Suppose <b>F</b> is a field, and n,p are integers with 1≤ p<n. Let M_n(<b> F</b>) be the multiplicative semigroup of all n× n matrices over <b> F</b>, and let M_n^p(<b> F</b>) be its subsemigroup consisting of all matrices with rank p at most. Assume that F and R are subsemigroups of M_n(<b> F</b>)
such that F\supseteq M_n^p(<b> F</b>). A map f:F \rightarrow R is called a homomorphism if f(AB)=f(A)f(B) for any A,B∈ F. In particular, f is called an endomorphism if F= R. The structure of all homomorphisms from F to R (respectively, all endomorphisms of M_n(<b> F</b>)) is described.