On finite lattices which are embeddable in subsemigroup lattices
A lattice \(L\) is called lower bounded, if there exists a homomorphism \(f\) of a free lattice onto \(L\) such that for each \(a \in L\) the set \(f^{- 1}(a)\) has a least element. The free commutative nilsemigroup of index two of finite rank \(n\) is denoted by \(T_ n\) and the free semilattice of finite rank \(n\) is denoted by \(E_ n\), their subsemigroup lattices are denoted by \(\text{Sub }T_ n\) and \(\text{Sub }E_ n\). The main theorem is the following: For a finite lattice \(L\) the following conditions are equivalent: (1) \(L\) is embeddable in the subsemigroup lattice of a free semigroup; (2) \(L\) is embeddable in the subsemigroup lattice of a free commutative semigroup; (3) \(L\) is embeddable in the subsemigroup lattice of the infinite cyclic semigroup; (4) \(L\) is embeddable in the subsemigroup lattice of a free commutative nilsemigroup of index two; (5) \(L\) is embeddable in \(\text{Sub }T_ n\) for some positive integer \(n\); (6) \(L\) is embeddable in the subsemigroup lattices of a free semilattice; (7) \(L\) is embeddable in \(\text{Sub }E_ n\) for some positive integer \(n\); (8) \(L\) is embeddable in the subsemigroup lattice of a finite semilattice; (9) \(L\) is embeddable in the subsemigroup lattice of a finite nilpotent semigroup; (10) \(L\) is lower bounded.
free commutative nilsemigroup of index two
scientific article; zbMATH DE number 164976