A property of meets in slim semimodular lattices and its application to retracts

Abstract

Slim semimodular lattices were introduced by G. Grätzer and E. Knapp in 2007, and they have intensively been studied since then. These lattices can be given by \(\mathcal{C}_1\)-diagrams, defined by the author in 2017. We prove that if x and y are incomparable elements in such a lattice L, then their meet has the property that the interval \([x \wedge y, x]\) is a chain, this chain is of a normal slope in every \(\mathcal{C}_1\)-diagram of L, and except possibly for x, the elements of this chain are meet-reducible.In the direct square K1 of the three-element chain, let X1 and A1 be the set of atoms and the sublattice generated by 0 and the coatoms, respectively. Denote by K2 the unique eight-element lattice embeddable in K1. Let A2 be the sublattice of K2 consisting of 0, 1, the meet-reducible atom, and the join-reducible coatom. Let X2 stand for the singleton consisting of the doubly reducible element of K2. For i = 1, 2, we apply the above-mentioned property of meets to prove that whenever Ki is a sublattice and Si is a retract of a slim semimodular lattice, then \(A_i \subseteq S_i\) implies that \(X_i \subseteq S_i\).