Department of Mathematics, University of Guilan, Rasht, Guilan, Iran
10.22091/maa.2026.15930.1059
Abstract
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 $S$.
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