Lattice isomorphisms of associative algebras

Let A be an associative algebra over the field F . We denote by ℒ( A ) the lattice of all subalgebras of A . By an ℒ-isomorphism (lattice isomorphism) of the algebra A onto an algebra B over the same field, we mean an isomorphism of ℒ( A ) onto ℒ( B ). We investigate the extent to which the algebra B is determined by the assumption that it is ℒ-isomorphic to a given algebra A . In this paper, we are mainly concerned with the case in which A is a finite- dimensional semi-simple algebra.