Showing papers on "Cancellative semigroup published in 1999"

The Semigroup of Ultrafilters Near 0

Abstract: of ultrafiliters on (0,1) that converge to 0 is a semigroup under the restriction of the usual operation + on BetaR d , the Stone-Cech compactification of the discrete semigroup (R d ,+). It is also a subsemigroup of Beta((0,1) d ,·). The interaction of these operations has recently yielded some strong results in Ramsey Theory. Since (0+,·) is an ideal of Beta((0,1) d ,·), much is known about the structure of (0+,·). On the other hand, (0+,+) is far from being an ideal of (BetaR d ,+) so little about its algebraic structure follows from known results. We characterize here the smallest ideal of (0+,+), its closure, and those sets "central" in (0+,+), that is, those sets which are members of minimal idempotents in (0+, +). We derive new combinatorial applications of those sets that are central in (0+,+).

Semigroup identities in 4-Engel groups

Abstract: We say that a group G or a variety of groups V satisfies a semigroup law, if it satisfies a nontrivial law of the form u(x1, . . . , xn) = v(x1, . . . xn), where u and v are words in the free semigroup freely generated by x1, . . . , xn. It follows from a result of J. Lewin and T. Lewin [2] that a variety V of groups which satisfies a semigroup law can be characterised by its semigroups laws. Furthermore, we have then a sufficient and necessary condition for a semigroup to be embeddable in some group in V . A semigroup S is embeddable in some group in V if and only if it is cancellative and it satisfies all the semigroup laws that hold in V . In other words we have that S is embeddable in some group in V if and only if S is a cancellative semigroup in the corresponding semigroup variety. In [4] B. H. Neumann and T. Taylor show that nilpotent groups satisfy semigroup laws. We will be using their work later on so we will now describe it in more details. Let F be a free group that is freely generated by the variables x, y, z1, z2, . . .. We define a sequence of words q1, q2, . . . in the variables x, y, z1, z2, . . . by induction as follows.

A polynomial time algorithm for local testability and its level

Abstract: A locally testable semigroup S is a semigroup with the property that for some non-negative integer k, called the order or level of local testability, two words u and v in some set of generators for S are equal in the semigroup if (1) the prefix and suffix of the words of length k coincide, and (2) the set of intermediate substrings of length k of the words coincide. The local testability problem for semigroups is, given a finite semigroup, to decide, if the semigroup is locally testable or not. Recently, we introduced a polynomial time algorithm for the local testability problem and to find the level of local testability for semigroups based on our previous description of identities of k-testable semigroups and the structure of locally testable semigroups. The first part of the algorithm we introduce solves the local testability problem. The second part of the algorithm finds the order of local testability of a semigroup. The algorithm is of order n2, where n is the order of the semigroup.

Identities of locally testable semigroups

Abstract: Let S be a semigroup of words over an alphabet ∑ . Suppose tliar every two words u and e over ∑ are equal in S if (1) the sets of subwords of length k of the words a and b coincide and are non-empty. (2) the prefix (suffix) of u of length k1 is equal to the prefix (suffix) of e. Then S is called k-testable. A semigroup is locally testable if it is k-testable for some k > 0. We present a finite basis of identities of the variety of A'-testable semigroups. The structure of k-testable semigroup is studied. Necessarv and sufficient conditions for local testability will be given. A solution to one problem from the survey of Shevrin and Sukhanov (1985) will be presented.

Compatible orders on the bicyclic semigroup

Abstract: In this paper we determine those compatible partial orders on the bicyclic semigroup B which turn it into a semilatticed semigroup. We shall see that there are exactly four distinct compatible total orderings on B. These are the only compatible orderings which turn B into a lattice ordered semigroup. On a group every compatible semilattice ordering is a lattice ordering. However this is not the case with inverse semigroups. Indeed, the situation regarding semilattice orderings on the bicyclic semigroups is much richer. There are four infinite families of compatible semilattce orderings on B. Two of these families turn B into a V-semilatticed semigroup; two of the families turn it into a ^-semilatticed semigroup.

Flatness Properties of Acts over Commutative, Cancellative Monoids

Abstract: This note presents a classification of commutative, cancellative monoids S by flatness properties of their associated S -acts.

On the adjoint semigroups of p-separable BCI-algebras

Abstract: We give some conditions under which the p-semisimple part SP(X) of a BCIalgebra X becomes an ideal, and prove that the adjoint semigroup of a p-separable BCI-algebra is a direct product of a negatively partially ordered semigroup and an abelian group.

Cancellative Superposition Decides the Theory of Divisible Torsion-Free Abelian Groups

Abstract: In divisible torsion-free abelian groups, the efficiency of the cancellative superposition calculus can be greatly increased by combining it with a variable elimination algorithm that transforms every clause into an equivadent clause without unshielded variables. We show that the resulting calculus is not only refutationally complete (even in the presence of eirbitrary free function symbols), but that it is also a decision procedure for the theory of divisible torsion-free abelisin groups.

Right Cancellative Monoids, Left Regular Bands and Left Regular Type A Monoids

Abstract: In this paper the structure of proper left regular type A monoids is established in terms of right cancellative monoids and leftregular bands. After it is proved that every left regular type A mon0id always has a P-cover, the structure of such cover is given.

Minimal quantum dynamical semigroup on a von neumann algebra

Abstract: Given a formal unbounded generator, the minimal quantum dynamical semigroup on a von Neumann algebra is constructed. A set of equivalent necessary and sufficient conditions for the conservativity of the minimal semigroup is given and in the case when it is not conservative, a distinguished family of conservative perturbations of the semigroup is studied. Finally, some of these results are applied to the classical Markov semigroup with arbitrary state space.

Representation of Finite Lattices by Annihilators of Completely 0-Simple Semigroups

Abstract: the explicit construction of a 0-simple Rees matrix semigroup is suggested such that the lattice of left annihilators of this semigroup is isomorphic to L.

The de Rham-Hodge semigroup on the path space: existence and L2-contractivity

Abstract: We construct the de Rham-Hodge semigroup on exact 1-forms on the path space of a Riemannian manifold and prove that it is L 2 -contractive.

The embedding of an ordered semigroup into an ordered group by means of pseudoorder

Abstract: It is shown that an ordered cancellative commutative semigroup can be embedded into an ordered commutative group. Bibliography: 6titles.

On the semigroup BetaS

Abstract: be an infinite, discrete, cancellative semigroup and let BetaS be the Stone-Cech compactification of S. Then BetaS is a semigroup with an operation which extends that of S and which is continous only in one variable. We generalize some algebraic properties known to hold for the additive semigroup of the integers.

On cancellative congruences for semigroups

Abstract: Given a semigroup identity u = v. We describe a smallest normal subgroup N in a free group F, such that F/N contains a relatively free cancellative semigroup which satisfies the identity u = v. Let F be a free group and F be a free semigroup (F 3 1), both generated by the same set M = {x, y, z, . . .}. A semigroup identity of a group G (or a semigroup S) is a nontrivial identity of the form u = v where u, v ∈ F , which holds under every substitution of generators by elements from G (elements from S). Research concerning semigroup identities was initiated by A.I. Mal’cev in early fifties and is still continued (see e.g. [10], [6], [18], [13] – [15], [11], [7]). One of the open questions concerning semigroup identities in a group G and a subsemigroup S generating G is whether an identity can hold in S without holding in G [1]. Partial answers to this question are given in [9], [5] and [2]. We give here some technical result which relates a cancellative congruence in F , providing an identity in a quotient semigroup, and a normal subgroup it defines in F. A congruence in F , providing the identity u = v has to contain (u, v) and the set which is the smallest invariant (under endomorphisms of F) reflexive, and symmetric closure of (u, v). We denote this set by (u, v). To extend (u, v) to a transitive relation we define a step. Two words a, b ∈ F are called connected by (u, v)-step if a = c1sc2, b = c1tc2, and (s, t) ∈ (u, v). It is clear that if (s, t) ∈ (u, v), then (a, b) ∈ (u, v). However if a = c1sc2 → c1tc2 = d1sd2 → d1td2 = b, where each arrow denotes (u, v)-step, then it is not necessary that (a, b) ∈ (u, v). A congruence in F , providing the identity u = v in the quotient semigroup without necessity to be cancellative is described in [4]. Namely, two words are congruent if and only if they are connected by a finite sequence of (u, v)-steps. AMS subject classification: Primary 20E10; Secondary 20M07.

Minimal quantum dynamical semiguoup on a von Neumann algebra

Abstract: Given a formal unbounded generator, the minimal quantum dynamical semigroup on a von Neumann algebra is constructed. A set of equivalent necessary and sufficient conditions for the conservativity of the minimal semigroup is given and in the case when it is not conservative, a distinguished family of conservative perturbations of the semigroup is studied. Finally, some of these results are applied to the classical Markov semigroup with arbitrary state space.