be expressed as the disjoint union of D(i)ʹs. Words in D(1) are called d-minimal words. d-Minimal words are often called non-overlapping words, dipolar words or unborded words. In this paper, we study the relationship between Di(1) and D(i) concerning the basic properties of decompositions and catenations of words. We give characterizations of words in D2(1)∩D(1) and D(2). We also show that sets Di(1)⧹D(j) and D(j)⧹Di(1) are disjunctive. It is known that every disjunctive language is dense but not regular. We obtain the results that X+D(1) and X+D(2) are regular but X+D(i) is disjunctive for every i ⩾ 4. Served as an example of disjunctive d-minimal context-free languages, a disjunctive d-minimal context-free language is constructed. Moreover, we show that the well-known Dyck language is a free semigroup generated by a d-minimal bifix code. The languages of which the catenations consist of d-minimal words are studied in this paper too. That is, some properties of d-minimality-annihilators of languages are investigated.