Maximal sublattices of finite distributive lattices

Algebraic properties of lattices of quotients of finite posets are considered. Using the known duality between the category of all finite posets together with all order-preserving maps and the category of all finite distributive (0, 1)-lattices together with all (0, 1)-lattice homomorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of finite distributive (0, 1)-lattices are thereby obtained.Presented by E. T. Schmidt.     

An algorithm for determining Φ(L) in finite distributive lattices

Following G. Birkhoff, the Frattini sublattice Φ(L) of a latticeL is defined as the intersection of all its maximal proper sublattices. Let ?(FD) be the class of all finite distributive lattices. The main aim in this note is to provide a new but elementary characterization of elements in Φ(L),L∈?(FD), and also an extremely simple algorithm for determining the Frattini sublattice of any finite distributive lattice. By applying this algorithm, it is shown that there is a new way to determine the Frattini subalgebra of a finite Stone algebra.     

No finite axiomatizations for posets embeddable into distributive lattices

Let m and n be cardinals with $3\le m,n\le \omega$. We show that the class of posets that can be embedded into a distributive lattice via a map preserving all existing meets and joins with cardinalities strictly less than m and n respectively cannot be finitely axiomatized.     

Structural matrix algebras related to finite distributive lattices

Abstract

In this paper, we investigate the relation between a structural matrix algebra and the lattice properties of its lattice of invariant subspaces, and reprove known results in a fresh and an explanatory way. Moreover, we also prove the theorem which is partially converse of Proposition 2.6 of [15].     

About polytopes of valuations on finite distributive lattices

Let L be a finite distributive lattice and V(L) the real vector space of all valuations on L. We verify the conjecture of Geissinger that the extreme points of the convex polytope M(L)={v L : 0 v 1} are precisely the 0–1 valuations.     

A characterization of MV-algebras free over finite distributive lattices

Mundici has recently established a characterization of free finitely generated MV-algebras similar in spirit to the representation of the free Boolean algebra with a countably infinite set of free generators as any Boolean algebra that is countable and atomless. No reference to universal properties is made in either theorem. Our main result is an extension of Mundici’s theorem to the whole class of MV-algebras that are free over some finite distributive lattice.      

An extremal problem for operators

Let A be a Banach algebra, F a compact set in the complex plane, and h a function holomorphic in some neighborhood of the set F. Thus h(a) is meaningful for each element a ε A whose spectrum σ(a) is contained in F, and it is possible to evaluate the norm |h(a)|. Problem: Compute the supremum of the norms |h(a) as a ranges over all elements of A with spectrum contained in F and whose norm does not exceed one; that is, compute sup{|h(a)|; a ε A, σ(a) ⊂ F, |a| ⩽ 1}. This problem was first formulated and treated by the author in the particular case where A is the algebra of all linear operators on a finite-dimensional Hilbert space and F is the disc {z; |z| ⩽ r} for a given positive number r<1. The paper discusses motivation, connections with complex function theory, convergence of iterative processes, critical exponents, and the infinite companion matrix.     

An extremal problem for polynomials

We give a solution to an extremal problem for polynomials, which asks for complex numbers α₀,…,α_n${\alpha }_0,\dots ,{\alpha }_n$ of unit magnitude that minimise the largest supremum norm on the unit circle for all polynomials of degree n whose k  -th coefficient is either α_k${\alpha }_k$ or −α_k$-{\alpha }_k$.     

A Euler-Poincaré characteristic for the open set lattices of finite topologies

Let $\text{T}$ be a finite topology. If P and Q are open sets of $\text{T}$ (Q may be the null set) then P is a minimal cover of Q provided Q ? P and there does not exist any open set R of $\text{T}$ such that Q ? R ? P. A subcollection $\text{D}$ of the open sets of $\text{T}$ is termed an i-discrete collection of $\text{T}$ provided $\text{D}$ contains every open O ∈ $\text{T}$ with the property that ? $\text{D}$ ? O ? ? $\text{D}$, $\text{D}$ contains exactly i minimal covers of ? $\text{D}$, and provided ?$\text{D}$ = ?{O | O ∈ $\text{D}$ and O is a minimal cover of ? $\text{D}$}. A single open set is a O-discrete collection. The number of distinct i-discrete collections of $\text{T}$ is denoted by p($\text{T}$, i). If there does not exist any i-discrete collection then p($\text{T}$,i) = 0, and this happens trivially for the case when i is greater than the number of points on which $\text{T}$ is defined. The object of this article is to establish the theorem: For any finite topology $\text{T}$, the quantity E($\text{T}$) = Σ_{i = 0}^∞ (?1)ⁱp($\text{T}$, i) = 1.