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On the Decidability of Iterated Semidirect Products With Applications to Complexity
, 1997
"... The notion of hyperdecidability has been introduced by the first author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidab ..."
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Cited by 17 (9 self)
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The notion of hyperdecidability has been introduced by the first author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidability of some iterated semidirect products. Roughly speaking, the decidability of a semidirect product follows from a mild, commonly verified property of the first factor plus the stronger property for all the other factors. A key role in this study is played by intermediate free semigroups (relatively free objects of expanded type lying between relatively free and relatively free profinite objects). As an application of the main results, the decidability of the KrohnRhodes (group) complexity is shown to follow from nonalgorithmic abstract properties likely to be satisfied by the pseudovariety of all finite aperiodic semigroups, thereby suggesting a new approach to answer (affirmativ...
The Lattice of Pseudovarieties of Idempotent Semigroups and a NonRegular Analogue
"... : We use classical results on the lattice L(B) of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice P s(DA) of subpseudovarieties of DA,  where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. ..."
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Cited by 13 (5 self)
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: We use classical results on the lattice L(B) of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice P s(DA) of subpseudovarieties of DA,  where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. We bring forward a lattice congruence on P s(DA), whose quotient is isomorphic to L(B), and whose classes are intervals with effectively computable least and greatest members. Also we characterize the proidentities satisfied by the members of an important family of subpseudovarieties of DA. Finally, letting V k be the pseudovariety generated by the kgenerated elements of DA (k 1), we use all our results to compute the position of the congruence class of V k in L(B). Introduction The lattice of pseudovarieties of finite semigroups has been the object of much attention over the past few decades, with motivations drawn not only from universal algebra, but also from theoretical computer scie...
Inverse monoids with a natural semilattice ordering
 Proc. London Math. Soc
, 1995
"... condition is equivalent to S both being (von Neumann) regular and having all idempotents commute. The classic example of such a semigroup is the symmetric inverse semigroup Ix of all partial bijections on a set X under the standard composition of partial functions on X. More generally, for any reaso ..."
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Cited by 11 (0 self)
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condition is equivalent to S both being (von Neumann) regular and having all idempotents commute. The classic example of such a semigroup is the symmetric inverse semigroup Ix of all partial bijections on a set X under the standard composition of partial functions on X. More generally, for any reasonably endowed mathematical object Q, the set /n of all partial symmetries of Q (bijections between subobjects of Q respecting all relevant structure) forms an inverse semigroup, the symmetric inverse semigroup of Q. Nearly all such semigroups, however, are more than mere inverse semigroups. Clearly such semigroups possess both an identity 1 and a zero 0, so that one has at least inverse monoids with zero; but of greater consequence is the fact that the natural partial order on the semigroup (given by x ^y if and only if y = yy ~ ] x =xy ~ ] y) usually possesses infima of arbitrary nonempty subsets. When this occurs, every element x has a fixed point idempotent f[x] = 1 AX which is maximal among those idempotents lying beneath x in the natural partial order. (For example, in the symmetric inverse semigroup on a set X, infima are given by intersections,
Hyperdecidability of Pseudovarieties of Orthogroups
 Glasgow Math. J
"... Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product B fl m V of the pseudovariety of bands with a pseudovariety of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that ..."
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Cited by 8 (8 self)
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Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product B fl m V of the pseudovariety of bands with a pseudovariety of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that decidability is preserved in case say only terms (i.e., terms involving only multiplication and the (! \Gamma 1)power) are considered. It is also shown that, if V is a hyperdecidable (respectively reducible) pseudovariety of groups, then so is W. 1 Introduction Motivated by the KrohnRhodes complexity problem [22], the search for uniform algorithms for computing semidirect products of pseudovarieties has led to substantial research in the theory of finite semigroups. Even though there is no universal solution, since the semidirect product of decidable pseudovarieties is not necessarily decidable [1], under suitable assumptions on the factors, the semidirect product might be decidable. The no...
Automorphisms of Endomorphism Monoids of Relatively Free Bands
"... For a set X and a variety V of bands, let BV(X) be the relatively free band in V on X. For an arbitrary band variety V and an arbitrary set X, we determine the group of automorphisms of End (BV(X)), the monoid of endomorphisms of BV(X). ..."
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Cited by 3 (3 self)
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For a set X and a variety V of bands, let BV(X) be the relatively free band in V on X. For an arbitrary band variety V and an arbitrary set X, we determine the group of automorphisms of End (BV(X)), the monoid of endomorphisms of BV(X).
An application of a theorem of Ash to finite covers
 Studia Logica, 78(12):45–57, 2004. 48 STUART MARGOLIS AND BENJAMIN STEINBERG
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A semigroup approach to wreathproduct extensions of Solomon’s descent algebras, arXiv:math/0710.2081. the electronic journal of combinatorics 16(2
, 1973
"... Abstract There is a wellknown combinatorial model, based on ordered set partitions, of the semigroup of faces of the braid arrangement. We generalize this model to obtain a semigroup F G n associated with G S n , the wreath product of the symmetric group S n with an arbitrary group G. Techniques o ..."
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Abstract There is a wellknown combinatorial model, based on ordered set partitions, of the semigroup of faces of the braid arrangement. We generalize this model to obtain a semigroup F G n associated with G S n , the wreath product of the symmetric group S n with an arbitrary group G. Techniques of Bidigare and Brown are adapted to construct an antihomomorphism from the S n invariant subalgebra of the semigroup algebra of F G n into the group algebra of G S n . The colored descent algebras of Mantaci and Reutenauer are obtained as homomorphic images when G is abelian.
Toroidal embeddings of right groups
 Thai J. Math
"... Abstract In this note we study embeddings of Cayley graphs of right groups on surfaces. We characterize those right groups which have a toroidal but no planar Cayley graph, such that the generating system of the right group has a minimal generating system of the group as a factor. ..."
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Abstract In this note we study embeddings of Cayley graphs of right groups on surfaces. We characterize those right groups which have a toroidal but no planar Cayley graph, such that the generating system of the right group has a minimal generating system of the group as a factor.