Semilattice Morphisms
The above algebraic definition of a semilattice suggests a notion of morphism between two semilattices. Given two join-semilattices 〈S, ∨〉 and 〈T, ∨〉, a homomorphism of (join-) semilattices is a function f: S → T such that
- f(x ∨ y) = f(x) ∨ f(y).
Hence f is just a homomorphism of the two semigroups associated with each semilattice. If S and T both include a least element 0, then f should also be a monoid homomorphism, i.e. we additionally require that
- f(0) = 0.
In the order-theoretic formulation, these conditions just state that a homomorphism of join-semilattices is a function that preserves binary joins and least elements, if such there be. The obvious dual—replacing ∧ with ∨ and 0 with 1—transforms this definition of a join-semilattice homomorphism into its meet-semilattice equivalent.
Note that any semilattice homomorphism is necessarily monotone with respect to the associated ordering relation. For an explanation see the entry preservation of limits.
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