Class Of Topological Semigroups Research Articles

Absolutely closed semigroups

Let C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document} be a class of topological semigroups. A semigroup X is called absolutely C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-closed if for any homomorphism h:X→Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h:X\\rightarrow Y$$\\end{document} to a topological semigroup Y∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y\\in {\\mathcal {C}}$$\\end{document}, the image h[X] is closed in Y. Let T1S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {1}}\ extsf {S}$$\\end{document}, T2S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {2}}\ extsf {S}$$\\end{document}, and TzS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {z}}\ extsf {S}$$\\end{document} be the classes of T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_1$$\\end{document}, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely TzS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {z}}\ extsf {S}$$\\end{document}-closed if and only if X is absolutely T2S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {2}}\ extsf {S}$$\\end{document}-closed if and only if X is chain-finite, bounded, group-finite and Clifford + finite. On the other hand, a commutative semigroup X is absolutely T1S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {1}}\ extsf {S}$$\\end{document}-closed if and only if X is finite. Also, for a given absolutely C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-closed semigroup X we detect absolutely C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-closed subsemigroups in the center of X.

Semigroup extensions of Abelian topological groups

We show that an Abelian topological group G is absolutely closed in the class of topological semigroups if and only if G is complete and there is $$n\in \mathbb {N}$$ such that the subgroup $$nG=\{nx:x\in G\}$$ is totally bounded. If for every $$n\in \mathbb {N}$$ , the subgroup nG is not totally bounded, then the topology of G can be extended to a semigroup topology $$\mathcal {T}$$ on $$G\times \mathbb {Z}^+$$ in which G is open and dense, and if G is locally compact, so can be chosen $$\mathcal {T}$$ . In particular, the topology of $$\mathbb {R}$$ can be extended to a locally compact semigroup topology on $$\mathbb {R}\times \mathbb {Z}^+$$ in which $$\mathbb {R}$$ is dense. We also show that the topology of G can be extended to a regular (equivalently, Tychonoff) semigroup topology on $$G\times \mathbb {Z}^+$$ in which G is open and dense if and only if there is a neighborhood U of $$0\in G$$ such that for every $$n\in \mathbb {N}$$ , the subgroup nG is U-unbounded.

Central Limit Properties of GZH-Semigroups and Their Applications in Probability Theory

A class of topological semigroups called GZH-semigroups is introduced. Conditions under which they have the property that limits of infinitesimal arrays are infinitely divisible are obtained. The convolution semigroup of all probability measures on a second countable LCA-group or on a real separable Hilbert space as well as the semigroup of all positive definite kernels defined on a countable set with complex values and with norms not greater than 1 are reduced to an extended form of Delphic semigroups.

Bochner's Theorem and the Hausdorff Moment Theorem on foundation Topological Semigroups

One of the most basic theorems in harmonic analysis on locally compact commutative groups is Bochner's theorem (see [16, p. 19]). This theorem characterizes the positive definite functions. In 1971, R. Lindhal and P. H. Maserick proved a version of Bochner's theorem for discrete commutative semigroups with identity and with an involution * (see [13]). Later, in 1980, C. Berg and P. H. Maserick in [6] generalized this theorem for exponentially bounded positive definite functions on discrete commutative semigroups with identity and with an involution *. In this work we develop these results, and also the Hausdorff moment theorem, for an extensive class of topological semigroups, the so-called “foundation topological semigroups”. We shall give examples to show that these theorems do not extend in the obvious way to general locally compact topological semigroups.

Left invariant measure in topological semigroups

AbstractWe consider the problem of the existence of a left invariant measure in a class of topological semigroups. Several authors have considered this and related problems on semigroups satisfying similar conditions, but the invariance they considered is right invariance. This paper is different in that it deals with left invariance.

A class of topological semigroups

A class of topological semigroups

Monotone Homomorphisms of Compact Semigroups

The problem of determining the class of homomorphic images of a given class of topological semigroups seems to have received little attention in the literature. In [4] Cohen and Krule determined the homomorphic images of a semigroup with zero on an interval. Anderson and Hunter in [1] proved several theorems in this direction. In general, the problem seems to be rather difficult. However, the difficulty is lessened somewhat if all of the homomorphisms of the semigroups in question must be monotone. Phillips, [7], showed that every homomorphism of a standard thread is monotone and hence every homomorphic image of a standard thread is either a standard thread or a point. In this paper a larger class of topological semigroups which admit only monotone homomorphisms is given. These results are used to determine the topological nature of the homomorphic images of certain classes of topological semigroups. These include products of standard threads with min threads, certain semilattices on a two-cell, and compact connected lattices in the plane.