Growth problems for avoidable words

The Unambiguity of Segmented Morphisms

“…This theorem provides a useful corollary for a class of patterns first described by Baker et al [1]. We call a pattern α ∈ N + locked if and only if |H(α)| = 1 and thus…”

“…In fact, these regularities can be described by the equivalence classes L ∼ i and R ∼ i on V (α) introduced by Freydenberger et al [5] as fundamental tools to construct tailor-made unambiguous morphisms σ su α . In the present paper, we describe these equivalence classes using an equivalent but simpler definition that is based on the adjacency graph of a pattern, a construction that has first been employed by Baker et al [1] to simplify the Bean-Ehrenfeucht-McNulty-Zimin characterisation of avoidable words, cf. Cassaigne [2].…”

Relatively Free Semigroups of Intermediate Growth

“…Besides Adian's paper 1 we want to mention the results of w x w x Frandendurg 5 and Brinkhuis 6 who independently proved that in the variety defined by the identity X 2 s 0 the 3-generated relatively free semigroup has exponential growth. A very nontrivial example of a relatively free semigroup of polynomial growth in the variety defined by the w x identity whose right side is a zero was given by Baker et al 2 .…”

“…where V is an integer word and 1 F h W F 3. We say that V is a 2 q-subword of W. Clearly, W begins with a letter c. 2 Let V be an arbitrary word over alphabet A that equals V in the semigroup F F . First note that W is shortlex reduced relative to F F and that 3 3 W contains the subwords ba and cb.…”

Combinatorics on Words – A Tutorial

“…This is a generalization of the notion of square-freeness and k-power-freeness. Interestingly, this notion has a rather deep relation to universal algebra, see [BMT89]. We consider two alphabets, the first denoted by A as usual, and the second one, by E, called the pattern alphabet.…”