Two embedding theorems for lower bounded lattices (Q1272234)

A finite lattice \(L\) is lower bounded if, for every homomorphism from a finitely generated free lattice into \(L,\) the preimage of an element of \(L\) has a smallest element whenever it is non-empty. Main results: Theorem 1. Every finite lower bounded lattice \(L\) can be embedded in a finite atomistic lower bounded lattice \(L_1\) with \(\big | J(L_1)\big | =\big | J(L)\big | .\) Theorem 2. A finite lattice \(L\) is lower bounded iff \(L\) can be embedded in the lattice \(\text{Sub}(P)\) of all subsemilattices of some finite semilattice \(P.\)