In this note we classify sequences according to whether they are morphic, pure morphic, uniform morphic, pure uniform morphic, primitive morphic, or pure primitive morphic, and for each possibility we either give an example or prove that no example is possible.

In this paper we generalize the notion of an ι-symmetric word, from an antimorphic involution, to an arbitrary involution ι as follows: a nonempty word w is said to be ι-symmetric if w = αβ = ι(βα) for some words α, β . We propose the notion of ιtwin-roots (x, y) of an ι-symmetric wordw. We prove the existence and uniqueness of the ι-twin-roots of an ι-symmetric word, and show that the left fac...

Let R be a ring. An element a in R is called left morphic (Nicholson and Sánchez Campos, 2004a) if l a R/Ra, where l a denotes the left annihilator of a in R. The ring itself is called a left morphic ring if every element is left morphic. Left morphic rings were first introduced by Nicholson and Sánchez Campos (2004a) and were discussed in great detail there and in Nicholson and Sánchez Campos ...

We study structure of pure morphic and morphic sequences and prove the following result: the subword complexity of arbitrary morphic sequence is either Θ(n) for some k ∈ N, or is

Multidimensional contractions of irreducible representations of the Cayley–Klein orthogonal algebras in Gel'fand–Zetlin basis are considered. Contracted over different parameters, algebras can turn out to be iso-morphic. In this case method of transitions describes the same reducible representations in different basises, say discrete and continuons ones.

In the paper we mainly deal with two well-known types of in nite words: morphic and uniformly recurrent (=almost periodic). We discuss the problem of nding criterion of uniform recurrence for morphic sequences and give e ective polynomial-time such criterion in two particular cases: pure morphic sequences and automatic sequences. We also prove that factor complexity of arbitrary uniformly recur...

We prove that the uniform recurrence of morphic sequences is decidable. For this we show that the number of derived sequences of uniformly recurrent morphic sequences is bounded. As a corollary we obtain that uniformly recurrent morphic sequences are primitive substitutive sequences.

This note is a response to one of the problems posed by Kwa´sniewski in [1, 2], see also [3] i.e. GCD-morphic Problem III. We show that any GCD-morphic sequence F is at the point product of primary GCD-morphic sequences and any GCD-morphic sequence is encoded by natural number valued sequence satisfying condition (C1). The problem of general importance-for example in number theory was formulate...