The homomorphism equation on semilattices. (English) Zbl 1448.39038

Given two semilattices \(S_1\) and \(S_2\) the author finds conditions under which a monotone mapping from \(S_1\) to \(S_2\) is a homomorphism. He shows that this happens only when either \(S_1\) is a chain or \(S_2\) is one-elemented. In the same spirit he poses the problem of understanding when a generic monotone function is a symmetric homomorphism from \(S_2 = C\times C\) to a semilattice \(S\). Here \(C\) is a chain. He shows that this is the case when this mapping is of the form \(f(x) \vee f(z)\) for some isotone mapping \(f\). Next he solves a similar problem, assuming that the monotonous mappings are restricted to principal filters.


39B52 Functional equations for functions with more general domains and/or ranges
06A12 Semilattices
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