The homomorphism equation on semilattices. (English) Zbl 1448.39038

Berrone, Lucio R.

Aequationes Math. 94, No. 5, 803-816 (2020).

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.

Reviewer: Ivan Chajda (Přerov)

MSC:

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

Keywords:

functional equation; semilattice; homomorphism; monotone mapping

References:

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