## 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.

### MSC:

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

### Keywords:

functional equation; semilattice; homomorphism; monotone mapping
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### References:

  Birkhoff, G., Lattice Theory (1973), Providence: The American Mathematical Society, Colloquium Publications, Providence  Chajda, T.; Halaš, R.; Kühr, J., Semilattices Structures, Research and Expositions in Mathematics (2007), Lemgo: Heldermann Verlag, Lemgo  Davey, BA; Priestley, HA, Introduction to Lattices and Order (2002), Cambridge: Cambridge University Press, Cambridge  Grätzer, G., Lattice Theory: Foundations (2010), Basel: Springer, Basel  Heubach, S.; Mansour, T., Combinatorics of Compositions of Words (2010), Boca Raton: Chapman & Hall, Boca Raton  Roman, S., Lattices and Ordered Sets (2008), New York: Springer, New York · Zbl 1154.06001  Szász, G., Introduction to Lattice Theory (1963), New York: Academic Press, New York · Zbl 0126.03703
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