Inverse Semigroups of Matrices
Zhu Yong · Journal of Mathematical Research and Exposition · 2008
We discuss some fundamental properties of inverse semigroups of matrices,and prove that the idempotents of such a semigroup constitute 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 2×2 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 2×2 matrices become inverse semigroups.