How can i create a Rees-matrix semigroup?

Question

I've already read the definition of Rees-matrix semigroups, but I still can not imagine how to create one. For example.: I shall create a M(G;I;Lambda;P) semigroup where G is a group with 2 elements, the sets of indexes containe 2 elements as well and the P sandwitch matrix has NO zero element.

It will generate 2x2 matrices, but which matrices, and the most important: HOW CAN I GET THEM?

Thank You for reading this

Answer 1

Let $I$ and $J$ be two sets, let $G$ be a group and let $P$ be a $J \times I$ matrix with entries in $G$. The Rees-matrix semigroup $M(G,I,J,P)$ is defined on the set $I \times G \times J$ by the product $$ (*) \quad (i,g,j)(i',g',j') = (i, gP_{j,i'}g', j') $$ It is also equal to the monoid of $I \times J$ matrices with a single entry in $G$ (the other entries being null), where the product of the matrices $M$ and $M'$ is the matrix $MPM'$ (which justifies the term sandwich matrix for $P$). For instance, if $G$ is the multiplicative group $\{1, -1\}$, if $I = J = \{1,2\}$ and $P = \pmatrix{1&-1\\1&1}$, then $$ M(G,I,J,P) = \biggl\{\pmatrix{1&0\\0&0}, \pmatrix{-1&0\\0&0}, \pmatrix{0&1\\0&0}, \pmatrix{0&-1\\0&0}, \pmatrix{0&0\\1&0}, \pmatrix{0&0\\-1&0}, \pmatrix{0&0\\0&1}, \pmatrix{0&0\\0&-1}\biggr\} $$

Answer 2

is that ok to generate a group G={e,a} where ea=a=ae and aa=e, adjungate an zero element, so G0={0,e,a}?? Then the semigroup elements are 2x2 matrices, with only one non-zero element. So (a_ij)= a or e for all i and j in I and Lambda. So we get 8 matrixes and the 0 matrix. The sandwitch matrix can be any of thoae 16 2x2 matrices which are includes only a and e elements.

IDK is that ok or not