Inverse Semigroups of Matrices

Abstract:

      We discuss some fundamental properties of inverse semigroups of matrices, and prove that the idempotents of such a semigroup constitute a subsemilattice of a finite Boolean lattice, and that the inverse semigroups of some matrices with the same rank are groups. At last, we determine completely the construction of the inverse semigroups of some $2\times 2$ matrices: such a semigroup is isomorphic to a linear group of dimension 2 or a null-adjoined group, or is a finite semilattice of Abelian linear groups of finite dimension, or satisfies some other properties. The necessary and sufficient conditions are given that the sets consisting of some $2\times 2$ matrices become inverse semigroups.