Different and distinct notions of regularity for modules exist in the literature. When these are restricted to commutative rings, they all coincide with well-known von-Neumann rings. We give new characterizations terms both (weakly-)morphic reduced modules. Furthermore, module theoretic settings established where general turn out be indistinguishable.

Circular D0L-systems are those with finite synchronizing delay. We introduce a tool called graph of overhangs which can be used to find the minimal value of synchronizing delay of a given D0L-system. By studying the graphs of overhangs, a general upper bound on the minimal value of a synchronizing delay of a circular D0L-system with a binary uniform morphism is given.

The aim of this paper is to prove that every polynomial function that maps the natural integers to the positive integers is the growth function of some D0L-system.

In this paper we prove the following result. Let s be an infinite word on a finite alphabet, and N ≥ 0 be an integer. Suppose that all left special factors of s longer than N are prefixes of s, and that s has at most one right special factor of each length greater than N . Then s is a morphic image, under an injective morphism, of a suitable standard Arnoux-Rauzy word.

We describe a simple algorithm which, for a given D0L-system, returns all words v such that v is a factor of the language of the system for all k. This algorithm can be used to decide whether a D0L-system is repetitive.

An infinite permutation can be defined as a linear ordering of the set of natural numbers. In particular, an infinite permutation can be constructed with an aperiodic infinite word over {0, . . . , q− 1} as the lexicographic order of the shifts of the word. In this paper, we discuss the question if an infinite permutation defined this way admits a canonical representative, that is, can be defin...