Isomorphisms and automorphisms of matrix algebras over lattices
In this paper, we consider the multiplicative groupoid of matrices with elements in a lattice with 0 and 1. Examples of such groupoids are the semigroup of binary relations and semigroups of minimax (fuzzy) relations. It is shown that every automorphism of a groupoid is the composition of an inner automorphism and the automorphism defined by an automorphism of the lattice. Despite the fact that, in general, the groupoid is not associative, it satisfies the UA-property: every multiplicative automorphism is an additive automorphism. Earlier, the realization of the UA-property has been considered mainly for associative rings and semirings. We describe the invertible matrices that define inner automorphisms.