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573
Timed Regular Expressions
 Journal of the ACM
, 2001
"... In this paper we define timed regular expressions, a formalism for specifying discrete behaviors augmented with timing information, and prove that its expressive power is equivalent to the timed automata of Alur and Dill. This result is the timed analogue of Kleene Theorem and, similarly to that re ..."
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Cited by 66 (21 self)
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In this paper we define timed regular expressions, a formalism for specifying discrete behaviors augmented with timing information, and prove that its expressive power is equivalent to the timed automata of Alur and Dill. This result is the timed analogue of Kleene Theorem and, similarly to that result, the hard part in the proof is the translation from automata to expressions. This result is extended from finite to infinite (in the sense of B uchi) behaviors. In addition to these fundamental results, we give a clean algebraic framework for two commonlyaccepted formalism for timed behaviors, timeevent sequences and piecewiseconstant signals. 1
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 fixed points on ..."
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Cited by 52 (13 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 fixed points on an inverse category associated with the tiling.
Partial actions of groups and actions of inverse semigroups
 Proc. AMS 126
, 1998
"... Abstract. Given a group G, we construct, in a canonical way, an inverse semigroup S(G) associated to G. The actions of S(G) areshowntobein onetoone correspondence with the partial actions of G, bothinthecaseof actions on a set, and that of actions as operators on a Hilbert space. In other words, G ..."
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Cited by 48 (11 self)
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Abstract. Given a group G, we construct, in a canonical way, an inverse semigroup S(G) associated to G. The actions of S(G) areshowntobein onetoone correspondence with the partial actions of G, bothinthecaseof actions on a set, and that of actions as operators on a Hilbert space. In other words, G and S(G) have the same representation theory. We show that S (G) governs the subsemigroup of all closed linear subspaces of a Ggraded C ∗algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A “partial ” version of the group C ∗algebra of a discrete group is introduced. While the usual group C ∗algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group C ∗algebra of the two commutative groups of order four, namely Z/4Z and Z/2Z ⊕ Z/2Z, are not isomorphic. 1.
Ash's type II theorem, profinite topology and Malcev products Part I
"... This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by t ..."
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Cited by 47 (10 self)
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This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture  also verified by Ash  it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH 1 H 2 \Delta \Delta \Delta Hn , where each H i is a finitely generated subgroup of G. This significantly extends classical results by M. Hall. Final...
INTUITIONISTIC FUZZY INTERIOR IDEALS OF SEMIGROUPS
, 2001
"... We consider the intuitionistic fuzzification of the concept of interior ideals in a semigroup S, and investigate some properties of such ideals. For any homomorphism f from a semigroup S to a semigroup T, if B = (µB,γB) is an intuitionistic fuzzy interior ideal of T, then the preimage f−1(B) = (f− ..."
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Cited by 31 (3 self)
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We consider the intuitionistic fuzzification of the concept of interior ideals in a semigroup S, and investigate some properties of such ideals. For any homomorphism f from a semigroup S to a semigroup T, if B = (µB,γB) is an intuitionistic fuzzy interior ideal of T, then the preimage f−1(B) = (f−1(µB),f−1(γB)) of B under f is an intuitionistic fuzzy interior ideal of S.
C ∗ crossed products by partial actions and actions of inverse semigroups
 J. Australian Math. Soc
, 1997
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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.
Geometric presentations for Thompson’s groups
 J. Pure Appl. Algebra
, 2005
"... Abstract. Starting from the observation that Thompson’s groups F and V are the geometry groups respectively of associativity, and of associativity together with commutativity, we deduce new presentations of these groups. These presentations naturally lead to introducing a new subgroup S • of V and a ..."
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Cited by 22 (6 self)
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Abstract. Starting from the observation that Thompson’s groups F and V are the geometry groups respectively of associativity, and of associativity together with commutativity, we deduce new presentations of these groups. These presentations naturally lead to introducing a new subgroup S • of V and a torsion free extension B • of S•. We prove that S • and B • are the geometry groups of associativity together with the law x(yz) = y(xz), and of associativity together with a twisted version of this law involving selfdistributivity, respectively. Previous work showed that associating to an algebraic law a socalled geometry group that captures some specific geometrical features gives useful information about that law: the approach proved instrumental for studying exotic laws like selfdistributivity x(yz) = (xy)(xz) [7] or x(yz) = (xy)(yz) [8]. In the case of associativity [6], the geometry group turns out to be Thompson’s group F, not a surprise as the connection of the latter with associativity has been known for long time [20]. In this paper, we develop a rather general method for constructing geometry groups and, chiefly, finding presentations for these groups, and we apply this method in the case of associativity—thus finding presentations of F —and of associativity plus commutativity, thus finding new
Hyperdecidable Pseudovarieties and the Calculation of Semidirect Products
 Internat. J. Algebra Comput
"... This note introduces the notion of a hyperdecidable pseudovariety. This notion appears naturally in trying to prove decidability of the membership problem of semidirect products of pseudovarieties of semigroups. It turns out to be a generalization of a notion introduced by C. J. Ash in connection ..."
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Cited by 21 (6 self)
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This note introduces the notion of a hyperdecidable pseudovariety. This notion appears naturally in trying to prove decidability of the membership problem of semidirect products of pseudovarieties of semigroups. It turns out to be a generalization of a notion introduced by C. J. Ash in connection with his proof of the "type II" theorem. The main results in this paper include a formulation of the definition of a hyperdecidable pseudovariety in terms of free profinite semigroups, the equivalence with Ash's property in the group case, the behaviour under the operator g of taking the associated global pseudovariety of semigroupoids, and the decidability of V W in case gV is decidable and has a given finite vertexrank and W is hyperdecidable. A further application of this notion which is given establishes that the join of a hyperdecidable pseudovariety with a locally finite pseudovariety with computable free objects is again hyperdecidable. 1. Introduction A typical problem in...