Crystal lattices are in nite periodic graphs that occur naturally in a variety of geometries and which are of fundamental importance in polymer science. Discrete models of protein folding use crystal lattices to de ne the space of protein conformations. Because various crystal lattices provide discretizations of the same physical phenomenon, it is reasonable to expect that there will exist \inv...

A mathematical model for conjectures in orthocomplemented lattices is presented. After defining when a conjecture is a consequence or a hypothesis, some operators of conjectures, consequences and hypotheses are introduced and some properties they show are studied. This is the case, for example, of being monotonic or non-monotonic operators. As orthocomplemented lattices contain orthomodular lat...

The worst-case hardness of finding short vectors in ideals of cyclotomic number fields (Ideal-SVP) is a central matter in lattice based cryptography. Assuming the worst-case hardness of Ideal-SVP allows to prove the Ring-LWE and Ring-SIS assumptions, and therefore to prove the security of numerous cryptographic schemes and protocols — including key-exchange, digital signatures, public-key encry...

We consider connections between similar sublattices and coincidence site lattices and, more generally, between similar submodules and coincidence site modules of general (free) Z-modules in R. In particular, we generalise results obtained by S. Glied and M. Baake on similarity and coincidence isometries of lattices and certain lattice-like modules called S-modules. An important result is that t...

In the book, General Lattice Theory, the first author raised the following problem (Problem II.18): Let L be a nontrivial lattice and let G be a group. Does there exist a lattice K such that K and L have isomorphic congruence lattices and the automorphism group of K is isomorphic to G? The finite case was solved, in the affirmative, by V.A. Baranskĭı and A. Urquhart in 1978, independently. In 1...

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that nitely-generated varieties of distributive lattices with operators are closed under canonical embedding...

1. Introduction. A typical result of this paper is the following. Let Lt, i e I be distributive lattices satisfying the countable chain condition. Then the free product L of these lattices also satisfies the countable chain condition. To be able to state the general result we need some notations. Let trt be an infinite cardinal. A poset (partially ordered set) P is said to satisfy the m-chain c...

The t-concept lattice is introduced as a set of triples associated to graded tabular information interpreted in a non-commutative fuzzy logic. Following the general techniques of formal concept analysis, and based on the works by Georgescu and Popescu, given a non-commutative conjunctor it is possible to provide generalizations of the mappings for the intension and the extension in two differen...

In this article, by using the method of finite difference time domain (FDTD) and PML boundary conditions, we have studied the photonic band gaps for TE and TM modes in square and triangular lattices consisting of air holes in dielectric medium and compared the results. In addition, the effect of nonlinearity of the photonic crystal background on the photonic band gaps and comparison with the re...

We consider general properties of lattices of relative colour-families and antivarieties. Several results generalise the corresponding assertions about colour-families of undirected loopless graphs, see [1]. Conditions are indicated under which relative colour-families form a lattice. We prove that such a lattice is distributive. In the class of lattices of antivarieties of relation structures ...