The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.

We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55-element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3-colorability problem.

Let us first recall some basic facts and definitions from universal algebra (see [Malcev], [Cohn]). Avarietyis a class of universal algebras given by identities, i.e. formulas of the type (∀x1,...,xn) u = v

For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W. This monoid is denoted by S(W). It is shown that for every finite set of words W, there are sets of words U ⊃ W and V ⊃ W such that the identities satisfied by S(V) are finitely based and those of S(U) are not finitely based (regardless of the situation for S(W)). The first examples of finitely based (not finitely based) aperiodic finite semigroups whose direct product is not finitely based (finitely based) are presented and it is shown that every monoid of the form S(W) with fewer than 9 elements is finitely based and that there is precisely one not finitely based 9 element example. 1

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Abstract. We present sufficient conditions for a unary semigroup variety to have no finite basis for its equational theory. In particular, we exhibit a 6-element involutory semigroup which is inherently non-finitely based as a unary semigroup. As applications we get several naturally arising unary semigroups without finite identity bases, for example: the semigroup of all complex 2 × 2-matrices endowed with Moore-Penrose inversion; the semigroup of all n × n-matrices (n ≥ 2) endowed with transposition over either a finite field or the Boolean semiring; various partition semigroups endowed with their natural involution, including the full partition semigroup Cn for n ≥ 2, the Brauer semigroup Bn for n ≥ 4 and the annular semigroup An for n ≥ 4, n even or a prime power. We also show that similar techniques apply to the finite basis problem for existence varieties of locally inverse semigroups. Contents