Editing Partially ordered set (section)

=== Correspondence of strict and non-strict partial order relations ===
[[File:PartialOrders redundencies svg.svg|thumb|upright=1.25|'''Fig. 2''' [[Commutative diagram]] about the connections between strict/non-strict relations and their duals, via the operations of reflexive closure (''cls''), irreflexive kernel (''ker''), and converse relation (''cnv''). Each relation is depicted by its [[logical matrix]] for the poset whose [[Hasse diagram]] is depicted in the center. For example <math>3 \not\leq 4</math> so row 3, column 4 of the bottom left matrix is empty.]]

Strict and non-strict partial orders on a set <math>P</math> are closely related. A non-strict partial order <math>\leq</math> may be converted to a strict partial order by removing all relationships of the form <math>a \leq a;</math> that is, the strict partial order is the set <math>< \; := \ \leq\  \setminus \  \Delta_P</math> where <math>\Delta_P := \{ (p, p) : p \in P \}</math> is the [[identity relation]] on <math>P \times P</math> and <math>\;\setminus\;</math> denotes [[set subtraction]]. Conversely, a strict partial order < on <math>P</math> may be converted to a non-strict partial order by adjoining all relationships of that form; that is, <math>\leq\; := \;\Delta_P\; \cup \;<\;</math> is a non-strict partial order. Thus, if <math>\leq</math> is a non-strict partial order, then the corresponding strict partial order < is the [[irreflexive kernel]] given by
<math display=block>a < b \text{ if } a \leq b \text{ and } a \neq b.</math>
Conversely, if < is a strict partial order, then the corresponding non-strict partial order <math>\leq</math> is the [[reflexive closure]] given by:
<math display=block>a \leq b \text{ if } a < b \text{ or } a = b.</math>