Multiplicative bases in matrix algebras

In a finite-dimensional algebra over a field F, a basis B is called a multiplicative basis provided that B ∪ {0} forms a semigroup. We will describe all multiplicative bases of Fn, the full algebra of n × n matrices over a subfield F of the real numbers. Every such basis is associated with a nonsingular zero–one matrix via a lattice order on Fn. This association is a one-to-one correspondence after identification of isomorphic semigroups and identification of the zero–one matrices that have just permuted rows and columns. This correspondence yields an enumeration method for nonequivalent multiplicative bases of Fn. The enumeration is done for n ⩽ 5.