Homomorphisms Between Multiplicative Semigroups of Matrices Over Fields

Suppose F is a field, and n, p are integers with 1≤ p < n. Let Mnp(F) be the multiplicative semigroup of all n × n matrices over F, and let Mnp(F) be its subsemigroup consisting of all matrices with rank p at most. Assume that F and R are subsemigroups of Mnp(F) such that . A map f:F→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 Mn(F)) is described.