ON f-DERIVATIONS FROM SEMILATTICES TO LATTICES
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In this paper, we introduce the notion of f-derivations from a semilattice S to a lattice L, as a generalization of derivation and f- derivation of lattices. Also, we define the simple f-derivation from S to L, and research the properties of them and the conditions for a lattice L to be distributive. Finally, we prove that a distributive lattice L is isomorphic to the class SDf(S,L) of all simple f-derivations on S to L for every Λ-homomorphism f : S → L such that f(x0) ∨ f(y0) = 1 for some x0, y0 ∈ S, in particular, L = SDf (S, L) for every Λ-homomorphism f : S → L such that f(x0) = 1 for some x0 ∈ S.
Abstract
In this paper, we introduce the notion of f-derivations from a semilattice S to a lattice L, as a generalization of derivation and f- derivation of lattices. Also, we define the simple f-derivation from S to L, and research the properties of them and the conditions for a lattice L to be distributive. Finally, we prove that a distributive lattice L is isomorphic to the class SDf(S,L) of all simple f-derivations on S to L for every Λ-homomorphism f : S → L such that f(x0) ∨ f(y0) = 1 for some x0, y0 ∈ S, in particular, L = SDf (S, L) for every Λ-homomorphism f : S → L such that f(x0) = 1 for some x0 ∈ S.
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