Partially ordered set
Notation

Given a set P {\displaystyle P} and a partial order relation, typically the non-strict partial order ≤ {\displaystyle \leq }, we may uniquely extend our notation to define four partial order relations ≤, {\displaystyle \leq,} <, {\displaystyle <,} ≥, {\displaystyle \geq,} and > {\displaystyle >}, where ≤ {\displaystyle \leq } is a non-strict partial order relation on P {\displaystyle P}, < {\displaystyle <} is the associated strict partial order relation on P {\displaystyle P} (the irreflexive kernel of ≤ {\displaystyle \leq }), ≥ {\displaystyle \geq } is the dual of ≤ {\displaystyle \leq }, and > {\displaystyle >} is the dual of < {\displaystyle <}. Strictly speaking, the term partially ordered set refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation, (P, ≤) {\displaystyle (P,\leq)} or (P, <) {\displaystyle (P,<)}, or, in rare instances, the non-strict and strict relations together, (P, ≤, <) {\displaystyle (P,\leq,<)}.

The term ordered set is sometimes used as a shorthand for partially ordered set, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than ≤ {\displaystyle \leq } such as ⊑ {\displaystyle \sqsubseteq } or ⪯ {\displaystyle \preceq } to distinguish partial orders from total orders.

When referring to partial orders, ≤ {\displaystyle \leq } should not be taken as the complement of > {\displaystyle >}. The relation > {\displaystyle >} is the converse of the irreflexive kernel of ≤ {\displaystyle \leq }, which is always a subset of the complement of ≤ {\displaystyle \leq }, but > {\displaystyle >} is equal to the complement of ≤ {\displaystyle \leq } if, and only if, ≤ {\displaystyle \leq } is a total order.
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