In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...

We have succeeded in applying Ziglin’s test on non-integrability to nonhomogeneous nonlinear lattices (Fermi-Pasta-Ulam lattices). By explicit calculations of the eigenvalues of the monodromy matrices concerning the normal variational equations of Lamé type and with the use of the non-resonance hypothesis about the eigenvalues, we obtained a theorem proving the nonexistence of additional analyt...

the starting point of this paper is given by priestley’s papers, where a theory of representation of distributive lattices is presented. the purpose of this paper is to develop a representation theory of fuzzy distributive lattices in the finite case. in this way, some results of priestley’s papers are extended. in the main theorem, we show that the category of finite fuzzy priestley space...

Based on an acyclic orientation of the Z-transformation graph, a finite distributive lattice (FDL for short)M(G) is established on the set of all 1-factors of a plane (weakly) elementary bipartite graph G. For an FDL L, if there exists a plane bipartite graph G such that L is isomorphic to M(G), then L is called a matchable FDL. A natural question arises: Is every FDL a matchable FDL? In this p...

Non-Hermitian photonic lattices combine the peculiar consequences of energy non-conservation with physics bandstructures, giving rise to a variety exotic properties not found in conventional materials or metamaterials. In this tutorial, we introduce key concepts design and implementation non-Hermitian lattices, including general features lattice Hamiltonians their role symmetries, topological c...

Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem  2.8}). We shall determine the structure of special elements (which are introduced after  Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of ...

Introduction and summary. In the theory of non-commutative rings certain distinguished subrings, one-sided and two-sided ideals, play the important roles. Ideals combine under crosscut, union and multiplication and hence are an instance of a lattice over which a non-commutative multiplication is defined.f The investigation of such lattices was begun by W. Krull (Krull [3]) who discussed decompo...

Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 subsets of 2 , under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of Pw. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of PM .