Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A characterization of permutability of 2-uniform tolerances on posets

Ivan Chajda, Helmut Länger

Abstract

Tolerance relations were investigated by several authors in various algebraic structures, see e.g. the monograph by I. Chajda. Recently G. Czédli studied so-called 2-uniform tolerances on lattices, i.e. tolerances that are compatible with the lattice operations and whose blocks are of cardinality 2. He showed that two such tolerances on a lattice containing no infinite chain permute if and only if they are amicable (a concept introduced in his paper). We extend this study to tolerances on posets. Since in posets we have no lattice operations, we must modify the notion of amicability. We modified it in such a way that in case of lattices it coincides with the original definition. With this new definition we can prove that two tolerances on a poset containing no infinite chain permute if and only if they are amicable in the new sense.

A characterization of permutability of 2-uniform tolerances on posets

Abstract

Tolerance relations were investigated by several authors in various algebraic structures, see e.g. the monograph by I. Chajda. Recently G. Czédli studied so-called 2-uniform tolerances on lattices, i.e. tolerances that are compatible with the lattice operations and whose blocks are of cardinality 2. He showed that two such tolerances on a lattice containing no infinite chain permute if and only if they are amicable (a concept introduced in his paper). We extend this study to tolerances on posets. Since in posets we have no lattice operations, we must modify the notion of amicability. We modified it in such a way that in case of lattices it coincides with the original definition. With this new definition we can prove that two tolerances on a poset containing no infinite chain permute if and only if they are amicable in the new sense.
Paper Structure (4 sections, 2 theorems, 5 equations)

This paper contains 4 sections, 2 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. A characterization of permutability of $2$-uniform tolerances on posets
  4. Examples

Key Result

Lemma 3.1

Let $\mathbf P=(P,\le)$ be a poset, $a,b\in P$ and $T$ be a $2$-uniform tolerance on $\mathbf P$. Then the following holds:

Theorems & Definitions (8)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Example 4.1
  • Example 4.2
  • Example 4.3