Image partition regularity of matrices whose entries are homomorphisms

Abstract

A finite matrix A is image partition regular on a semigroup $$(S,+)$$ with identity 0 provided that whenever S is finitely colored, there must be some $$\vec {x}$$ with entries from $$S{\setminus }\{0\}$$ such that all entries of $$A\vec {x}$$ are in some color class. Our aim in this article is to define homomorphism image partition regular and the first entries condition for matrices with entries from homomorphisms. Also we state a conclusion far stronger than the assertion that matrices with entries from homomorphisms satisfying the first entries condition are homomorphism image partition regular. In particular, we represent and work with geometric progressions by means of matrices with entries from homomorphisms.