# 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 [10].

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 [2]. 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 [9]. Weakly-uniform monothetic semi-groups are either compact or discrete. There is an example [13] 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 [12] 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.

#### References

[1] | A.B. Paalman-de Miranda, "Topological semigroups" , Math. Centre , Amsterdam (1964) |

[2] | K.H. Hofmann, P.S. Mostert, "Elements of compact semigroups" , C.E. Merrill (1966) |

[3] | J. Berglund, K. Hofmann, "Compact semitopological semigroups and weakly almost periodic functions" , Springer (1967) |

[4] | K. Hofmann, M. Mislove, A. Stralka, "The Pontryagin duality of compact 0-dimensional semilattices and its application" , Springer (1974) |

[5] | K. Hofmann, A. Stralka, "The algebraic theory of compact Lawson semilattices. Applications of Galois connections to compact semilattices" Diss. Math. , 137 (1976) |

[6] | K. Hofmann, "Topological semigroups: history, theory, applications" Jahresber. Deutsch. Math.-Verein. , 78 (1976) pp. 9–59 |

[7] | A.D. Wallace, "The structure of topological semigroups" Bull. Amer. Math. Soc. , 61 (1955) pp. 95–112 |

[8] | J.H. Williamson, "Harmonic analysis on semigroups" J. London Math. Soc. , 42 (1967) pp. 1–41 |

[9] | E. Hewitt, "Compact monothetic semigroups" Duke Math. J. , 23 (1956) pp. 447–457 |

[10] | L.B. Shneperman, "The Rees theorem for weakly uniform semigroups" Semigroup Forum , 23 (1981) pp. 261–273 |

[11] | D. Day, "Expository lectures on topological semigroups" M.A. Arbib (ed.) , Algebraic Theory of Machines, Languages and Semigroups , Acad. Press (1968) pp. 269–296 |

[12] | L.B. Shneperman, "On the theory of characters of locally bicompact topological semigroups" Math. USSR Sb. , 6 : 4 (1968) pp. 471–492 Mat. Sb. , 77 : 4 (1968) pp. 508–532 |

[13] | E.G. Zelen'yuk, "On Pontryagin's alternative for topological semigroups" Mat. Zametki , 44 : 3 (1988) pp. 402–403 (In Russian) |

#### Comments

In the years since 1970, the study of topological semi-groups has followed various main trends: compact semi-topological and right- (respectively, left-) topological semi-groups, compact semi-lattices and continuous lattices (cf. Continuous lattice) and the Lie theory of semi-groups.

A right-topological semi-group is a semi-group in which all translations $ x \mapsto xs $ are continuous. (Some authors use the opposite notation.) Compact semi-topological semi-groups and compact right-topological semi-groups, like topological semi-groups, contain idempotents and possess completely simple kernels (minimal two-sided ideals), but, in contrast to compact topological semi-groups, these need no longer be closed. The existence of a kernel in a compact topological semi-group has been used in probability theory on topological groups and semi-groups (cf. [a9], [a10]). Compact semi-topological semi-groups occur as semi-groups of linear operators in the strong operator topology and are crucial in the theory of weakly almost-periodic functions on a topological group or semi-group (cf. [3], [a1], [a2], and Almost-periodic function on a group), and they arise as compactifications of Lie groups (cf. [a9], [a11], and Lie group). Harmonic analysis and representation theory call for semi-topological semi-groups too (cf. [a3], [a4]). Right-topological semi-groups emerge in topological dynamics (cf. [a5], [a9], [a11]), and, since the Stone–Čech compactification $ \beta \mathbf N $ of the additive semi-group of natural numbers (cf. Cech–Stone compactification of omega) is a right-topological semi-group, in number theory (Ramsey theory, cf. Ramsey theorem). The existence of idempotents in $ \beta \mathbf N $ has been used for a new proof of the van der Waerden theorem on arithmetic progressions (cf. [a9]).

In [5] it was recognized that the concept of a compact semi-lattice in which every element has a neighbourhood base of sub-semi-lattices agrees with the concept of a continuous lattice. Therefore, the theory of compact semi-lattices is linked with the theory of continuous lattices and its generalizations.

The Lie theory of semi-groups deals with sub-semi-groups of Lie groups and with topological semi-groups which can be imbedded into a Lie group, at least locally about their identity element (cf. [a8], [a9]). If $ S $ is a sub-semi-group of a Lie group $ G $ with Lie algebra $ \mathfrak g = L( G) $, then the set $ L( S) $ of all $ X \in \mathfrak g $ with $ \mathop{\rm exp} t \cdot X \in \overline{S}\; $ for all $ t \geq 0 $ is a convex cone $ W $ satisfying $ e ^ { \mathop{\rm ad} X } W = W $ for all $ X \in W \cap - W $, where $ ( \mathop{\rm ad} X )( Y) = [ X, Y] $. Such cones are called Lie wedges. If $ W $ generates $ \mathfrak g $ as a Lie algebra, then the semi-group algebraically generated by $ \mathop{\rm exp} W $ in $ \overline{S}\; $ contains inner points with respect to $ G $. If $ S $ is invariant under all inner automorphisms, then $ e ^ { \mathop{\rm ad} X } W = W $ for all $ X \in \mathfrak g $. Such cones are called invariant. Invariant pointed cones $ W $ with inner points exist in a Lie algebra $ \mathfrak g $ only if $ \mathfrak g $ contains a compactly imbedded Cartan subalgebra $ \mathfrak h $; in this case they can be classified with the aid of the intersections $ W \cap \mathfrak h $( cf. [a9]). S. Lie's fundamental theorems have analogues in the Lie theory of semi-groups (cf. [a9], [a11]). The Lie theory of semi-groups is applied in such areas as chronogeometry in general relativity (cf. [a7]), non-linear control theory on manifolds and Lie groups (cf. [a9]) and representation theory (cf. [a9]).

#### References

[a1] | J.F. Berglund, H.D. Junghenn, P. Milnes, "Compact right topological semigroups and generalizations of almost periodicity" , Lect. notes in math. , 663 , Springer (1978) |

[a2] | J.F. Berglund, H.D. Junghenn, P. Milnes, "Analysis on semigroups" , Wiley (1989) |

[a3] | C.D. Dunkl, D. Ramirez, "Representations of commutative semitopological semigroups" , Lect. notes in math. , 435 , Springer (1975) |

[a4] | H.A.M. Dzinotyiweyi, "The analogue of the group algebra for topological semigroups" , Pitman (1984) |

[a5] | R. Ellis, "Lectures in topological dynamics" , Benjamin (1969) |

[a6] | G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.V. Mislove, D.S. Scott, "A compendium of continuous lattices" , Springer (1980) ISBN 3-540-10111-X Zbl 0452.06001 |

[a7] | J. Hilgert, K.H. Hofmann, "The causal structure of homogeneous manifolds" Math. Scand. , 67 (1990) pp. 119–144 |

[a8] | J. Hilgert, K.H. Hofmann, J.D. Lawson, "Lie groups, convex cones, and topological theory of semigroups" , Oxford Univ. Press (1989) |

[a9] | K.H. Hofmann (ed.) J.D. Lawson (ed.) J.S. Pym (ed.) , The analytical and topological theory of semigroups , de Gruyter (1990) |

[a10] | A. Mukherjea, N. Tserpes, "Measures on topological semigroups" , Lect. notes in math. , 547 , Springer (1976) |

[a11] | W.A.F. Ruppert, "Compact semitopological semigroups: an intrinsic theory" , Lect. notes in math. , 1079 , Springer (1984) |

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Topological semi-group.

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