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)' s. 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 D i (1) and D( i) concerning the basic properties of decompositions and catenations of words. We give characterizations of words in D 2(1)∩ D(1) and D(2). We also show that sets D i (1)⧹ D( j) and D( j)⧹ D i (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.