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Tameness of pseudovariety joins involving R
, 2004
"... In this paper, we establish several decidability results for pseudovariety joins of the form V ∨ W, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all Jtrivial (resp. Rtrivial) semigroups. In particular, we show that the pseudovariety V ∨ ..."
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In this paper, we establish several decidability results for pseudovariety joins of the form V ∨ W, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all Jtrivial (resp. Rtrivial) semigroups. In particular, we show that the pseudovariety V ∨ W is (completely) κtame when V is a subpseudovariety of J and W is (completely) κtame. Moreover, if W is a κtame pseudovariety which satisfies the pseudoidentity x1 · · · xry ω+1 zt ω = x1 · · · xryzt ω, then we prove that R ∨ W is also κtame. In particular the joins R ∨ Ab, R ∨ G, R ∨ OCR, and R ∨ CR are decidable.
THE EQUATIONAL THEORY OF ωTERMS FOR FINITE Rtrivial Semigroups
, 2005
"... A new topological representation for free profinite Rtrivial semigroups in terms of spaces of vertexlabeled complete binary trees is obtained. Such a tree may be naturally folded into a finite automaton if and only if the element it represents is an ωterm. The variety of ωsemigroups generated by ..."
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Cited by 3 (3 self)
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A new topological representation for free profinite Rtrivial semigroups in terms of spaces of vertexlabeled complete binary trees is obtained. Such a tree may be naturally folded into a finite automaton if and only if the element it represents is an ωterm. The variety of ωsemigroups generated by all finite Rtrivial semigroups, with the usual interpretation of the ωpower, is then studied. A simple infinite basis of identities is exhibited and a lineartime solution of the word problem for relatively free ωsemigroups is presented. This work is also compared with recent work of Bloom and Choffrut on transfinite words.
Complete reducibility of systems of equations with respect to R
"... It is shown that the pseudovariety R of all finite Rtrivial semigroups is completely reducible with respect to the canonical signature. Informally, if the variables in a finite system of equations with rational constraints may be evaluated by pseudowords so that each value belongs to the closure of ..."
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Cited by 3 (3 self)
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It is shown that the pseudovariety R of all finite Rtrivial semigroups is completely reducible with respect to the canonical signature. Informally, if the variables in a finite system of equations with rational constraints may be evaluated by pseudowords so that each value belongs to the closure of the corresponding rational constraint and the system is verified in R, then there is some such evaluation which is “regular”, that is one in which, additionally, the pseudowords only involve multiplications and ωpowers.
Profinite semigroups and applications
 IN STRUCTURAL THEORY OF AUTOMATA, SEMIGROUPS, AND UNIVERSAL ALGEBRA
, 2003
"... Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which ..."
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Cited by 2 (1 self)
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Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which for pseudovarieties play the role of free algebras in the theory of varieties. Combinatorial problems on rational languages translate into algebraictopological problems on profinite semigroups. The aim of these lecture notes is to introduce these topics and to show how they intervene in the most recent developments in the area.
COMPLETE REDUCIBILITY OF PSEUDOVARIETIES
"... Abstract. The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results establishing this and the stronger property of complete reducibility ..."
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Abstract. The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results establishing this and the stronger property of complete reducibility for specific pseudovarieties. 1.
Some Key Problems on Finite Semigroups
, 2001
"... The paper starts from a historical perspective of the theory of finite semigroups, namely of its most developed part, the theory of pseudovarieties and its connections with the theories of finite automata and rational languages. It goes on to review some of the most important developments and to pre ..."
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The paper starts from a historical perspective of the theory of finite semigroups, namely of its most developed part, the theory of pseudovarieties and its connections with the theories of finite automata and rational languages. It goes on to review some of the most important developments and to present some of the key problems that have guided the theory for the past 35 years. Several open problems that may in turn play a significant role in years to come are also discussed.
The word problem for free profinite semigroups over the pseudovariety CR and kappareducibility
, 2000
"... Necessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. ..."
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Necessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. A strengthened version of Ash's inevitability theorem (reducibility of the pseudovariety G of all finite groups) is proposed as an open problem and it is shown that, if this stronger version holds, then CR is also reducible and, therefore, hyperdecidable. 1 Introduction Word problems (or rather the decidability thereof) have long played an important role in various branches of Mathematics. In some contexts a property can be associated with a decision problem by which the problem can be reduced in the sense that if it has a solution in an enlarged universe then it has a solution in the restricted universe. The first author and Steinberg [7] (see also [8]) have shown that two such properti...
Tameness Of Pseudovarieties Of Semigroups
, 2000
"... Tameness is a property introduced in 1997 by Steinberg and the author in connection with the KrohnRhodes complexity problem in order to establish the decidability of semidirect products of pseudovarieties of semigroups. Since then a number of works have been dedicated to proving tameness of pseudov ..."
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Tameness is a property introduced in 1997 by Steinberg and the author in connection with the KrohnRhodes complexity problem in order to establish the decidability of semidirect products of pseudovarieties of semigroups. Since then a number of works have been dedicated to proving tameness of pseudovarieties. This paper is a survey of work in this area.