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21
Extending partial automorphisms and the profinite topology on free groups
 Tran. AMS
, 2000
"... Abstract. A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1,p2,...,pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and ..."
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Abstract. A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1,p2,...,pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and automorphisms α1,α2,...,αn of C3 extending the pi. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskiĭ stating that a finite product of finitely generated subgroups is closed for this topology. 1.
Profinite Methods in Semigroup Theory
 Int. J. Algebra Comput
, 2000
"... this paper. The extended bibliography given below shows other important contributions by Azevedo, Costa, Delgado, Pin, Teixeira, Volkov, Weil and Zeitoun. ..."
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Cited by 23 (3 self)
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this paper. The extended bibliography given below shows other important contributions by Azevedo, Costa, Delgado, Pin, Teixeira, Volkov, Weil and Zeitoun.
Finite state automata: A geometric approach
 Trans. Amer. Math. Soc
"... Abstract. Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various probl ..."
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Cited by 22 (13 self)
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Abstract. Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various problems on the overlap between group theory and monoid theory. For instance, we give a geometric algorithm for computing the closure of a rational language in the profinite topology of a free group. We introduce some geometric notions for automata and show that certain important classes of monoids can be described in terms of the geometry of their Cayley graphs. A long standing open question, to which the answer was only known in the simplest of cases (and even then was nontrivial), is whether it is true, for a pseudovariety of groups H, thataJtrivial coextension of a group in H must divide a semidirect product of a Jtrivial monoid and a group in H. We show the answer is affirmative if H is closed under extension, and may be negative otherwise. 1.
The pseudovariety J is hyperdecidable
 THEORETICAL INFORMATICS AND APPLICATIONS 31
, 1997
"... This article defines the notion of hyperdecidability for a class of finite semigroups, which is closely connected to the notion of decidability. It then proves that the pseudovariety J of Jtrivial semigroups is hyperdecidable. 1 ..."
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Cited by 15 (11 self)
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This article defines the notion of hyperdecidability for a class of finite semigroups, which is closely connected to the notion of decidability. It then proves that the pseudovariety J of Jtrivial semigroups is hyperdecidable. 1
On the extension problem for partial permutations
 PROC. AMER. MATH. SOC
, 2003
"... A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no nontrivial group identity and a metabelian pseudovariety Q. For each of these pseudovarietie ..."
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Cited by 14 (2 self)
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A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no nontrivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl∗V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators ∗, ∗∗, m○, ♦, ♦n, andP do not preserve decidability. In addition, several joins, including A ∨ U, are shown to be undecidable.
Abelian Kernels of Some Monoids of Injective Partial Transformations and an Application
, 1999
"... In this paper we compute the abelian kernels of the monoids POI n and POPI n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI n (n 2 N ) ..."
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Cited by 11 (10 self)
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In this paper we compute the abelian kernels of the monoids POI n and POPI n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI n (n 2 N ) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI n (n 2 N ) with the pseudovariety Ab of all finite abelian groups.
Aperiodic pointlikes and beyond
 Internat. J. Algebra Comput
"... Dedicated to the memory of Bret Tilson Abstract. We prove that if π is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are πgroups. In particular, when π is the empty set, we obtain Henckell’s decidability of aperiodic pointlikes. Our ..."
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Cited by 9 (1 self)
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Dedicated to the memory of Bret Tilson Abstract. We prove that if π is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are πgroups. In particular, when π is the empty set, we obtain Henckell’s decidability of aperiodic pointlikes. Our proof, restricted to the case of aperiodic semigroups, is simpler than the original proof. 1.
Complete reducibility of the pseudovariety LSl
 Int. J. Algebra Comput
"... In this paper we prove that the pseudovariety LSl of local semilattices is completely κreducible, where κ is the implicit signature consisting of the multiplication and the ωpower. Informally speaking, given a finite equation system with rational constraints, the existence of a solution by pseudo ..."
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Cited by 6 (2 self)
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In this paper we prove that the pseudovariety LSl of local semilattices is completely κreducible, where κ is the implicit signature consisting of the multiplication and the ωpower. Informally speaking, given a finite equation system with rational constraints, the existence of a solution by pseudowords of the system over LSl implies the existence of a solution by κwords of the system over LSl satisfying the same constraints.
A profinite approach to stable pairs
 Internat. J. Algebra Comput
"... Dedicated to the memory of Bret Tilson Abstract. We give a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids. Stable pairs are also described for the pseudovariety of all finite monoids. 1. ..."
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Cited by 5 (5 self)
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Dedicated to the memory of Bret Tilson Abstract. We give a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids. Stable pairs are also described for the pseudovariety of all finite monoids. 1.
The geometry of profinite graphs with applications to free groups and finite monoids
 TRANS AMER. MATH. SOC
, 2003
"... We initiate the study of the class of profinite graphs Γ defined by the following geometric property: for any two vertices v and w of Γ, there is a (unique) smallest connected profinite subgraph of Γ containing them; such graphs are called treelike. Profinite trees in the sense of Gildenhuys and ..."
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Cited by 5 (1 self)
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We initiate the study of the class of profinite graphs Γ defined by the following geometric property: for any two vertices v and w of Γ, there is a (unique) smallest connected profinite subgraph of Γ containing them; such graphs are called treelike. Profinite trees in the sense of Gildenhuys and Ribes are treelike, but the converse is not true. A profinite group is then said to be dendral if it has a treelike Cayley graph with respect to some generating set; a BassSerre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition. We define a pseudovariety of groups H to be arboreous if all finitely generated free proH groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties H, aproH analog of the Ribes and Zalesskiĭ product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions H to the much studied pseudovariety equation J ∗ H = J m ○ H.