The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: A ⊂ B,C ⊂ D. A diamondfree family in the n-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements B and C may or may not be related. There is a diamond-free family in the n-dimensional Boolean lattice of size (2−o(1)) (...

Many tiling spaces such as domino tilings of fixed figures have an underlying lattice structure. This lattice structure corresponds to the dynamics induced by flips. In this paper, we further investigate the properties of these lattices of tilings. In particular, we point out a stability property: the set of all the shortest sequences of flips joining to fixed tilings also have a lattice struct...

The syntactic concept lattice is a residuated lattice associated with a given formal language; it arises naturally as a generalisation of the syntactic monoid in the analysis of the distributional structure of the language. In this paper we define the syntactic concept lattice and present its basic properties, and its relationship to the universal automaton and the syntactic congruence; we cons...

We investigate how the chain property for the associated primes of monomial degenerations of toric (or lattice) ideals can be generalized to arbitrary A-graded ideals. The generalization works in dimension d = 2, but it fails for d ≥ 3.

In various dimensional euclidean lattice gauge theories, we examine a compatibility of the Majorana decomposition and the charge conjugation property of lattice Dirac operators. In 8n and 1+8n dimensions, we find a difficulty to decompose a classical lattice action of the Dirac fermion into a system of the Majorana fermion and thus to obtain a factrized form of the Dirac determinant. Similarly,...

in this paper, our purpose is twofold. firstly, the tensor andresiduum operations on $l-$nested systems are introduced under thecondition of complete residuated lattice. then we show that$l-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. thus the new representation theorem of$l-$subsets on complete re...

The study of the lattice E of computably enumerable sets under inclusion has been one of the central tasks of computability theory since the 1960's. We investigate initial segments of the lattice L of 0 1 classes (of sets) under inclusion and we compare this lattice with E. It was recently proved by Nies that the theory of each interval of the lattice E which is not a boolean algebra interprets...

Through the current work, usefulness of concept architectured rod lattices based on unit cell motifs designed at mesoscale is demonstrated. Specifically, 2D triangular with cells containing different numbers rods are considered. Combinations two types provide explored a greater complexity and versatility. For mesocells large number variable parameters, it virtually impossible to calculate entir...