# Topological semi-group

A set equipped with both the algebraic structure of a semi-group and the structure of a topological Hausdorff space, such that the semi-group operation is continuous in the given topology. Any semi-group is a topological semi-group in the discrete topology. There exist semi-groups which admit only the discrete topology. Any Hausdorff space can be made into a topological semi-group, e.g. by giving it a left-singular or zero multiplication.

Several independent branches of topological semi-groups have emerged: the general theory of compact semi-groups (cf. Compactness); homotopy properties of topological semi-groups; the study of semi-groups on concrete topological spaces; harmonic analysis on topological semi-groups; and semi-groups of continuous transformations of topological spaces. Besides, the study of topological semi-groups began in connection with the consideration of all closed sub-semi-groups.

A natural class of topological semi-groups, which includes the compact and discrete semi-groups, is that of the locally compact semi-groups. However, many properties which hold for compact and discrete semi-groups cease to hold for arbitrary locally compact semi-groups. Hence one usually imposes additional restrictions of algebraic or topological character. An important condition of this type is weak uniformity: A locally compact semi-group $S$ is called weakly uniform if, for any $a, b \in S$( one of these elements may be the empty symbol) and any subsets $Y, W \subseteq S$, where $W$ is an open subset with compact closure $\overline{W}\;$ and $\overline{ {aYb }}\; \subseteq W$ or $\overline{ {aYb }}\; \subseteq S \setminus \overline{W}\;$, there exist neighbourhoods $V ( a)$ and $V ( b)$ of $a$ and $b$ such that $V ( a) YV ( b) \subset W$, respectively $V ( a) YV ( b) \subset S \setminus \overline{W}\;$. The class of weakly-uniform semi-groups contains all compact semi-groups, discrete semi-groups and locally compact groups. If a locally compact semi-group $S$ is a group, then the mapping of taking the inverse is continuous, i.e. $S$ is a topological group. In a locally compact inverse semi-group, this mapping (cf. Regular element) is continuous if and only if $S$ is weakly uniform. In a weakly-uniform semi-group the maximal subgroups are closed. This property need not hold in an arbitrary locally compact semi-group.

An arbitrary compact semi-group $S$ contains a closed kernel $M ( S)$( cf. Kernel of a semi-group), which is a completely-simple semi-group. In particular, $S$ has idempotents. The structure of compact, completely-simple (completely $0$- simple) semi-groups is described by a theorem analogous to Rees' theorem on discrete completely-simple (completely $0$- simple) semi-groups (cf. Rees semi-group of matrix type). The analogue of Rees' theorem holds for weakly-uniform semi-groups, but not, in general, for locally compact semi-groups .

A semi-group $S$ is called a thread if $S$ can be linearly ordered in such a way that $S$ becomes a connected topological semi-group under the order (interval) topology. A semi-group $S$ with zero 0 and identity $e$ is called a standard thread (or $I$- semi-group) if $S$ is a thread and if 0 and $e$ are the least and largest elements of $S$. There is a complete description of standard threads . A compact semi-group with identity $e$ is called irreducible if it is connected and does not contain a proper connected closed sub-semi-group $T$ for which $e \in T$ and $T \cap M ( S) \neq \emptyset$. Connected compact semi-groups with identity contain irreducible semi-groups as closed sub-semi-groups. The irreducible semi-groups can be described as follows: An irreducible semi-group $S$ is commutative, the Green equivalence relation ${\mathcal H}$( cf. Green equivalence relations) is a closed congruence on $S$, and $S/ {\mathcal H}$ is a standard thread.

The "minimal blocks" of a topological semi-group are the closures of its monogenic sub-semi-groups, called monothetic semi-groups. For a compact monothetic semi-group $S$ the kernel $M ( S)$ is a compact monothetic group. The compact monothetic semi-groups have been completely described . Weakly-uniform monothetic semi-groups are either compact or discrete. There is an example  of a monothetic locally compact semi-group which is neither discrete nor compact.

A character of a commutative topological semi-group with identity is a non-zero continuous homomorphism into the multiplicative semi-group of complex numbers of modulus $\leq 1$. The set of all characters $S ^ {*}$ forms a commutative topological semi-group with identity with respect to pointwise multiplication (cf. Character of a semi-group) and the compact-open topology. One says that the (Pontryagin) duality theorem holds for a commutative topological semi-group with identity if the canonical homomorphism from $S$ into the semi-group of characters of $S ^ {*}$ is a topological isomorphism "onto" . The duality theorem is true for a commutative compact semi-group $S$ with identity if and only if $S$ is an inverse semi-group and its sub-semi-group of idempotents forms a totally-disconnected space. Necessary and sufficient conditions have been found  for the duality theorem to hold for a commutative locally compact semi-group. One of the necessary conditions is that the semi-group be weakly uniform.

An important subclass of commutative compact semi-groups are the compact semi-lattices (cf. Idempotents, semi-group of). A compact semi-lattice admits a unique topology, up to a homeomorphism. The description of certain types of topological semi-groups leads to metric semi-groups. A metric $d$ on a topological semi-group $S$ is called invariant if $d ( ax, ay) \leq d ( x, y)$ and $d ( xa, ya) \leq d ( x, y)$ for all $a, x, y \in S$. A topological semi-group is called metric if there exists an invariant metric on $S$ inducing the topology on $S$. Every compact semi-group is a projective limit of compact metric semi-groups. Every totally-disconnected compact semi-group is a projective limit of finite semi-groups.

Certain generalizations of topological semi-groups have been considered: semi-groups with a non-Hausdorff space, and semi-topological semi-groups, that is, a topological space on which there is defined an associative binary operation such that all left and right inner translations are continuous mappings.

How to Cite This Entry:
Topological semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_semi-group&oldid=53001
This article was adapted from an original article by B.P. TananaL.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article