Lattices embeddable in subsemigroup lattices. I. Semilattices

Author: Semenova M.  

Publisher: Springer Publishing Company

ISSN: 0002-5232

Source: Algebra and Logic, Vol.45, Iss.2, 2006-03, pp. : 124-133

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Abstract

V. B. Repnitskii showed that any lattice embeds in some subsemilattice lattice. In his proof, use was made of a result by D. Bredikhin and B. Schein, stating that any lattice embeds in the suborder lattice of a suitable partial order. We present a direct proof of Repnitskii’s result, which is independent of Bredikhin—Schein’s, giving the answer to a question posed by L. N. Shevrin and A. J. Ovsyannikov. We also show that a finite lattice is lower bounded iff it is isomorphic to the lattice of subsemilattices of a finite semilattice that are closed under a distributive quasiorder.