Distributivity of strongly representable lattices (Q1239176)

A lattice is called finitely strongly representable if (1) there is an embedding of it into a finite equivalence lattice and (2) the image of every embedding of it, preserving 0 and 1, in a finite equivalence lattice is a congruence lattice of an algebra. The aim of the paper is to prove the following theorem: If a lattice is finitely strongly representable then it is distributive. It adds, to the previous result of \textit{R. Quackenbush} and \textit{B. Wolk} [Algebra Univers. 1, 165--166 (1971; Zbl 0231.06006)], that the property of finite strong representability is equivalent to distributivity for finite lattices. In the proof, an embedding violating condition (2) is constructed for every non-distributive lattice satisfying (1). This is achieved by joining several copies of the original embedding together.