On the graph of partial orders

: Any binary relation

X σ ⊆ (where X is an arbitrary set) generates a characteristic function on the set 2 X : If ( , ) x y σ ∈ , then ( , ) 1 x y σ = , otherwise ( , ) 0 x y σ = . In terms of characteristic functions on the set of all binary relations of the set X we introduced the concept of a binary of reflexive relation of adjacency and determined the algebraic system consisting of all binary relations of a set X and all unordered pairs of various adjacent binary relations. If X 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 ( ) G X of partial orders. On the set of 2 2 X all sets of binary relations on the set X we introduce a binary reflexive adjacency. (1) The relation τ is called adjacent with a relation σ .

Remark 1.3
From the definition it follows that if the relation τ adjacent with a relation σ , then σ adjacent with a relation τ , and this fact we write in the form of a diagram

‫اﻟﻌﺎﺷﺮ‬ ‫اﻟﺪوﻟﻲ‬ ‫اﻟﻌﻠﻤﻲ‬ ‫اﻟﻤﺆﺗﻤﺮ‬ ‫واﺳﻂ‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺘﺮﺑﯿﺔ‬ ‫ﻛﻠﯿﺔ‬
Here and elsewhere in the diagrams we mark for the value of the characteristic functions at those points which are known a priori. For example, in the block Y Z × for the relation σ we write generalized  zero, and this means that ( , ) 0 x y σ = for all ( , ) x y Y Z ∈ × , And in the same block for the relation τ we write 1
In the terms of the characteristic we have: (5) (6) Proof. By symmetry, it suffices to prove this implication = , then the reflexivity relation τ obviously.
x z τ = in all cases, we have the equality ( , ) 1.
x z τ = Thus, the set X generates a pair ( ), ( ) V X E X 〈 〉, where ( ) V X -is a set of vertices, consist of all partial orders of the set X and ( ) E X -is a set of edges, consist of all unordered distinct pairs of adjacent partial orders of the set will be called (undirected) graph of partial orders of the set X .

Definition 2.2
The partial orders σ and τ belong to the same connected component of the graph ( ) G X ,if there is a finite sequence of partial orders 1 2 , , , m σ = σ σ σ = τ  , in which the relations is the connected component of the graph ( ) G X , which contains the partial order σ .

3-On the features of the structure of the graph of partial orders.
We fix the partial order ( ) V X σ ∈ and an element x X ∈ . For σ we have the representation: Lemma 3.1 The following statements holds: 1. Since 2. Let ( , ) x x y z I K ∈ × .

‫اﻟﻌﺎﺷﺮ‬ ‫اﻟﺪوﻟﻲ‬ ‫اﻟﻌﻠﻤﻲ‬ ‫اﻟﻤﺆﺗﻤﺮ‬ ‫واﺳﻂ‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺘﺮﺑﯿﺔ‬ ‫ﻛﻠﯿﺔ‬
without loss of generality, we can also assume that x J ∈ (if x J ∈ in the calculations presented below the relation σ and τ changing places. For σ have the representation I J

‫اﻟﻌﺎﺷﺮ‬ ‫اﻟﺪوﻟﻲ‬ ‫اﻟﻌﻠﻤﻲ‬ ‫اﻟﻤﺆﺗﻤﺮ‬ ‫واﺳﻂ‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺘﺮﺑﯿﺔ‬ ‫ﻛﻠﯿﺔ‬
Visual comparison x σ and x τ shows their equality. If card X < ∞ then there exist a one -to-one between the set 0 ( ) V X and the set of all labeled of transitive graph define on the set X (see example [1, p28]) and there exist a one-to -one between these set and the set of 0 T − topology define on the set X (see example [2, p256]) and the number is called (support of partial order) σ (or support set). a fact that we write in the form.  S σ ≥