Showing papers on "Bicyclic semigroup published in 2007"

The quiver of the semigroup algebra of a left regular band

Abstract: Recently it has been noticed that many interesting combinatorial objects belong to a class of semigroups called left regular bands, and that random walks on these semigroups encode several well-known random walks. For example, the set of faces of a hyperplane arrangement is endowed with a left regular band structure. This paper studies the module structure of the semigroup algebra of an arbitrary left regular band, extending results for the semigroup algebra of the faces of a hyperplane arrangement. In particular, a description of the quiver of the semigroup algebra is given and the Cartan invariants are computed. These are used to compute the quiver of the face semigroup algebra of a hyperplane arrangement and to show that the semigroup algebra of the free left regular band is isomorphic to the path algebra of its quiver.

Orthogonal Completions of the Polycyclic Monoids

Abstract: We introduce the notion of an orthogonal completion of an inverse monoid with zero. We show that the orthogonal completion of the polycyclic monoid on n generators is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated right ideals of the free monoid on n generators, and so we can make a direct connection with the Thompson groups Vn,1.

Dichotomies in the Complexity of Solving Systems of Equations over Finite Semigroups

Abstract: We consider the problem of testing whether a given system of equations over a fixed finite semigroup S has a solution. For the case where S is a monoid, we prove that the problem is computable in polynomial time when S is commutative and is the union of its subgroups but is NP-complete otherwise. When S is a monoid or a regular semigroup, we obtain similar dichotomies for the restricted version of the problem where no variable occurs on the right-hand side of each equation. We stress connections between these problems and constraint satisfaction problems. In particular, for any finite domain D and any finite set of relations Γ over D, we construct a finite semigroup SΓ such that CSP(Γ) is polynomial-time equivalent to the satifiability problem for systems of equations over SΓ.

Representation of Numerical Semigroups by Dyck Paths

Abstract: We introduce square diagrams that represent numerical semigroups and we obtain an injection from the set of numerical semigroups into the set of Dyck paths.

Semigroups and Toposes

Abstract: We establish a semigroup presentation of the classifying topos of an inverse semigroup. For this purpose we consider a category larger than inverse semigroups and prehomomorphisms, whose objects we call left *-semigroups. We show that any etendue has a left *-semigroup presentation. We show that any left *-semigroup may be codensely embedded in an inverse semigroup. We also establish a strictly topos description of E-unitary inverse semigroups.

Is $A^{-1}$ an infinitesimal generator?

Abstract: In this paper we study the question whether $A^{-1}$ is the infinitesimal generator of a bounded $C_0$-semigroup if $A$ generates a bounded $C_0$-semigroup. If the semigroup generated by $A$ is analytic and sectorially bounded, then the same holds for the semigroup generated by $A^{-1}$. However, we construct a contraction semigroup with growth bound minus infinity for which $A^{-1}$ does not generate a bounded semigroup. Using this example we construct an infinitesimal generator of a bounded semigroup for which its inverse does not generate a semigroup. Hence we show that the question posed by deLaubenfels in 1988 must be answered negatively. All these examples are on Banach spaces. On a Hilbert space the question whether the inverse of a generator of a bounded semigroup also generates a bounded semigroup still remains open.

A note on the quasi-antiorder in a semigroup

Abstract: Connections between quasi-antiorder on a semigroup with apartness and a naturally deflned quasi-antiorder relation on factor semi- group (according to congruence and anti-congruence) are presented.

Automorphisms of the semigroup of invertible matrices with nonnegative elements

Abstract: In this paper, we prove that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly ordered associative ring on some specially defined subgroup coincides with the composition of an inner automorphism of the semigroup, an order-preserving automorphism of the ring, and a central homothety

Quantum markov semigroups: product systems and subordination

Abstract: We show that if a product system comes from a quantum Markov semigroup, then it carries a natural Borel structure with respect to which the semigroup may be realized in terms of a measurable representation. We show, too, that the dual product system of a Borel product system also carries a natural Borel structure. We apply our analysis to study the order interval consisting of all quantum Markov semigroups that are subordinate to a given one.

Every numerical semigroup is one half of a symmetric numerical semigroup

Abstract: Let S be a numerical semigroup. Then there exists a symmetric numerical semigroup S such that S = {n ∈ N 2n ∈ S}.

A Generalization of Symmetric Inverse Semigroups

Abstract: The set of difunctional binary relations DX plays a special role in representing inverse semigroups by binary relations. However, DX is not an inverse semigroup either with the standard operation ∘, or with an alternative operation introduced in [6]. We introduce a new binary operation ⋄ on the set BX of binary relations. We demonstrate that (DX, ⋄) is an inverse semigroup, and the symmetric inverse semigroup (IX, ∘) is a subsemigroup of (DX,⋄).

A Note on Numerical Semigroups

Abstract: This correspondence is a short extension to the previous article Bras-Amoroacutes, 2004. In that work, some results were given on one-point codes related to numerical semigroups. One of the crucial concepts in the discussion was the so-called nu-sequence of a semigroup. This sequence has been used in the literature to derive bounds on the minimum distance as well as for defining improvements on the dimension of existing codes. It was proven in that work that the nu-sequence of a semigroup uniquely determines it. Here this result is extended to another object related to a semigroup, the oplus operation. This operation has also been important in the literature for defining other classes of improved codes. It is also proven here that, although the infinite set of values in the nu-sequence (resp. the oplus values) uniquely determines the associated semigroup, no finite part of it can determine it, because it is shared by infinitely many semigroups. In that reference the proof of the fact that the nu-sequence of a numerical semigroup uniquely determines it is constructive. The result here presented shows that, however, that construction can not be performed as an algorithm with finite input

On Diagonal Acts of Monoids

Abstract: In this paper we discuss what is known so far about diagonal acts of monoids. The first results that will be discussed comprise an overview of some work done on determining whether or not the diagonal act can be finitely generated or cyclic when looking at specific classes of monoids. This has been a topic of interest to a handful of semigroup theorists over the past seven years. We then move on to discuss some results pertaining to flatness properties of diagonal acts. The theory of flatness properties of acts over monoids has been of major interest over the past two decades, but so far there are no papers published on this subject that relate specifically to diagonal acts. We attempt to shed some light on this topic as well as present some new problems.

On binary quadratic forms with the semigroup property

Abstract: A quadratic form f is said to have the semigroup property if its values at the points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with the semigroup property. If there is an integer bilinear map s such that f(s(x,y)) = f(x)f(y) for all vectors x and y from the integer two-dimensional lattice, then the form f has the semigroup property. We give an explicit integer parameterization of all pairs (f,s) with the property stated above. We do not know any other examples of forms with the semigroup property.

R-Unipotent Congruences on Eventually Regular Semigroups

Abstract: A semigroup S is called an eventually regular semigroup if for every a ∈ S, there exists a positive integer n such that an is regular. In this paper, the R-unipotent, inverse semigroup and group congruences on an eventually regular semigroup S are described by means of certain congruence pairs (ξ, K), where ξ is a normal congruence on the subsemigroup 〈E(S)〉 generated by E(S), and K is a normal subsemigroup of S.

Semilattice ordered inverse semigroups

Abstract: or, equivalently, ∨ is compatible with multiplication on both the left and the right. Such a semigoup is necessarily partially ordered; that is the partial order associated with ∨ is compatible with multiplication on both the left and the right. When S is a partially ordered group and a semilattice under the partial ordering, it is necessarily a semilatticed semigroup, in fact it is lattice ordered in the sense that it is a lattice in fact a distributive lattice under the partial ordering and both lattice operations are compatible with multiplication. In this case we say that the group is an l-group. Thus, for groups, there is a very strong relationship between the order and the structure of the group. For example, it is well known, that

Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero

Abstract: We give a characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero.

The wreath product of atoms of the lattice of semigroup varieties

Abstract: A semigroup variety is called a Cross variety if it is finitely based, is generated by a finite semigroup, and has a finite lattice of subvarieties. It is established in which cases the wreath product of two semigroup varieties each of which is an atom of the lattice of semigroup varieties is a Cross variety. Furthermore, for all the pairs of atoms U and V for which this is possible, either a finite basis of identities for the wreath product UwV is given explicitly, a finite semigroup generating this variety is found and the lattice of subvarieties is described, or it is proved that such a finite characterization is impossible. 1. Statement of the problem and formulation of the main results This paper is devoted to a systematic study of the wreath products of atoms of the lattice of semigroup varieties. We consider three classical conditions for varieties to be finite: having a finite basis of identities, being generated by a finite semigroup, and having a finite lattice of subvarieties. For all the pairs of atoms U and V for which this is possible, a finite basis of identities for the wreath product UwV is given explicitly, a finite semigroup generating this variety is found and the lattice of subvarieties is described. Definition 1.1. A semigroup variety is called a Cross variety if it is finitely based, is generated by a finite semigroup and has a finite lattice of subvarieties. The atoms of the lattice L of all semigroup varieties are well known [16]. These are precisely the varieties N2 of all semigroups with zero multiplication, Sl of all semilattices, L1 of all semigroups of left zeros, R1 of all semigroups of right zeros and Ap of all Abelian groups of prime exponent p. The main result of the paper can be stated as follows. Theorem 1.2. If U and V are atoms of the lattice of semigroup varieties, then the wreath product UwV is a Cross variety, except in the following cases: 1) U = V = Ap; here the variety ApwAp = Ap is finitely based but is not generated by a finite semigroup and has an infinite lattice of subvarieties; 2) U = V = Sl and U = Sl,V = R1; here each of the varieties SlwSl = Sl and SlwR1 is finitely based, is generated by a finite semigroup and has an infinite lattice of subvarieties; 3) U = Sl,V = Ap; here the variety SlwAp is essentially infinitely based, is generated by a finite semigroup and has an infinite lattice of subvarieties. As a by-product, it becomes possible to estimate how big the difference is between the monoid wreath product and the lattice join of two semigroup varieties in the case where the varieties involved in the wreath product are atoms in the lattice of semigroup 2000 Mathematics Subject Classification. Primary 20M07; Secondary 20E22.

A rank for right congruences on inverse semigroups

Abstract: The S-rank (where 'S' abbreviates 'sandwich') of a right congruence p on a semigroup 5 is the Cantor-Bendixson rank of p in the lattice of right congruences HC of 5 with respect to a topology we call the finite type topology. If every p e 1ZC possesses Srank, then 5 is ranked. It is known that every right Noetherian semigroup is ranked and every ranked inverse semigroup is weakly right Noetherian. Moreover, if S is ranked, then so is every maximal subgroup of S. We show that a Brandt semigroup B°(G, I) is ranked if and only if G is ranked and / is finite. We establish a correspondence between the lattice of congruences on a chain E, and the lattice of right congruences contained within the least group congruence on any inverse semigroup 5 with semilattice of idempotents E(S) =* E. Consequently we argue that the (inverse) bicyclic monoid B is not ranked; moreover, a ranked semigroup cannot contain a bicyclic »7-class. On the other hand, B is weakly right Noetherian, and possesses trivial (hence ranked) subgroups. Our notion of rank arose from considering stability properties of the theory Ts of existentially closed (right) S-sets over a right coherent monoid S. The property of right coherence guarantees that the existentially closed 5-sets form an axiomatisable class. We argue that B is right coherent. As a consequence, it follows from known results that TB is a theory of B-sets that is superstable but not totally transcendental.

On lax-phillips semigroups

Abstract: Lax-Phillipsevolutionsaredescribedbytwo-spacescatteringsys- tems. The canonical identification operator is characterized for Lax-Phillips evolutions, whose outgoing and incoming projections commute. In this case a (generalized) Lax-Phillips semigroup can be introduced and its spectral the- ory is considered. In the special case, originally considered by Lax and Phillips (where the outgoing and incoming subspaces are mutually orthogonal), this semigroup coincides with that introduced by Lax and Phillips. The basic con- nection of the Lax-Phillips semigroup to the so-called characteristic semigroup of the reference evolution is emphasized.

Continuous extension of a densely parameterized semigroup

Abstract: Let S be a dense sub-semigroup of the positive real numbers, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over S can be extended to a weakly continuous semigroup over the positive real numbers. We obtain similar results for non-linear, non-expansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over the positive real numbers.

Derivations on Certain Semigroup Algebras

Abstract: In the present paper we give a partially negative answer to a conjecture of Ghahramani, Runde and Willis. We also discuss the derivation problem for both foundation semigroup algebras and Clifford semigroup algebras. In particular, we prove that if S is a topological Clifford semigroup for which Es is finite, then H1(M(S),M(S))={0}.

A new method in the finite basis problem with applications to rank 2 transformation semigroups

Abstract: We prove that the semigroup of all transformations of a 3-element set with rank at most 2 does not have a finite basis of identities. This gives a negative answer to a question of Shevrin and Volkov. It is worthwhile to notice that the semigroup of transformations with rank at most 2 of an n-element set, where n > 4, has a finite basis of identities. A new method of constructing finite non-finitely based semigroups is developed.