On the graph of partial orders

Dr Khalid Shea Khairalla Aljabri

doi:10.31185/eduj.vol2.iss25.2746

Abstract

Any binary relation (where is an arbitrary set) generates a charac-teristic function on the set : If , then , otherwise . In terms of characteristic functions on the set of all binary rela-tions of the set we introduced the concept of a binary of reflexive relation of adjacency and determined the algebraic system consisting of all binary re-lations of a set and all unordered pairs of various adjacent binary rela-tions. If is finite set then this algebraic system is a graph “ a graph of graphs” in this work we investigated some features of the structures of the graph of partial orders.

Highlights

On the set of 2X 2 all sets of binary relations on the set X we introduce a binary reflexive adjacency
Remark 1.3 From the definition it follows that if the relation τ adjacent with a relation σ, σ adjacent with a relation τ, and this fact we write in the form of a diagram σ ← Y×Z → τ : Y Z
And elsewhere in the diagrams we mark for the value of the characteristic functions at those points which are known a priori

Summary

Introduction

Any characteristic function χ : X 2 → B generates a binary relation The relation σ ⊆ X 2 belongs in the set V ( X ) , if satisfies the following axioms: 1) reflexivity: (x, x) ∈ σ ; 2) transitivity: if (x, y) ∈ σ , ( y, z) ∈ σ , (x, z) ∈ σ ; 3) antisymmetry: if (x, y) ∈ σ , ( y, x) ∈ σ , x = y . Theorem 2.1 Let σ и τ – are adjacent relations( i.e σ ← Y×Z → τ ).

Results

Conclusion