Rees semi-group of matrix type

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)