partition regularity of finite sets of homomorphisms

‎in this paper‎, ‎we first establish a necessary and sufficient condition for a finite family of functions to be a ramsey family‎. ‎more precisely‎, ‎we prove that a non-empty finite set $f$ of functions from $t$ to $s$ is a ramsey family if and only if the system of equations‎‎ [‎‎f^beta(p) = y‎, ‎qquad forall f in f‎‎]‎ ‎takes equal values at some point $p in beta s$‎, ‎where $f^beta‎: ‎beta s to beta t$ is the unique continuous extension of $f‎: ‎s to t$‎. ‎we then investigate monochromatic structures associated with homomorphisms‎. ‎furthermore‎, ‎this paper generalizes theorem 15.5 in [4] to homomorphisms on a commutative semigroup $s$‎.