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18
Tiling Semigroups
 11th ICALP, Lecture Notes in Computer Science 199
, 1999
"... It has recently been shown how to construct an inverse semigroup from any tiling: a construction having applications in Ktheoretical gaplabelling. In this paper, we provide the categorical basis for this construction in terms of an appropriate group acting partially and without xed points on an ..."
Abstract

Cited by 36 (10 self)
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It has recently been shown how to construct an inverse semigroup from any tiling: a construction having applications in Ktheoretical gaplabelling. In this paper, we provide the categorical basis for this construction in terms of an appropriate group acting partially and without xed points on an inverse category associated with the tiling. 1 Introduction In [6] and [7], the rst author showed how to construct an inverse semigroup from any tiling of Euclidean space; we call such semigroups tiling semigroups. This work was motivated by questions in solidstate physics, particularly by those concerning quasicrystals. Our motivation here is to understand the mathematical nature of tiling semigroups. We show that tiling semigroups can best be understood in terms of `partial actions of groups'. Such partial group actions were introduced by Exel [2] and their theory further developed in [11]. The fact that the group acts only partially is accounted for by the fact that the tilings of ...
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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Cited by 23 (11 self)
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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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Cited by 20 (2 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.
Algorithmic problems in groups, semigroups and inverse semigroups
 Semigroups, Formal Languages and Groups
, 1995
"... ..."
Inverse Automata And Profinite Topologies On A Free Group
 J. Pure and Applied Algebra
, 1999
"... This paper gives an elementary, selfcontained proof that a finite product of finitely generated subgroups of a free group is closed in the profinite topology. The proof uses inverse automata (immersions) and inverse monoid theory. Generalizations are given to other topologies. In particular, we obt ..."
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Cited by 7 (3 self)
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This paper gives an elementary, selfcontained proof that a finite product of finitely generated subgroups of a free group is closed in the profinite topology. The proof uses inverse automata (immersions) and inverse monoid theory. Generalizations are given to other topologies. In particular, we obtain the new result that for arborescent pseudovarieties, the product of two closed finitely generated subgroups is again closed. An application to monoid theory is given.
The word and geodesic problem in free solvable groups
"... Abstract. We study the computational complexity of the Word Problem (WP) in free solvable groups Sr,d, wherer ≥ 2 is the rank and d ≥ 2isthe solvability class of the group. It is known that the Magnus embedding of Sr,d into matrices provides a polynomial time decision algorithm for WP in a fixed gro ..."
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Cited by 6 (0 self)
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Abstract. We study the computational complexity of the Word Problem (WP) in free solvable groups Sr,d, wherer ≥ 2 is the rank and d ≥ 2isthe solvability class of the group. It is known that the Magnus embedding of Sr,d into matrices provides a polynomial time decision algorithm for WP in a fixed group Sr,d. Unfortunately, the degree of the polynomial grows together with d, so the uniform algorithm is not polynomial in d. In this paper we show that WP has time complexity O(rn log 2 n)inSr,2, andO(n 3 rd) inSr,d for d ≥ 3. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in Sr,2 is NPcomplete. We prove also that one can compute Fox derivatives of elements from Sr,d in time O(n 3 rd); in particular, one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as on relatively new geometric ideas; in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.
Combinatorial Group Theory, Inverse Monoids, Automata, And Global Semigroup Theory
 Geometric and Combinatorial Methods in Group Theory and Semigroup Theory
, 2000
"... This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = hAjRi be a group presentation and BA;R its standard 2complex. Suppose X is a 2complex with a morphism to BA;R which restricts to an immersion on the 1skeleton. Then we ..."
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Cited by 6 (2 self)
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This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = hAjRi be a group presentation and BA;R its standard 2complex. Suppose X is a 2complex with a morphism to BA;R which restricts to an immersion on the 1skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A;R) to the presentation hAjRi with the property that pointed subgraphs of covers of BA;R are classi ed by closed inverse submonoids of M(A;R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to nongeodesic combings. We also obtain a formal language...
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 4 (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.
Partially commutative inverse monoids
 PROCEEDINGS OF THE 31TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS 2006), BRATISLAVE (SLOVAKIA), NUMBER 4162 IN LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algo ..."
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Cited by 2 (2 self)
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Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NPcompleteness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.