REDUCING THE LENGTHS OF SLIM PLANAR SEMIMODULAR LATTICES WITHOUT CHANGING THEIR CONGRUENCE LATTICES

Following G. Gr & auml;tzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no M- 3 as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset P is said to be JConSPS-representable if there is an SPS lattice L such that P is isomorphic to the poset J(ConL) of join-irreducible congruences of L. We prove that if 1 < n is an element of N and P is an n- element JConSPS-representable poset, then there exists a slim rectangular lattice L such that J(ConL) congruent to = P, the length of L is at most 2n(2), and |L|<= 4n(4). This offers an algorithm to decide whether a finite poset P is JConSPS-representable (or a finite distributive lattice is "ConSPS-representable"). This algorithm is slow as G. Cz & eacute;dli, T. D & eacute;k & aacute;ny, G. Gyenizse, and J. Kulin proved in 2016 that there are asymptotically 1/2 (k - 2)! e(2) slim rectangular lattices of a given length k, where e is the famous constant approximate to 2.71828. The known properties and constructions of JConSPS-representable posets can accelerate the algorithm; we present a new construction.