Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices

Let be a {0, 1}-homomorphism of a finite distributive lattice D into the congruence lattice Con L of a rectangular (whence finite, planar, and semimodular) lattice L. We prove that L is a filter of an appropriate rectangular lattice K such that ConK is isomorphic with D and is represented by the restriction map from Con K to Con L. The particular case where is an embedding was proved by E.T. Schmidt. Our result implies that each {0, 1}-lattice homomorphism between two finite distributive lattices can be represented by the restriction of congruences of an appropriate rectangular lattice to a rectangular filter.     

Approximate unitary equivalence of completely distributive commutative subspace lattices

In this note, we classify completely distributive CSLs up to approximate unitary equivalence. Our proof uses a new characterization of complete distributivity and leads to a generalization of a result of Arveson on ordered group lattices. As consequences of our results, we obtain a similarity theorem for hyperreflexive, completely distributive CSLs and some new perturbation results.Partially supported by an NSERC grantResearch partially supported by NSF grant DMS-9204811.     

Independence and graph homomorphisms graph homomorphisms

A graph with n vertices that contains no triangle and no 5-cycle and minimum degree exceeding n/4 contains an independent set with at least (3n)/7 vertices. This is best possible. The proof proceeds by producing a homomorphism to the 7-cycle and invoking the No Homomorphism Lemma. For k ≥ 4, a graph with n vertices, odd girth 2k+1, and minimum degree exceeding n/(k+1) contains an independent set with at least kn/(2k+1) vertices; however, we suspect this is not best possible. © 1993 John Wiley & Sons, Inc.     

A note on asymptotic homomorphisms

Using the notion of asymptotic homomorphism due to Connes and Higson we construct bivariant homology-cohomology theories for separableC*-algebras, which satisfy general excision axioms and are nonperiodic.     

Prefix-primitivity-preserving homomorphisms

This paper aims to investigate homomorphisms which preserve p-primitive languages. A characterization of p-primitivity-preserving homomorphisms can be detected within finite steps. Also the set of square-freeness-preserving homomorphisms is shown to be a proper subfamily of the set of p-primitivity-preserving homomorphisms. For homomorphisms over an alphabet X with |X|=2, it is also shown that the set of p-primitivity-preserving homomorphisms is a proper subfamily of the set of primitivity-preserving homomorphisms. But it is conjectured to also hold for homomorphisms over an alphabet with more than two letters.     

Fully idempotent homomorphisms

For arbitrary modules A and B we introduce and study the notion of a fully idempotent Hom (A, B). As a corollary we obtain some well-known properties of fully idempotent rings and modules.     

Geometric graph homomorphisms

A geometric graph is a simple graph drawn on points in the plane, in general position, with straightline edges. A geometric homomorphism from to is a vertex map that preserves adjacencies and crossings. This work proves some basic properties of geometric homomorphisms and defines the geochromatic number as the minimum n so that there is a geometric homomorphism from to a geometric n‐clique. The geochromatic number is related to both the chromatic number and to the minimum number of plane layers of . By providing an infinite family of bipartite geometric graphs, each of which is constructed of two plane layers, which take on all possible values of geochromatic number, we show that these relationships do not determine the geochromatic number. This article also gives necessary (but not sufficient) and sufficient (but not necessary) conditions for a geometric graph to have geochromatic number at most four. As a corollary, we get precise criteria for a bipartite geometric graph to have geochromatic number at most four. This article also gives criteria for a geometric graph to be homomorphic to certain geometric realizations of K_{2, 2} and K_{3, 3}. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:97‐113, 2012     

Approximate <Emphasis Type="Italic">C</Emphasis>*-ternary ring homomorphisms

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation

On Blaschke-Minkowski homomorphisms

In this article we establish a Brunn-Minkowski-type inequality for mixed Blaschke-Minkowski homomorphisms.