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Simpler characterizations of total orderization invariant maps

Christopher Michael Schwanke

Abstract

Given a finite subset $A$ of a distributive lattice, its total orderization $to(A)$ is a natural transformation of $A$ into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms.

Simpler characterizations of total orderization invariant maps

Abstract

Given a finite subset of a distributive lattice, its total orderization is a natural transformation of into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms.
Paper Structure (4 sections, 16 theorems, 66 equations)

This paper contains 4 sections, 16 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterizations of Total Orderization Invariant Maps
  4. Applications to Multi-Homomorphisms

Key Result

Theorem 1.1

BenAmorBoyRySnigaBusSch-characterizingKusaKusa-on the rep of orth addsSchSch-toi Let $E$ be a uniformly complete Archimedean vector lattice, let $Y$ be a real separated bornological space, put $r,n\in\mathbb{N}\setminus\{1\}$, let $T\colon E^n\to Y$ be a bounded symmetric $n$-linear map, and let $P_

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.3
  • proof
  • ...and 18 more