The classification of maximal inverse monoids of matrices.

Let F be a field. The set M n (F) of all n × n matrices over F forms a monoid under usual matrix multiplication. We explore and classify all maximal inverse monoids in M n (F) . Given a partition of { 1 , ... , n } , we construct a maximal matrix inverse monoid as a direct sum of some block matrices and show that every maximal inverse monoid is such a matrix inverse monoid associated to a partition of { 1 , ... , n } up to isomorphism. [ABSTRACT FROM AUTHOR]