The graph product is an operator mixing direct and free products. It is already known that free products and direct products of automatic monoids are automatic. The main aim of this paper is to prove that graph products of automatic monoids of finite geometric type are still automatic. A similar result for prefix-automatic monoids is established.

Natural algorithms to compute rational expressions for recognizablelanguages, even those which work well in practice, may produce very longexpressions. So, aiming towards the computation of the commutative image of arecognizable language, one should avoid passing through an expressionproduced this way.We modify here one of those algorithms in order to compute directly a semilinear expression for the commutative imageof a recognizable language. We also give a secondmodification of the algorithm which allows the direct computation of theclosure in the profinite topology of the commutative image. As anapplication, we give a modification of an algorithm for computing the Abelian kernel of a finite monoid obtainedby the author in 1998 which is much more efficient in practice.

A morphism f is k-power-free if and only if f(w) is k-power-free whenever w is a k-power-free word.A morphism f is k-power-free up to m if and onlyif f(w) isk-power-free whenever w is a k-power-free word of length at most m.Given an integer k ≥ 2,we prove that a binary morphism is k-power-freeif and only if it is k-power-free up to k 2.This bound becomes linear for primitive morphisms:a binary primitive morphism is k-power-freeif and only if it is k-power-free up to 2k+1

We prove that for each positive integer n, the finite commutative languageEn = c(a 1 a 2...an ) possesses a test set of size at most 5n. Moreover, it is shown that each test set for E n has at least n-1 elements. The result is then generalized to commutative languages L containing a word w such that (i) alph(w) = alph}(L); and (ii) each symbol a ∈ alph}(L) occurs at least twice in w if it occurs at least twice in some word of L: each such L possesses a test set of size 11n, where n = Card(alph(L)). The considerations rest on the analysis of some basic types of word equations.

We investigate the succinctness of several kinds of unary automata by studying their state complexity in accepting the family {Lm } of cyclic languages, where Lm = akm | k ∈ N. In particular, we show that, for any m, the number of states necessary and sufficient for accepting the unary language L m with isolated cut point on one-way probabilistic finite automata is $p_1^{\alpha_1}+ p_2^{\alpha_2} +\cdots +p_s^{\alpha_s}$ , with $p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_s^{\alpha_s}$ being the factorization of m. To prove this result, we give a general state lower bound for accepting unary languages with isolated cut point on the one-way probabilistic model. Moreover, we exhibit one-way quantum finite automata that, for any m, accept L m with isolated cut point and only two states. These results are settled within a survey on unary automata aiming to compare the descriptional power of deterministic, nondeterministic, probabilistic and quantum paradigms.