Locally compact subgroups of the spectrum of the measure algebra III

The maximal ideal space of the measure algebra of a locally compact abelian (LCA) group has the structure of a compact commutative semitopological semigroup (separately continuous multiplication). Idempotents in the semigroup correspond to certain algebraic projections on the measure algebra. In this paper we study the maximal groups about certain idempotents. This research was partially supported by NSF contract number GP-19852 and GP-31483X.     

On the embeddability of certain infinitely divisible probability measures on Lie groups

We describe certain sufficient conditions for an infinitely divisible probability measure on a Lie group to be embeddable in a continuous one-parameter semigroup of probability measures. A major class of Lie groups involved in the analysis consists of central extensions of almost algebraic groups by compactly generated abelian groups without vector part. This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain connected Lie groups, including the so called Walnut group. The embeddability is concluded also under certain other conditions. Our methods are based on a detailed study of actions of certain nilpotent groups on special spaces of probability measures and on Fourier analysis along the fibering of the extension.     

Compact totally disconnected k-divisible semigroups

One of the early results [5] regarding divisibility in semigroups states that no finite non-degenerate group is divisible. A sequel to this (which in view of well-known results on compact semigroups is a generalization) is that a compact semigroup is divisible if and only if each component is a divisible subsemigroup [2]. Consequently, a finite semigroup is divisible if and only if it is an idempotent semigroup. However, it is of some interest to know which finite semigroups are k-divisible for a given positive integerk≥2. In this note we present a complete characterization of finitek-divisible semigroups, and use this along with a result of K. Numakura [8], to characterize compact totally disconnected k-divisible semigroups     

紧致交换矩阵半群

通过将矩阵同时对角化或同时上三角化的方法,给出有关紧致Abel矩阵半群以及紧致Hermite矩阵半群中矩阵的特征值的一些很好的刻画,证明了由可逆的Hermite矩阵构成的紧致矩阵半群中每个矩阵的特征值都是±1,Hermite矩阵单半群相似于对角矩阵半群,紧致交换矩阵半群的谱半径不超过1,等等.     

The congruence extension property in compact semigroups having precisely one regular D-class

It is proved that each compact semigroup with precisely one regular D -class has the congruence extension property if and only if it has the algebraic congruence extension property.     

The Congruence Extension Property in Compact Semigroups Having Precisely One Regular D -class

It is proved that each compact semigroup with precisely one regular D -class has the congruence extension property if and only if it has the algebraic congruence extension property.     

ON THE STRUCTURES OF RANDOM MEASURE AND POINT PROCESS CONVOLUTION SEMIGROUPS

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Ordered semigroups which are both right commutative and right cancellative

In this paper we prove that each right commutative, right cancellative ordered semigroup (S,.,??) can be embedded into a right cancellative ordered semigroup (T,??,?) such that (T,??) is left simple and right commutative. As a consequence, an ordered semigroup S which is both right commutative and right cancellative is embedded into an ordered semigroup T which is union of pairwise disjoint abelian groups, indexed by a left zero subsemigroup of?T.     

On affine translation divisible designs

We prove that a translation divisible design (TDD) with an abelian translation group can be embedded in PG (n,q) for some n2. Moreover we study affine TDD's showing that they have an (elementary) abelian translation group. A construction for TDD's with an abelian translation group which are not affine is given too.     

On Infinitely Divisible Distributions on Locally Compact Abelian Groups

Our aim in this paper is to characterize some classes of infinitely divisible distributions on locally compact abelian groups. Firstly infinitely divisible distributions with no idempotent factor on locally compact abelian groups are characterized by means of limit distributions of sums of independent random variables. We introduce semi-selfdecomposable distributions on topological fields, and in case of totally disconnected fields we give a limit theorem for them. We also give a characterization of semistable laws on p-adic field and show that semistable processes are constructed as scaling limits of sums of i.i.d.     

Divisible groups that are brauer groups ∗

Throughour this paper G denotes an abelian divisible torsion group. It is not unreasonable to conjecture that such a G must occur as the Brauer group B(K) of some field K. Some evidence to support this conjecture is provided in [3]; it is proved there that if G is countable then G ? B(K) for some K algebraic over the rational field Q [3, Theorem 2]. In this note we provide still more evidence in support of htis conjecture.     

Embedding of a uniquely divisible Abelian semigroup in a convex cone

It is proved that every uniquely divisible Abelian semigroup admits an injective subadditive embedding in a convex cone. As an application, the classical theory of generators of one-parameter operator semigroups is generalized to the case in which the parameter ranges over a uniquely divisible semigroup.     

On compact quantum semigroup QS red

We study some properties of a reduced semigroup C*-algebra of a semigroup S. For the semigroup C*-algebra generated by the deformation of the algebra of continuous functions on a compact abelian group we obtain a structure of a compact quantum semigroup. We also consider morphisms of constructed compact quantum semigroups.     

Cellular covers of totally ordered abelian groups

Cellular covers of groups, and in particular, those of divisible abelian groups, were studied in [FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of groups, J. Pure Appl. Algebra 208, (2007), 61–76], [CHA-CHÓLSKI, W.—FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of divisible abelian groups. In: Contemp. Math. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 77–97], and continued in [FUCHS, L.—GÖBEL, R.: Cellular covers of abelian groups, Results Math. 53, (2009), 59–76] for abelian groups in general. In this note we are investigating cellular covers in the category of totally ordered abelian groups (called o-cellular covers; for definition see Section 2). Some results are similar to those on torsion-free abelian groups (unordered), while others are completely different. For instance, though kernels of o-cellular covers can not be non-zero divisible groups (Lemma 3.1), they may contain non-zero divisible subgroups (Example 3.2); however, the divisible part can not be much larger than the reduced part (Theorem 3.4). There are o-groups, even among the additive subgroups of the rationals, whose o-cellular covers form a proper class (Theorem 4.3).     

Algebraic endomorphisms of LCA-groups that preserve closed subgroups

The object of this paper is the classification of those algebraic (i.e. not necessarily continuous) endomorphisms of a locally compact abelian group leaving invariant all closed subgroups. In a canonical way they turn out to form again a locally compact abelian group which can be determined up to isomorphism. If the group is totally disconnected or not periodic all endomorphisms with this property are continuous and form a topological ring.     

Classes of abelian groups related to sequential convergence

We study classes of abelian groups related to sequential com¬pactness and its generalizations (completeness, coarseness and sequential pre-compactness) in convergence groups. In particular, we describe the algebraic structure of the abelian groups on which every coarse convergence is complete and we prove that: i) every abelian group admits a sequentially precompact convergence; ii) every algebraically compact abelian group admits a sequen¬tially compact convergence.     

Convergence of approximate martingale arrays to mixtures of infinitely divisible distributions in locally compact abelian groups

Sufficient conditions are given for the stable weak convergence of the row sums of an approximate martingale triangular array to a mixture of infinitely divisible distributions on a locally compact abelian group.     

On the intersection of the maximal ideals

Some additional properties of the intersection of the maximal ideals of a compact semigroup are developed here based on results in [1] and [3]. Throughout S denotes a compact usually connected semigroup with at least one maximal proper ideal. The set of all such maximal ideals is denoted by ℳ, the intersection of the members of ℳ by R, the idempotents by E and the minimal ideal by K. Some proofs are more algebraic and in a few cases we do not need S connected. Key facts are that members of ℳ are open and dense, complements of distinct maximal ideals are disjoint and the union of any two such is S. After some generalization of results in [1] and [3], we investigate R relative to the topology of S. Necessary and sufficient conditions are found for R to be compact hence closed and for R to be open. Unlike the situation with the minimal ideal, R can be closed or open largely depending on the position of E relative to R. The following theorem summarizes necessary preliminaries from [1] and [3]. Adapted from material in Chapter 3 of the author's dissertation, written under the co-direction of Dr. John Mack and Dr. John Selden at the University of Kentucky and supported by the National Science Foundation. Support to organize and prepare this paper was provided by Mount Vernon Nazarene College.     

Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

In 2011, Nathanson proposed several questions on minimal complements in a group or a semigroup. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

     

On VC-minimal fields and dp-smallness

In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.