Quasi-uniformities on topological semigroups and bicompletion

We study the bicompletion of the quasi-uniformities that are induced in a natural way on a topological semigroup which has a neutral element. In particular, we show that if X is a topological semigroup, with neutral element, for which the left translations are open, then the bicompletion of the left quasi-uniformity of X can be considered a topological semigroup which contains the topological space X as a sup-dense subsemigroup. The bicompletion in the case that the left translations are not necessarily open is also discussed. In particular, both Abelian and left-cancellable topological semigroups are considered. For semigroups which are (left-)cancellable or which are locally totally bounded, theorems similar to those known from the classical theory of (para)topological groups are established. July 1, 1999     

Congruences on generalized inverse semigroups

A generalized inverse semigroup is a regular semigroup whose idempotents satisfy a permutation identity X₁ X₂...X_n=X_p1 X_p2...X_pn, where (P₁, P₂..., P_n) is a nontrivial permutation of (1, 2,..., n). Yamada [4] has given a complete classification of generalized inverse semigroups in terms of inverse semigroups, left normal bands, and right normal bands. In this paper we show that every congruence on a generalized inverse semigroup is uniquely determined by a congruence on its associated inverse semigroup, left normal band, and right normal band. A converse is also provided. This paper is extracted from the doctoral thesis of the author written at Monash University under the direction of Professor G. B. Preston. The research was carried out while the author held a Commonwealth Postgraduate Award.     

The free idempotent generated locally inverse semigroup

In this paper we construct a model for the free idempotent generated locally inverse semigroup on a set X. The elements of this model are special vertex-labeled bipartite trees with a pair of distinguished vertices. To describe this model, we need first to introduce a variation of a model for the free pseudosemilattice on a set X presented in Auinger and Oliveira (On the variety of strict pseudosemilattices. Stud Sci Math Hungarica 50:207–241, 2013). A construction of a graph associated with a regular semigroup was presented in Brittenham et al. (Subgroups of free idempotent generated semigroups need not be free. J Algebra 321:3026–3042, 2009) in order to give a first example of a free regular idempotent generated semigroup on a biordered set E with non-free maximal subgroups. If G is the graph associated with the free pseudosemilattice on X, we shall see that the models we present for the free pseudosemilattice on X and for the free idempotent generated locally inverse semigroup on X are closely related with the graph G.     

Convolution semigroups of states

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C ₀-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Note on the locally product and almost locally product structures

This paper is concerned with a study of some of the properties of locally product and almost locally product structures on a differentiable manifold X _n of class C ^k . Every locally product space has certain almost locally product structures which transform the local tangent space to X _n at an arbitrary point P in a set fashion: this is studied in Theorem (2.2). Theorem (2.3) considers the nature of transformations that exist between two co-ordinate systems at a point whenever an almost locally product structure has the same local representation in each of these co-ordinate systems. A necessary and sufficient condition for X _n to be a locally product manifold is obtained in terms of the pseudo-group of co-ordinate transformations on X _n and the subpseudo-groups [cf., Theoren (2.1)]. Section 3 is entirely devoted to the study of integrable almost locally product structures.     

Number theory of matrix semigroups

Unlike factorization theory of commutative semigroups which are well-studied, very little literature exists describing factorization properties in noncommutative semigroups. Perhaps the most ubiquitous noncommutative semigroups are semigroups of square matrices and this article investigates the factorization properties within certain subsemigroups of M_n(Z), the semigroup of n×n matrices with integer entries. Certain important invariants are calculated to give a sense of how unique or non-unique factorization is in each of these semigroups.     

Constructing numerical semigroups of a given genus

Let n _g denote the number of numerical semigroups of genus g. Bras-Amorós conjectured that n _g possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree. We offer a new, simpler approach to counting numerical semigroups of a given genus. Our method gives direct constructions of families of numerical semigroups, without referring to the generators or the semigroup tree. In particular, we give an improved asymptotic lower bound for n _g .     

A Generalization of Symmetric Inverse Semigroups

The set of difunctional binary relations D_X plays a special role in representing inverse semigroups by binary relations. However, D_X 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 B_X of binary relations. We demonstrate that (D_X, ⋄) is an inverse semigroup, and the symmetric inverse semigroup (I_X, ∘) is a subsemigroup of (D_X,⋄).     

Bases of identities for semigroups of bounded rank transformations of a set

We complete the series of results by M. V. Sapir, M. V. Volkov and the author solving the Finite Basis Problem for semigroups of rank ≤ k transformations of a set, namely based on these results we prove that the semigroup T _k (X) of rank ≤ k transformations of a set X has no finite basis of identities if and only if k is a natural number and either k = 2 and |X| ∈ «3, 4» or k ≥ 3. A new method for constructing finite non-finitely based semigroups is developed. We prove that the semigroup of rank ≤ 2 transformations of a 4-element set has no finite basis of identities but that the problem of checking its identities is tractable (polynomial).     

On algebras of <Emphasis Type="Italic">P</Emphasis>-Ehresmann semigroups and their associate partial semigroups

P-Ehresmann semigroups are introduced by Jones as a common generalization of Ehresmann semigroups and regular \(*\)-semigroups. Ehresmann semigroups and their semigroup algebras are investigated by many authors in literature. In particular, Stein shows that under some finiteness condition, the semigroup algebra of an Ehresmann semigroup with a left (or right) restriction condition is isomorphic to the category algebra of the corresponding Ehresmann category. In this paper, we generalize this result to P-Ehresmann semigroups. More precisely, we show that for a left (or right) P-restriction locally Ehresmann P-Ehresmann semigroup \(\mathbf{S}\), if its projection set is principally finite, then we can give an algebra isomorphism between the semigroup algebra of \(\mathbf{S}\) and the partial semigroup algebra of the associate partial semigroup of \(\mathbf{S}\). Some interpretations and necessary examples are also provided to show why the above isomorphism dose not work for more general P-Ehresmann semigroups.     

Perfect semigroups and aliens

A topologized semigroup is called perfect if its multiplication is a perfect map (= a closed continuous mapping such that the inverse image of every point is compact). Thus a locally compact topological semigroup is perfect if and only if its multiplication is closed and each of its elements is compactly divided, that is, its divisors form a compact set. In the present paper we study compactly and non-compactly divided elements in the contexts of general locally compact semigroups, subsemigroups of groups, Lie semigroups and subsemigroups of Sl(2, ?).     

Semigroup actions on posets and preimage quasi-orders

Structures consisting of a semigroup of (partial) functions on a set X, a?poset of subsets of X, and a preimage operation linking the two, arise commonly throughout mathematics. The poset may be equipped with one or more set operations, up to Boolean algebra structure. Such structures are finitely axiomatized here in terms of order-preserving semigroup actions on posets. This generalises Schein??s axiomatization of semigroups of partial functions equipped with the first projection quasi-order.     

Further results in the theory of generalised inflations of semigroups

We determine the structure of semigroups that satisfy xyzw∈{xy,xw,zy,zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F(X) on a set X. In the latter case the semigroup must be an inflation of F(X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band.     

Die Stoppverteilungen eines Markoff-Prozesses mit lokalendlichem Potential

Consider a Markovian standard semigroup P_t, t≥o, with potential kernel U=∫P_tdt on a locally compact space E. Let μ be a finite measure on E with locally finite potential μ^U and X_t, t≥O, the process having (P_t) as transition semigroup and μ as initial law. Then for a measure ν on E the following two statements are equivalent:

1. μ^U≥νU;
2. there exists a “randomized” stopping time T such that X_T is distributed according to ν.

     

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.     

Sulľapplicazione della funzione ellitticap u alla teoria dei poligoni di poncelet

Abstract  In this paper we study strongly continuous positive semigroups on particular classes of weighted continuous function space on a locally compact Hausdorff space X having a countable base. In particular we characterize those positive semigroups which are the transition semigroups of suitable Markov processes. Some applications are also discussed. Keywords Positive semigroup, Markov transition function, Markov process, Weighted continuous function space, Degenerate second order differential operator Mathematics Subject Classification (2000) 47D06, 47D07, 60J60