On representation of lattices by lattices of semigroups (Q677747)

A class \(K\) of algebras is said to be lattice-wise universal, if every lattice is embeddable into the lattice of subalgebras of a certain algebra from \(K\). The main result of the paper is the theorem which shows three classes of semigroups which are lattice-wise universal; namely the class of all commutative nilsemigroups of index 2, the class of all semilattices and the class of all commutative semigroups without idempotents with reducing and univalent root extracting. A corollary of this theorem says that also the class of all commutative linearly orderable semigroups without idempotents is lattice-wise universal.