In the early forties, R. P. Dilworth proved his famous result: Every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In one of our early papers, we presented the first published proof of this result; in fact we proved: Every finite distributive lattice D can be represented as the congruence lattice of a finite sectionally complemented lattice L....

in this article, we suggested a novel design of polarization splitter based on coupler waveguide on inp substrate at 1.55mm wavelength. photonic crystal structure is consisted of two dimensional (2d) air holes embedded in inp/ingaasp material with an effective refractive index of 3.2634 which is arranged in a hexagonal lattice. the photonic band gap (pbg) of this structure is determined using t...

Let L be a finite geometric lattice of rank n with rank function r. (For definitions, see e.g., [3, Chapter 2], [4], or [1, Chapter 4].) An element x s L is called a modular element if it forms a modular pair with every y e L , i.e., if a<~y then a V ( x A y ) = (a v x )Ay . Recall that in an upper semimodular lattice (and thus in a geometric lattice) the relation of being a modular pair is sym...

We study abstract properties of intervals in the complete lattice of all meet-closed subsets ( -subsemilattices) of a -(meet-)semilattice S, where is an arbitrary cardinal number. Any interval of that kind is an extremally detachable closure system (that is, for each closed set A and each point x outside A, deleting x from the closed set generated by A and x leaves a closed set). Such closure s...

Introduction. Let L be a complete, orthocomplemented lattice. We say that L is a dimension lattice if L is weakly modular and there is an equivalence relation on L satisfying the axioms A,B,C, and D' of Loomis [5]. We say that L is locally finite if every element of L is the join of finite elements. If L is a dimension lattice in which every element is finite, then L is modular. Conversely, Kap...

Flow in planar graphs has been extensively studied, and very efficient algorithms have been developed to compute max-flows, min-cuts, and circulations. Intimate connections between solutions to the planar circulation problem and with "consistent" potential functions in the dual graph are shown. It is also shown that the set of integral circulations in a planar graph very naturally forms a distr...

Introduction. The property of the Petersen graph of being hypohamiltonian is notorious, and known for a long time. The existence of hypotraceable graphs, on the contrary, was discovered much later. Before that, in 1966, Gallai [1] had asked the easier question whether (connected) graphs do exist such that each vertex is missed by some longest path. Such graphs have been discovered by Walther in...

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence latt...