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

Slim semimodular lattices were introduced by G. Gratzer and E. Knapp in 2007, and they have intensively been studied since then. These lattices can be given by 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 boolean AND y, x] is a chain, this chain is of a normal slope in every C-1-diagram of L, and except possibly for x, the elements of this chain are meet-reducible. In the direct square K-1 of the three-element chain, let X-1 and A(1) be the set of atoms and the sublattice generated by 0 and the coatoms, respectively. Denote by K-2 the unique eight-element lattice embeddable in K-1. Let A(2) be the sublattice of K-2 consisting of 0, 1, the meet-reducible atom, and the join-reducible coatom. Let X-2 stand for the singleton consisting of the doubly reducible element of K-2. For i = 1, 2, we apply the above-mentioned property of meets to prove that whenever K-i is a sublattice and S-i is a retract of a slim semimodular lattice, then A(i) subset of S-i implies that X-i subset of S-i.