In this paper, we study d-primitive words and D(1)-concatenated words. First we show that neither D(1), the set of all d-primitive words, nor D(1)D(1), the set of all D(1)-concatenated words, is regular. Next we show that for u, v, w ∈Σ+ with |u|=|w|, uvw ∈ D(1) if and only if uv+w ⊆ D(1). It is also shown that every d-primitive word, with the length of two or more, is D(1)-concatenated.

We consider the following three statements for a code L. (P1) For every n 2, both wn Ln and wn+1 Ln+1 imply w L. (P2) For every n 2, if wn Ln, then w L. (P3) For every m, k 2 with m k, wk Lm implies w L. First we show that for every code L, P1 holds. Next we show that for every infix code L, P2 holds, and that a code L is an infix code iff P2 holds and L is a weakly infix code. Last we show that for every strongly infix code L, P3 holds, and that a code L is a strongly infix code iff P3 holds and L is a hyper infix code.

Let Q be the set of all primitive words over a finite alphabet having at least two letters. In this paper, we study the language D(1) of all non-overlapping (d-primitive) words, which is a proper subset of Q. We show that D(1) is a context-sensitive langauage but not a deterministic context-free language. Further it is shown that [D(1)]n is not regular for n ≥ 1.

We consider syntactic congruences of some codes. As a main result, for an infix code L, it is proved that the following (i) and (ii) are equivalent and that (iii) implies (i), where PL is the syntactic congruence of L. (i) L is a PL2-class. (ii) Lm is a PLk-class, for given two integers m and k with 1 m k. (iii)L* is a PL*-class. Next we show that every (i), (ii) and (iii) holds for a strongly infix code L. Moreover we consider properties of syntactic conguences of a residue W(L) for a strongly outfix code L.

In this paper, we give some resuts on primitive words, square-free words and disjunctive languages. We show that for a word u ∈Σ+, every element of λ(cp(u)) is d-primitive iff it is square-free, where cp(u) is the set of all cyclic-permutations of u, and λ(cp(u)) is the set of all primitive roots of it. Next we show that pmqn is a primitive word for every n, m ≥1 and primitive words p, q, under the condition that |p| = |q| and (m, n) ≠ (1, 1). We also give a condition of disjunctiveness for a language.

In this paper, we introduce a syntactically embedded (s-embedded) language, and consider its principal congruence. The following three results are proved, where PL is the principal congruence of a language L, and W(L) is the residual of L. (1) For a language K, s-embedded in M, K is equal to a PM class. (2) For a language K, s-embedded in an infix language M, K is equal to a PW(M) class. (3) For a nonempty s-embedded language L, if L is double-unitary, then L is equal to a PW(M) class. From the above results, we can obtain those for principal congruence of some codes. For example, Ln is equal to a PLn+1 class for an inter code L of index n.

In this paper we study the ploblem whether the language D(1) of all d-primitive words can be generated by a contextual grammar. It is proved that D(1) can be generated neither by an external contextual grammar nor by an internal contextual grammar, and that it can be generated by a total contextual grammar with choice.

In this note, we present some results about parses of codes. First we present a sufficient condition of a bifix code to have the bounded indicator. Next we consider a proper parse, introduced notion. We prove that for a strongly infix code, the number of proper parses is at most three under some condition. We also prove that if a code X has a unique proper parse for each word under the same condition, then X is a strongly infix code.

We introduce a strongly infix code. A code X is a strongly infix code if X is an infix code and any catenation of two words in X has no proper factor in X, which is neither a left factor nor a right factor. We show that the class of strongly infix codes is closed under composition, and, as the dual result, that the property to be strongly infix is inherited by a component of a decomposition.

In this paper, we study partial words in relation with pcodes, compatibility, and containment. First, we introduce C⊂(L), the set of all partial words contained by elements of L, and C⊃(L), the set of all partial words containing elements of L, for a set L of partial words. We discuss the relation between C(L), the set of all partial words compatible with elements of the set L, C⊂(L), and C⊃(L). Next, we consider the condition for C(L), C⊂(L), and C⊃(L) to be a pcode when L is a pcode. Furthermore, we introduce some classes of pcodes. An infix pcode and a comma-free pcode are defined, and the inclusion relation among these classes is established.