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Profinite Categories, Implicit Operations and Pseudovarieties of Categories
 J. Pure Appl. Algebra
, 1996
"... The last decade has seen two methodological advances of particular direct import for the theory of finite monoids and indirect import for that of rational languages. The first has been the use of categories (considered as "algebras over graphs") as a framework in which to study monoids ..."
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The last decade has seen two methodological advances of particular direct import for the theory of finite monoids and indirect import for that of rational languages. The first has been the use of categories (considered as "algebras over graphs") as a framework in which to study monoids and their homomorphisms, the second has been the use of implicit operations to study pseudovarieties of monoids. Still more recent work has emphasized the role of profiniteness in finite monoid theory. This paper fuses these three topics by means of a general study of profinite categories, with applications to Cvarieties (pseudovarieties of categories) in general, to those Cvarieties arising from Mvarieties (pseudovarieties of monoids) in particular, to implicit operations on categories and to recognizable languages over graphs. Before discussing the contents of the paper in detail, we provide some brief background on each of the three topics mentioned above. In an expository paper [21], B....
Free Profinite Semigroups Over Semidirect Products
, 1995
"... We give a general description of the free profinite semigroups over a semidirect product of pseudovarieties. More precisely,\Omega A (V W) is described as a closed subsemigroup of a profinite semidirect product of the form\Omega \Omega AW\ThetaA V \Omega AW. As a particular case, the free pro ..."
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We give a general description of the free profinite semigroups over a semidirect product of pseudovarieties. More precisely,\Omega A (V W) is described as a closed subsemigroup of a profinite semidirect product of the form\Omega \Omega AW\ThetaA V \Omega AW. As a particular case, the free profinite semigroup over J 1 V is described in terms of the geometry of the Cayley graph of the free profinite semigroup over V (here J 1 is the pseudovariety of semilattice monoids). Applications are given to the calculations of the free profinite semigroup over J 1 Nil and of the free profinite monoid over J 1 G (where Nil is the pseudovariety of finite nilpotent semigroups and G is the pseudovariety of finite groups). The latter free profinite monoid is compared with the free profinite inverse monoid, which is also calculated here.
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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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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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 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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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.
Semidirect Products With the Pseudovariety of All Finite Groups
, 2000
"... This is a survey of recent results related to semidirect products of an arbitrary pseudovariety with the pseudovariety of all nite groups. The main avour is the establishment of links between various operators on pseudovarieties, some obviously computable, others known not to be so. This not only le ..."
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This is a survey of recent results related to semidirect products of an arbitrary pseudovariety with the pseudovariety of all nite groups. The main avour is the establishment of links between various operators on pseudovarieties, some obviously computable, others known not to be so. This not only leads to decidability results but does so in a sort of uniform way which has a structural tint even though the arguments are mostly syntactical.
UNIVERSIDAD NACIONAL DE EDUCACiÓN A DISTANCIA DI~EQTACIONE~
"... fUNDAMENTALE~ Virtually free prop groups P. A. Zalesskii* In this paper We overview recent results on virtuaily free prop groups, i.e., prop groups having open free subgroups. We describe also applications of these results to the study of automorphisms of finite orderof free prop groups. The fir ..."
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fUNDAMENTALE~ Virtually free prop groups P. A. Zalesskii* In this paper We overview recent results on virtuaily free prop groups, i.e., prop groups having open free subgroups. We describe also applications of these results to the study of automorphisms of finite orderof free prop groups. The first result on virtually free prop groups was obtained by Serrein a seminal paper [Serre 65]: Theorem 1 • [ Serre 1965] Let G be a propgroup and F an open free prop subgroup of G. Jf G is torsion free, then G is also a free prop group. This led Serre to conjecture that a (discrete) torsion free virtuaily free group is free. With this conjecture he essential1y initiated the study of groups of cohomological dimension 1 in the discrete case. He showed that any such group has cohomological dimension at most 1 over Z. Subsequently, [Stallings 1968] proved thatevery finitely generated group of cohomological dimension 1 over Z is free,and then [Swan 1969] eliminated the assumption