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Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... 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. ..."
Abstract

Cited by 22 (3 self)
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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.
Finite Basis Problems and Results For Quasivarieties
, 2004
"... Let K be a finite collection of finite algebras of finite signature such that SP(K) has meet semidistributive congruence lattices. We prove that there exists a finite collection K1 of finite algebras of the same signature, K1 ⊇ K, such that SP(K1) is finitely axiomatizable. We show also that if HS ..."
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Cited by 4 (2 self)
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Let K be a finite collection of finite algebras of finite signature such that SP(K) has meet semidistributive congruence lattices. We prove that there exists a finite collection K1 of finite algebras of the same signature, K1 ⊇ K, such that SP(K1) is finitely axiomatizable. We show also that if HS(K) ⊆ SP(K), then SP(K) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates.
Interpreting graph colorability in finite semigroups
 MR MR2217645 (2006m:20081
"... We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3colorabil ..."
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Cited by 3 (0 self)
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We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3colorability problem.
Quasiidentities of Finite Semigroups and Symbolic Dynamics
, 1993
"... 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 ..."
Abstract

Cited by 2 (0 self)
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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
Finitely based, finite sets of words
 Internat. J. Algebra Comput
"... 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 ..."
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Cited by 2 (1 self)
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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
EQUATIONAL THEORIES OF SEMIGROUPS WITH ENRICHED SIGNATURE
, 902
"... Abstract. We present sufficient conditions for a unary semigroup variety to have no finite basis for its equational theory. In particular, we exhibit a 6element involutory semigroup which is inherently nonfinitely based as a unary semigroup. As applications we get several naturally arising unary s ..."
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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 6element involutory semigroup which is inherently nonfinitely 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 × 2matrices endowed with MoorePenrose inversion; the semigroup of all n × nmatrices (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
FINITE AXIOMATIZABILITY OF CONGRUENCE RICH VARIETIES
"... In this paper we introduce the notion of a congruence rich variety of algebras, and investigate which locally finite subvarieties of such a variety are relatively finitely based. We apply the results obtained to investigate the finite axiomatizability of an interesting variety generated by a particu ..."
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In this paper we introduce the notion of a congruence rich variety of algebras, and investigate which locally finite subvarieties of such a variety are relatively finitely based. We apply the results obtained to investigate the finite axiomatizability of an interesting variety generated by a particular fiveelement directoid.