Difference between revisions of "Rees semi-group of matrix type"
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− | + | A semi-group theoretical construction defined as follows. Let $ S $ | |
+ | be an arbitrary [[Semi-group|semi-group]], let $ I $ | ||
+ | and $ \Lambda $ | ||
+ | be (index) sets and let $ P = ( p _ {\lambda i } ) $ | ||
+ | be a $ ( \Lambda \times I) $- | ||
+ | matrix over $ S $, | ||
+ | i.e. a mapping from the Cartesian product $ \Lambda \times I $ | ||
+ | into $ S $. | ||
+ | The following formula defines an operation on the set $ M = I \times S \times \Lambda $: | ||
− | + | $$ | |
+ | ( i, s, \lambda ) ( j, t, \mu ) = ( i, sp _ {\lambda j } t, \mu ). | ||
+ | $$ | ||
− | + | Then $ M $ | |
+ | is a semi-group, called a Rees semi-group of matrix type over $ S $ | ||
+ | and denoted by $ {\mathcal M} ( S; I, \Lambda ; P) $; | ||
+ | the matrix $ P $ | ||
+ | is called the sandwich matrix of $ {\mathcal M} ( S; I, \Lambda ; P) $. | ||
+ | If $ S $ | ||
+ | is a semi-group with zero 0, then $ Z = \{ {( i, 0, \lambda ) } : {i \in I, \lambda \in \Lambda } \} $ | ||
+ | is an ideal in $ M = {\mathcal M}( S; I, \Lambda ; P) $ | ||
+ | and the Rees quotient semi-group $ M/Z $( | ||
+ | see [[Semi-group|Semi-group]]) is denoted by $ {\mathcal M} ^ {0} ( S; I, \Lambda ; P) $; | ||
+ | in the case when $ S = G ^ {0} $ | ||
+ | is a group $ G ^ {0} $ | ||
+ | with an adjoined zero, instead of $ {\mathcal M} ^ {0} ( G ^ {0} ; I, \Lambda ; P) $ | ||
+ | one writes $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ | ||
+ | and calls it a Rees semi-group of matrix type over the group $ G ^ {0} $ | ||
+ | with an adjoined zero. The group $ G $ | ||
+ | is called the structure group for the semi-groups $ {\mathcal M} ( G; I, \Lambda ; P) $ | ||
+ | and $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $. | ||
− | where on the right-hand side is the "ordinary" matrix product. This set becomes a semi-group with respect to this operation. The mapping | + | Another representation of the Rees semi-group of matrix type over a semi-group $ S $ |
+ | with zero and $ ( \Lambda \times I) $- | ||
+ | sandwich matrix $ P $ | ||
+ | is realized in the following way. An $ ( I \times \Lambda ) $- | ||
+ | matrix over $ S $ | ||
+ | is called a Rees matrix if it does not contain more than one non-zero element. Let $ \| a \| _ {i \lambda } $ | ||
+ | be the Rees matrix over $ S $ | ||
+ | that has $ a $ | ||
+ | in the $ i $- | ||
+ | th row and $ \lambda $- | ||
+ | th column, and zeros in all other places. On the set of all $ ( I \times \Lambda ) $- | ||
+ | Rees matrices over $ S $ | ||
+ | one can define an operation | ||
+ | |||
+ | $$ \tag{1 } | ||
+ | A \circ B = APB, | ||
+ | $$ | ||
+ | |||
+ | where on the right-hand side is the "ordinary" matrix product. This set becomes a semi-group with respect to this operation. The mapping $ \| a \| _ {i \lambda } \mapsto ( i, a, \lambda ) $ | ||
+ | is an isomorphism between this semi-group and the semi-group $ {\mathcal M} ^ {0} ( S; I, \Lambda ; P) $; | ||
+ | the notation $ {\mathcal M} ^ {0} ( S; I, \Lambda ; P) $ | ||
+ | is used for both of these semi-groups. Formula (1) provides an explanation of the term "sandwich matrix" for $ P $. | ||
+ | If $ G $ | ||
+ | is a group, then the semi-group $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ | ||
+ | is regular if and only if each row and each column of the matrix $ P $ | ||
+ | contains a non-zero element; any semi-group $ {\mathcal M} ( G; I, \Lambda ; P) $ | ||
+ | is completely simple (cf. [[Completely-simple semi-group|Completely-simple semi-group]]), any [[Regular semi-group|regular semi-group]] $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ | ||
+ | is completely $ 0 $- | ||
+ | simple. The converse of the last two statements gives the main content of Rees's theorem [[#References|[1]]]: Any completely-simple (completely $ 0 $- | ||
+ | simple) semi-group can be isomorphically represented as a Rees semi-group of matrix type over a group (as a regular Rees semi-group of matrix type over a group with an adjoined zero). If $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ | ||
+ | and $ {\mathcal M} ^ {0} ( G ^ \prime ; I ^ \prime , \Lambda ^ \prime ; P ^ \prime ) $ | ||
+ | are isomorphic, then the groups $ G $ | ||
+ | and $ G ^ \prime $ | ||
+ | are isomorphic, $ I $ | ||
+ | and $ I ^ \prime $ | ||
+ | have the same cardinality, and $ \Lambda $ | ||
+ | and $ \Lambda ^ \prime $ | ||
+ | have the same cardinality. Necessary and sufficient conditions for isomorphy of the semi-groups $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ | ||
+ | and $ {\mathcal M} ^ {0} ( G ^ \prime ; I ^ \prime , \Lambda ^ \prime ; P ^ \prime ) $ | ||
+ | are known, and together with the just-mentioned conditions they include a quite definite relation between the sandwich matrices $ P $ | ||
+ | and $ P ^ \prime $( | ||
+ | see [[#References|[1]]]–[[#References|[3]]]). In particular, any completely $ 0 $- | ||
+ | simple semi-group can be isomorphically represented as a Rees semi-group of matrix type in whose sandwich matrix $ P $ | ||
+ | each element in a given row and a given column is either 0 or the identity element of the structure group; such a sandwich matrix is called normalized. Similar properties are valid for completely-simple semi-groups. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Rees, "On semi-groups" ''Proc. Cambridge Philos. Soc.'' , '''36''' (1940) pp. 387–400</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc. (1961–1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Rees, "On semi-groups" ''Proc. Cambridge Philos. Soc.'' , '''36''' (1940) pp. 387–400</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc. (1961–1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)</TD></TR></table> |
Latest revision as of 08:10, 6 June 2020
A semi-group theoretical construction defined as follows. Let $ S $
be an arbitrary semi-group, let $ I $
and $ \Lambda $
be (index) sets and let $ P = ( p _ {\lambda i } ) $
be a $ ( \Lambda \times I) $-
matrix over $ S $,
i.e. a mapping from the Cartesian product $ \Lambda \times I $
into $ S $.
The following formula defines an operation on the set $ M = I \times S \times \Lambda $:
$$ ( i, s, \lambda ) ( j, t, \mu ) = ( i, sp _ {\lambda j } t, \mu ). $$
Then $ M $ is a semi-group, called a Rees semi-group of matrix type over $ S $ and denoted by $ {\mathcal M} ( S; I, \Lambda ; P) $; the matrix $ P $ is called the sandwich matrix of $ {\mathcal M} ( S; I, \Lambda ; P) $. If $ S $ is a semi-group with zero 0, then $ Z = \{ {( i, 0, \lambda ) } : {i \in I, \lambda \in \Lambda } \} $ is an ideal in $ M = {\mathcal M}( S; I, \Lambda ; P) $ and the Rees quotient semi-group $ M/Z $( see Semi-group) is denoted by $ {\mathcal M} ^ {0} ( S; I, \Lambda ; P) $; in the case when $ S = G ^ {0} $ is a group $ G ^ {0} $ with an adjoined zero, instead of $ {\mathcal M} ^ {0} ( G ^ {0} ; I, \Lambda ; P) $ one writes $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ and calls it a Rees semi-group of matrix type over the group $ G ^ {0} $ with an adjoined zero. The group $ G $ is called the structure group for the semi-groups $ {\mathcal M} ( G; I, \Lambda ; P) $ and $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $.
Another representation of the Rees semi-group of matrix type over a semi-group $ S $ with zero and $ ( \Lambda \times I) $- sandwich matrix $ P $ is realized in the following way. An $ ( I \times \Lambda ) $- matrix over $ S $ is called a Rees matrix if it does not contain more than one non-zero element. Let $ \| a \| _ {i \lambda } $ be the Rees matrix over $ S $ that has $ a $ in the $ i $- th row and $ \lambda $- th column, and zeros in all other places. On the set of all $ ( I \times \Lambda ) $- Rees matrices over $ S $ one can define an operation
$$ \tag{1 } A \circ B = APB, $$
where on the right-hand side is the "ordinary" matrix product. This set becomes a semi-group with respect to this operation. The mapping $ \| a \| _ {i \lambda } \mapsto ( i, a, \lambda ) $ is an isomorphism between this semi-group and the semi-group $ {\mathcal M} ^ {0} ( S; I, \Lambda ; P) $; the notation $ {\mathcal M} ^ {0} ( S; I, \Lambda ; P) $ is used for both of these semi-groups. Formula (1) provides an explanation of the term "sandwich matrix" for $ P $. If $ G $ is a group, then the semi-group $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ is regular if and only if each row and each column of the matrix $ P $ contains a non-zero element; any semi-group $ {\mathcal M} ( G; I, \Lambda ; P) $ is completely simple (cf. Completely-simple semi-group), any regular semi-group $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ is completely $ 0 $- simple. The converse of the last two statements gives the main content of Rees's theorem [1]: Any completely-simple (completely $ 0 $- simple) semi-group can be isomorphically represented as a Rees semi-group of matrix type over a group (as a regular Rees semi-group of matrix type over a group with an adjoined zero). If $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ and $ {\mathcal M} ^ {0} ( G ^ \prime ; I ^ \prime , \Lambda ^ \prime ; P ^ \prime ) $ are isomorphic, then the groups $ G $ and $ G ^ \prime $ are isomorphic, $ I $ and $ I ^ \prime $ have the same cardinality, and $ \Lambda $ and $ \Lambda ^ \prime $ have the same cardinality. Necessary and sufficient conditions for isomorphy of the semi-groups $ {\mathcal M} ^ {0} ( G; I, \Lambda ; P) $ and $ {\mathcal M} ^ {0} ( G ^ \prime ; I ^ \prime , \Lambda ^ \prime ; P ^ \prime ) $ are known, and together with the just-mentioned conditions they include a quite definite relation between the sandwich matrices $ P $ and $ P ^ \prime $( see [1]–[3]). In particular, any completely $ 0 $- simple semi-group can be isomorphically represented as a Rees semi-group of matrix type in whose sandwich matrix $ P $ each element in a given row and a given column is either 0 or the identity element of the structure group; such a sandwich matrix is called normalized. Similar properties are valid for completely-simple semi-groups.
References
[1] | D. Rees, "On semi-groups" Proc. Cambridge Philos. Soc. , 36 (1940) pp. 387–400 |
[2] | A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) |
[3] | E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) |
Rees semi-group of matrix type. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rees_semi-group_of_matrix_type&oldid=12298