Difference between revisions of "Extension of a semi-group"
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− | + | A semi-group $ S $ | |
+ | containing the given semi-group $ A $ | ||
+ | as a sub-semi-group. One is usually concerned with extensions that are in some way related to the given semi-group $ A $. | ||
+ | The most well-developed theory is that of ideal extensions (those semi-groups containing $ A $ | ||
+ | as an ideal). To each element $ s $ | ||
+ | of an ideal extension $ S $ | ||
+ | of a semi-group $ A $ | ||
+ | are assigned its left and right translations $ \lambda _ {s} $, | ||
+ | $ \rho _ {s} $: | ||
+ | $ \lambda _ {s} x = sx $, | ||
+ | $ x \rho _ {s} = xs $( | ||
+ | $ x \in A $); | ||
+ | let $ \tau = \tau _ {s} = ( \lambda _ {s} , \rho _ {s} ) $. | ||
+ | The mapping $ \tau $ | ||
+ | is a homomorphism of $ S $ | ||
+ | into the translation hull $ T ( A) $ | ||
+ | of $ A $, | ||
+ | and is an isomorphism in the case when $ A $ | ||
+ | is weakly reductive (see [[Translations of semi-groups|Translations of semi-groups]]). The semi-group $ \tau S $ | ||
+ | is called the type of the ideal extension $ S $. | ||
+ | Among the ideal extensions $ S $ | ||
+ | of $ A $, | ||
+ | one can distinguish strong extensions, for which $ \tau S = \tau A $, | ||
+ | and pure extensions, for which $ \tau ^ {-} 1 \tau A = A $. | ||
+ | Every ideal extension of $ A $ | ||
+ | is a pure extension of one of its strong extensions. | ||
− | + | An ideal extension $ S $ | |
+ | of $ A $ | ||
+ | is called dense (or essential) if every homomorphism of $ S $ | ||
+ | that is injective on $ A $ | ||
+ | is an isomorphism. $ A $ | ||
+ | has a maximal dense ideal extension $ D $ | ||
+ | if and only if $ A $ | ||
+ | is weakly reductive. In this case, $ D $ | ||
+ | is unique up to an isomorphism and is isomorphic to $ T ( A) $. | ||
+ | Also, in this case, $ A $ | ||
+ | is called a densely-imbedded ideal in $ D $. | ||
+ | The sub-semi-groups of $ T ( A) $ | ||
+ | containing $ \tau A $, | ||
+ | and only these, are isomorphic to dense ideal extensions of a weakly reductive semi-group $ A $. | ||
− | Another broad area in the theory of extensions of semi-groups is concerned with various problems on the existence of extensions of a semi-group | + | If $ S $ |
+ | is an ideal extension of $ A $ | ||
+ | and if the quotient semi-group $ S/A $ | ||
+ | is isomorphic to $ Q $, | ||
+ | then $ S $ | ||
+ | is called an extension of $ A $ | ||
+ | by $ Q $. | ||
+ | The following cases have been studied extensively: ideal extensions of completely-simple semi-groups, of a group by a completely $ O $- | ||
+ | simple semi-group, of a commutative semi-group with cancellation by a group with added zero, etc. In general, the problem of describing all ideal extensions of a semi-group $ A $ | ||
+ | by $ Q $ | ||
+ | is far from being solved. | ||
+ | |||
+ | Among other types of extensions of $ A $ | ||
+ | one can mention semi-groups that have a congruence with $ A $ | ||
+ | as one of its classes, and in particular the so-called Schreier extensions of a semi-group with identity [[#References|[1]]], which are analogues of Schreier extensions of groups. In studying the various forms of extensions of a semi-group (in particular, for inverse semi-groups), one uses cohomology of semi-groups. | ||
+ | |||
+ | Another broad area in the theory of extensions of semi-groups is concerned with various problems on the existence of extensions of a semi-group $ A $ | ||
+ | that belong to a given class. Thus, any semi-group $ A $ | ||
+ | can be imbedded in a complete semi-group, in a simple semi-group (relative to congruences), or in a bi-simple semi-group with zero and an identity (see [[Simple semi-group|Simple semi-group]]), and any finite or countable semi-group can be imbedded in a semi-group with two generators. Conditions are known under which a semi-group $ A $ | ||
+ | can be imbedded in a semi-group without proper left ideals, in an inverse semi-group (cf. [[Inversion semi-group|Inversion semi-group]]), in a group (see [[Imbedding of semi-groups|Imbedding of semi-groups]]), etc. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1''' , Amer. Math. Soc. (1961)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Petrich, "Introduction to semigroups" , C.E. Merrill (1973)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1''' , Amer. Math. Soc. (1961)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Petrich, "Introduction to semigroups" , C.E. Merrill (1973)</TD></TR></table> |
Latest revision as of 19:38, 5 June 2020
A semi-group $ S $
containing the given semi-group $ A $
as a sub-semi-group. One is usually concerned with extensions that are in some way related to the given semi-group $ A $.
The most well-developed theory is that of ideal extensions (those semi-groups containing $ A $
as an ideal). To each element $ s $
of an ideal extension $ S $
of a semi-group $ A $
are assigned its left and right translations $ \lambda _ {s} $,
$ \rho _ {s} $:
$ \lambda _ {s} x = sx $,
$ x \rho _ {s} = xs $(
$ x \in A $);
let $ \tau = \tau _ {s} = ( \lambda _ {s} , \rho _ {s} ) $.
The mapping $ \tau $
is a homomorphism of $ S $
into the translation hull $ T ( A) $
of $ A $,
and is an isomorphism in the case when $ A $
is weakly reductive (see Translations of semi-groups). The semi-group $ \tau S $
is called the type of the ideal extension $ S $.
Among the ideal extensions $ S $
of $ A $,
one can distinguish strong extensions, for which $ \tau S = \tau A $,
and pure extensions, for which $ \tau ^ {-} 1 \tau A = A $.
Every ideal extension of $ A $
is a pure extension of one of its strong extensions.
An ideal extension $ S $ of $ A $ is called dense (or essential) if every homomorphism of $ S $ that is injective on $ A $ is an isomorphism. $ A $ has a maximal dense ideal extension $ D $ if and only if $ A $ is weakly reductive. In this case, $ D $ is unique up to an isomorphism and is isomorphic to $ T ( A) $. Also, in this case, $ A $ is called a densely-imbedded ideal in $ D $. The sub-semi-groups of $ T ( A) $ containing $ \tau A $, and only these, are isomorphic to dense ideal extensions of a weakly reductive semi-group $ A $.
If $ S $ is an ideal extension of $ A $ and if the quotient semi-group $ S/A $ is isomorphic to $ Q $, then $ S $ is called an extension of $ A $ by $ Q $. The following cases have been studied extensively: ideal extensions of completely-simple semi-groups, of a group by a completely $ O $- simple semi-group, of a commutative semi-group with cancellation by a group with added zero, etc. In general, the problem of describing all ideal extensions of a semi-group $ A $ by $ Q $ is far from being solved.
Among other types of extensions of $ A $ one can mention semi-groups that have a congruence with $ A $ as one of its classes, and in particular the so-called Schreier extensions of a semi-group with identity [1], which are analogues of Schreier extensions of groups. In studying the various forms of extensions of a semi-group (in particular, for inverse semi-groups), one uses cohomology of semi-groups.
Another broad area in the theory of extensions of semi-groups is concerned with various problems on the existence of extensions of a semi-group $ A $ that belong to a given class. Thus, any semi-group $ A $ can be imbedded in a complete semi-group, in a simple semi-group (relative to congruences), or in a bi-simple semi-group with zero and an identity (see Simple semi-group), and any finite or countable semi-group can be imbedded in a semi-group with two generators. Conditions are known under which a semi-group $ A $ can be imbedded in a semi-group without proper left ideals, in an inverse semi-group (cf. Inversion semi-group), in a group (see Imbedding of semi-groups), etc.
References
[1] | A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1 , Amer. Math. Soc. (1961) |
[2] | M. Petrich, "Introduction to semigroups" , C.E. Merrill (1973) |
Extension of a semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_semi-group&oldid=17640