Laterally closed lattice homomorphisms

Let A and B be two Archimedean vector lattices and let be a lattice homomorphism. We call that T is laterally closed if T(D) is a maximal orthogonal system in the band generated by T(A) in B, for each maximal orthogonal system D of A. In this paper we prove that any laterally closed lattice homomorphism T of an Archimedean vector lattice A with universal completion Au into a universally complete vector lattice B can be extended to a lattice homomorphism of Au into B, which is an improvement of a result of M. Duhoux and M. Meyer [M. Duhoux and M. Meyer, Extended orthomorphisms and lateral completion of Archimedean Riesz spaces, Ann. Soc. Sci. Bruxelles 98 (1984) 3-18], who established it for the order continuous lattice homomorphism case. Moreover, if in addition Au and B are with point separating order duals ^′(Au) and B^′ respectively, then the laterally closedness property becomes a necessary and sufficient condition for any lattice homomorphism to have a similar extension to the whole Au. As an application, we give a new representation theorem for laterally closed d-algebras from which we infer the existence of d-algebra multiplications on the universal completions of d-algebras.     

Domination properties of lattice homomorphisms

Let E and F be Banach lattices, T, K : E → F be such that 0 ≤ T ≤ K and T is either a lattice homomorphism or almost interval-preserving. In this paper we will deduce that (1) If K is AM-compact then T also is AM-compact; (2) If either E′ or F has an order continuous norm and K is compact, then T is compact as well; (3) If K is weakly compact then so is T. Some related results are also obtained.      

A characterization of complex lattice homomorphisms

In this paper we discuss the problem of how to recognize a complex lattice homomorphism on the complexification of a real vector lattice L from its behavior on a small subset of .      

On weak lattice and frame homomorphisms

In the context of distributive lattices, frames, or -frames, a join homomorphism preserving the unit and those binary meets which are zero often preserves all binary meets. This paper analyzes this phenomenon.     

OnC 0-semigroups of lattice homomorphisms on a Banach lattice

Ohne Zusammenfassung     

Sums and extensions of vector lattice homomorphisms

This paper gives a summary account, with minimal or no proofs, first of some results which characterize order bounded linear operators which are sums of lattice homomorphisms, or more generally of orthomorphisms; and secondly of theorems concerning extensions of vector lattice homomorphisms (theorems of Hahn-Banach type if you will). In all cases we assume that domain and range are vector lattices and that the range is Dedekind complete. The results vary from historical (pre-1940) to recent (1990). The most recent work, on sums of lattice homomorphisms, is covered in §1 and the more classical work on extension theorems is dealt with in §2.     

An alternative proof of a Radon-Nikodym theorem for lattice homomorphisms

We give a new proof of the Luxemburg-Schep theorem for lattice homomorphisms.     

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.     

Fine structure of the homomorphisms of the lattice of uniformly continuous functions on the line

Positivity - We provide a representation of the homomorphisms $$U\longrightarrow {\mathbb {R}}$$, where U is the lattice of all uniformly continuous functions on the line The resulting picture is...     

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