View source for 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) $.

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====
<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>