Inverse Semigroups of Matrices
Received:July 18, 2006  Revised:March 22, 2007
Key Words: matrix semigroup   inverse semigroup   Green's relation   Clifford semigroup   semilattice.  
Fund Project:the National Natural Science Foundation of China (No.10571005).
Author NameAffiliation
ZHU Yong Wen School of Mathematics and Information Science, Yantai University, Shandong 264005, China 
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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.
Citation:
DOI:10.3770/j.issn:1000-341X.2008.03.013
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