# 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 : 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 ). 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.

How to Cite This Entry:
Rees semi-group of matrix type. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rees_semi-group_of_matrix_type&oldid=48466
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article