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A set equipped with both the algebraic structure of a semi-group and the structure of a topological [[Hausdorff space|Hausdorff space]], such that the semi-group operation is continuous in the given topology. Any [[Semi-group|semi-group]] is a topological semi-group in the [[Discrete topology|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.
 
A set equipped with both the algebraic structure of a semi-group and the structure of a topological [[Hausdorff space|Hausdorff space]], such that the semi-group operation is continuous in the given topology. Any [[Semi-group|semi-group]] is a topological semi-group in the [[Discrete topology|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|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.
 
Several independent branches of topological semi-groups have emerged: the general theory of compact semi-groups (cf. [[Compactness|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931201.png" /> is called weakly uniform if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931202.png" /> (one of these elements may be the empty symbol) and any subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931204.png" /> is an open subset with compact closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931206.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931207.png" />, there exist neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t0931209.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312012.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312013.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312014.png" /> is a group, then the mapping of taking the inverse is continuous, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312015.png" /> is a [[Topological group|topological group]]. In a locally compact inverse semi-group, this mapping (cf. [[Regular element|Regular element]]) is continuous if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312016.png" /> 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.
+
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|topological group]]. In a locally compact inverse semi-group, this mapping (cf. [[Regular element|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312017.png" /> contains a closed kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312018.png" /> (cf. [[Kernel of a semi-group|Kernel of a semi-group]]), which is a completely-simple semi-group. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312019.png" /> has idempotents. The structure of compact, completely-simple (completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312020.png" />-simple) semi-groups is described by a theorem analogous to Rees' theorem on discrete completely-simple (completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312021.png" />-simple) semi-groups (cf. [[Rees semi-group of matrix type|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 [[#References|[10]]].
+
An arbitrary compact semi-group $  S $
 +
contains a closed kernel $  M ( S) $(
 +
cf. [[Kernel of a semi-group|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|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 [[#References|[10]]].
  
A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312022.png" /> is called a thread if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312023.png" /> can be linearly ordered in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312024.png" /> becomes a connected topological semi-group under the order (interval) topology. A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312025.png" /> with zero 0 and identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312026.png" /> is called a standard thread (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312028.png" />-semi-group) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312029.png" /> is a thread and if 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312030.png" /> are the least and largest elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312031.png" />. There is a complete description of standard threads [[#References|[2]]]. A compact semi-group with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312032.png" /> is called irreducible if it is connected and does not contain a proper connected closed sub-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312033.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312035.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312036.png" /> is commutative, the Green equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312037.png" /> (cf. [[Green equivalence relations|Green equivalence relations]]) is a closed congruence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312039.png" /> is a standard thread.
+
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 [[#References|[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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312040.png" /> the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312041.png" /> is a compact monothetic group. The compact monothetic semi-groups have been completely described [[#References|[9]]]. Weakly-uniform monothetic semi-groups are either compact or discrete. There is an example [[#References|[13]]] of a monothetic locally compact semi-group which is neither discrete nor compact.
+
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 [[#References|[9]]]. Weakly-uniform monothetic semi-groups are either compact or discrete. There is an example [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312042.png" />. The set of all characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312043.png" /> forms a commutative topological semi-group with identity with respect to pointwise multiplication (cf. [[Character of a semi-group|Character of a semi-group]]) and the [[Compact-open topology|compact-open topology]]. One says that the (Pontryagin) duality theorem holds for a commutative topological semi-group with identity if the canonical homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312044.png" /> into the semi-group of characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312045.png" /> is a topological isomorphism  "onto" . The duality theorem is true for a commutative compact semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312046.png" /> with identity if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312047.png" /> is an inverse semi-group and its sub-semi-group of idempotents forms a [[Totally-disconnected space|totally-disconnected space]]. Necessary and sufficient conditions have been found [[#References|[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.
+
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|Character of a semi-group]]) and the [[Compact-open topology|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|totally-disconnected space]]. Necessary and sufficient conditions have been found [[#References|[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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312048.png" /> on a topological semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312049.png" /> is called invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312051.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312052.png" />. A topological semi-group is called metric if there exists an invariant metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312053.png" /> inducing the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312054.png" />. 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.
+
An important subclass of commutative compact semi-groups are the compact semi-lattices (cf. [[Idempotents, semi-group of|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.
 
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.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.B. Paalman-de Miranda,  "Topological semigroups" , Math. Centre , Amsterdam  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.H. Hofmann,  P.S. Mostert,  "Elements of compact semigroups" , C.E. Merrill  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Berglund,  K. Hofmann,  "Compact semitopological semigroups and weakly almost periodic functions" , Springer  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Hofmann,  M. Mislove,  A. Stralka,  "The Pontryagin duality of compact 0-dimensional semilattices and its application" , Springer  (1974)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K. Hofmann,  A. Stralka,  "The algebraic theory of compact Lawson semilattices. Applications of Galois connections to compact semilattices"  ''Diss. Math.'' , '''137'''  (1976)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Hofmann,  "Topological semigroups: history, theory, applications"  ''Jahresber. Deutsch. Math.-Verein.'' , '''78'''  (1976)  pp. 9–59</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.D. Wallace,  "The structure of topological semigroups"  ''Bull. Amer. Math. Soc.'' , '''61'''  (1955)  pp. 95–112</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.H. Williamson,  "Harmonic analysis on semigroups"  ''J. London Math. Soc.'' , '''42'''  (1967)  pp. 1–41</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  E. Hewitt,  "Compact monothetic semigroups"  ''Duke Math. J.'' , '''23'''  (1956)  pp. 447–457</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  L.B. Shneperman,  "The Rees theorem for weakly uniform semigroups"  ''Semigroup Forum'' , '''23'''  (1981)  pp. 261–273</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  D. Day,  "Expository lectures on topological semigroups"  M.A. Arbib (ed.) , ''Algebraic Theory of Machines, Languages and Semigroups'' , Acad. Press  (1968)  pp. 269–296</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  E.G. Zelen'yuk,  "On Pontryagin's alternative for topological semigroups"  ''Mat. Zametki'' , '''44''' :  3  (1988)  pp. 402–403  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.B. Paalman-de Miranda,  "Topological semigroups" , Math. Centre , Amsterdam  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.H. Hofmann,  P.S. Mostert,  "Elements of compact semigroups" , C.E. Merrill  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Berglund,  K. Hofmann,  "Compact semitopological semigroups and weakly almost periodic functions" , Springer  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Hofmann,  M. Mislove,  A. Stralka,  "The Pontryagin duality of compact 0-dimensional semilattices and its application" , Springer  (1974)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K. Hofmann,  A. Stralka,  "The algebraic theory of compact Lawson semilattices. Applications of Galois connections to compact semilattices"  ''Diss. Math.'' , '''137'''  (1976)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Hofmann,  "Topological semigroups: history, theory, applications"  ''Jahresber. Deutsch. Math.-Verein.'' , '''78'''  (1976)  pp. 9–59</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.D. Wallace,  "The structure of topological semigroups"  ''Bull. Amer. Math. Soc.'' , '''61'''  (1955)  pp. 95–112</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.H. Williamson,  "Harmonic analysis on semigroups"  ''J. London Math. Soc.'' , '''42'''  (1967)  pp. 1–41</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  E. Hewitt,  "Compact monothetic semigroups"  ''Duke Math. J.'' , '''23'''  (1956)  pp. 447–457</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  L.B. Shneperman,  "The Rees theorem for weakly uniform semigroups"  ''Semigroup Forum'' , '''23'''  (1981)  pp. 261–273</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  D. Day,  "Expository lectures on topological semigroups"  M.A. Arbib (ed.) , ''Algebraic Theory of Machines, Languages and Semigroups'' , Acad. Press  (1968)  pp. 269–296</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  E.G. Zelen'yuk,  "On Pontryagin's alternative for topological semigroups"  ''Mat. Zametki'' , '''44''' :  3  (1988)  pp. 402–403  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====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|Continuous lattice]]) and the Lie theory of semi-groups.
 
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|Continuous lattice]]) and the Lie theory of semi-groups.
  
A right-topological semi-group is a semi-group in which all translations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312055.png" /> 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. [[#References|[a9]]], [[#References|[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. [[#References|[3]]], [[#References|[a1]]], [[#References|[a2]]], and [[Almost-periodic function on a group|Almost-periodic function on a group]]), and they arise as compactifications of Lie groups (cf. [[#References|[a9]]], [[#References|[a11]]], and [[Lie group|Lie group]]). Harmonic analysis and representation theory call for semi-topological semi-groups too (cf. [[#References|[a3]]], [[#References|[a4]]]). Right-topological semi-groups emerge in [[Topological dynamics|topological dynamics]] (cf. [[#References|[a5]]], [[#References|[a9]]], [[#References|[a11]]]), and, since the Stone–Čech compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312056.png" /> of the additive semi-group of natural numbers is a right-topological semi-group, in number theory (Ramsey theory, cf. [[Ramsey theorem|Ramsey theorem]]). The existence of idempotents in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312057.png" /> has been used for a new proof of the [[Van der Waerden theorem|van der Waerden theorem]] on arithmetic progressions (cf. [[#References|[a9]]]).
+
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. [[#References|[a9]]], [[#References|[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. [[#References|[3]]], [[#References|[a1]]], [[#References|[a2]]], and [[Almost-periodic function on a group|Almost-periodic function on a group]]), and they arise as compactifications of Lie groups (cf. [[#References|[a9]]], [[#References|[a11]]], and [[Lie group|Lie group]]). Harmonic analysis and representation theory call for semi-topological semi-groups too (cf. [[#References|[a3]]], [[#References|[a4]]]). Right-topological semi-groups emerge in [[Topological dynamics|topological dynamics]] (cf. [[#References|[a5]]], [[#References|[a9]]], [[#References|[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|Ramsey theorem]]). The existence of idempotents in $  \beta \mathbf N $
 +
has been used for a new proof of the [[Van der Waerden theorem|van der Waerden theorem]] on arithmetic progressions (cf. [[#References|[a9]]]).
  
In [[#References|[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|continuous lattice]]. Therefore, the theory of compact semi-lattices is linked with the theory of continuous lattices and its generalizations.
+
In [[#References|[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|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. [[#References|[a8]]], [[#References|[a9]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312058.png" /> is a sub-semi-group of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312059.png" /> with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312060.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312061.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312062.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312063.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312064.png" /> is a convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312065.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312066.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312068.png" />. Such cones are called Lie wedges. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312069.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312070.png" /> as a Lie algebra, then the semi-group algebraically generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312071.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312072.png" /> contains inner points with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312073.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312074.png" /> is invariant under all inner automorphisms, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312075.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312076.png" />. Such cones are called invariant. Invariant pointed cones <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312077.png" /> with inner points exist in a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312078.png" /> only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312079.png" /> contains a compactly imbedded [[Cartan subalgebra|Cartan subalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312080.png" />; in this case they can be classified with the aid of the intersections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093120/t09312081.png" /> (cf. [[#References|[a9]]]). S. Lie's fundamental theorems have analogues in the Lie theory of semi-groups (cf. [[#References|[a9]]], [[#References|[a11]]]). The Lie theory of semi-groups is applied in such areas as chronogeometry in general relativity (cf. [[#References|[a7]]]), non-linear control theory on manifolds and Lie groups (cf. [[#References|[a9]]]) and representation theory (cf. [[#References|[a9]]]).
+
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. [[#References|[a8]]], [[#References|[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|Cartan subalgebra]] $  \mathfrak h $;  
 +
in this case they can be classified with the aid of the intersections $  W \cap \mathfrak h $(
 +
cf. [[#References|[a9]]]). S. Lie's fundamental theorems have analogues in the Lie theory of semi-groups (cf. [[#References|[a9]]], [[#References|[a11]]]). The Lie theory of semi-groups is applied in such areas as chronogeometry in general relativity (cf. [[#References|[a7]]]), non-linear control theory on manifolds and Lie groups (cf. [[#References|[a9]]]) and representation theory (cf. [[#References|[a9]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.F. Berglund,  H.D. Junghenn,  P. Milnes,  "Compact right topological semigroups and generalizations of almost periodicity" , ''Lect. notes in math.'' , '''663''' , Springer  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.F. Berglund,  H.D. Junghenn,  P. Milnes,  "Analysis on semigroups" , Wiley  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.D. Dunkl,  D. Ramirez,  "Representations of commutative semitopological semigroups" , ''Lect. notes in math.'' , '''435''' , Springer  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.A.M. Dzinotyiweyi,  "The analogue of the group algebra for topological semigroups" , Pitman  (1984)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Ellis,  "Lectures in topological dynamics" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Gierz,  K.H. Hofmann,  K. Keimel,  J.D. Lawson,  M.V. Mislove,  D.S. Scott,  "A compendium of continuous lattices" , Springer  (1980)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Hilgert,  K.H. Hofmann,  "The causal structure of homogeneous manifolds"  ''Math. Scand.'' , '''67'''  (1990)  pp. 119–144</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Hilgert,  K.H. Hofmann,  J.D. Lawson,  "Lie groups, convex cones, and topological theory of semigroups" , Oxford Univ. Press  (1989)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  K.H. Hofmann (ed.)  J.D. Lawson (ed.)  J.S. Pym (ed.) , ''The analytical and topological theory of semigroups'' , de Gruyter  (1990)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A. Mukherjea,  N. Tserpes,  "Measures on topological semigroups" , ''Lect. notes in math.'' , '''547''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  W.A.F. Ruppert,  "Compact semitopological semigroups: an intrinsic theory" , ''Lect. notes in math.'' , '''1079''' , Springer  (1984)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.F. Berglund,  H.D. Junghenn,  P. Milnes,  "Compact right topological semigroups and generalizations of almost periodicity" , ''Lect. notes in math.'' , '''663''' , Springer  (1978)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.F. Berglund,  H.D. Junghenn,  P. Milnes,  "Analysis on semigroups" , Wiley  (1989)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  C.D. Dunkl,  D. Ramirez,  "Representations of commutative semitopological semigroups" , ''Lect. notes in math.'' , '''435''' , Springer  (1975)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  H.A.M. Dzinotyiweyi,  "The analogue of the group algebra for topological semigroups" , Pitman  (1984)</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Ellis,  "Lectures in topological dynamics" , Benjamin  (1969)</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  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}}</TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Hilgert,  K.H. Hofmann,  "The causal structure of homogeneous manifolds"  ''Math. Scand.'' , '''67'''  (1990)  pp. 119–144</TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Hilgert,  K.H. Hofmann,  J.D. Lawson,  "Lie groups, convex cones, and topological theory of semigroups" , Oxford Univ. Press  (1989)</TD></TR>
 +
<TR><TD valign="top">[a9]</TD> <TD valign="top">  K.H. Hofmann (ed.)  J.D. Lawson (ed.)  J.S. Pym (ed.) , ''The analytical and topological theory of semigroups'' , de Gruyter  (1990)</TD></TR>
 +
<TR><TD valign="top">[a10]</TD> <TD valign="top">  A. Mukherjea,  N. Tserpes,  "Measures on topological semigroups" , ''Lect. notes in math.'' , '''547''' , Springer  (1976)</TD></TR>
 +
<TR><TD valign="top">[a11]</TD> <TD valign="top">  W.A.F. Ruppert,  "Compact semitopological semigroups: an intrinsic theory" , ''Lect. notes in math.'' , '''1079''' , Springer  (1984)</TD></TR>
 +
</table>

Latest revision as of 14:36, 19 March 2023


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)
How to Cite This Entry:
Topological semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_semi-group&oldid=12760
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