poset.cppm File Reference

Preorders, partial orders, directed sets, chains. More...

#include <algorithm>
#include <compare>
#include <concepts>
#include <functional>
import dedekind.sets;
import dedekind.category;

Include dependency graph for poset.cppm:

Namespaces
namespace	dedekind::sets

Concepts
concept	dedekind::order::IsPreOrdered
	The most basic relation: Reflexive and Transitive.
concept	dedekind::order::IsDirectedSet
	A Pre-ordered structure where any two elements have a common successor.
concept	dedekind::order::IsPartiallyOrdered
	A refinement of the Pre-Order that satisfies Antisymmetry (Identity).
concept	dedekind::order::IsDirectedPoset
	The "Convergent Identity": A Poset that is also a Directed Set.
concept	dedekind::order::HasPartialOrderOperators
	Pure syntactic shape: T supports the partial-order operators <, <=, >, >= with results in the truth-value carrier L::Ω of a chosen logical species.
concept	dedekind::order::HasPartialOrderOperatorsWith
	Heterogeneous shape: T and U can be compared via the four partial-order operators in both directions, yielding results in the truth-value carrier L::Ω.

Functions
template<typename C> requires dedekind::order::IsPartiallyOrdered<C>
constexpr bool	dedekind::sets::admits (On< C >, std::less<>, bool)
template<typename C> requires dedekind::order::IsPartiallyOrdered<C>
constexpr bool	dedekind::sets::admits (On< C >, std::less_equal<>, bool)
template<typename C> requires dedekind::order::IsPartiallyOrdered<C>
constexpr bool	dedekind::sets::admits (On< C >, std::greater<>, bool)
template<typename C> requires dedekind::order::IsPartiallyOrdered<C>
constexpr bool	dedekind::sets::admits (On< C >, std::greater_equal<>, bool)

Detailed Description

Preorders, partial orders, directed sets, chains.

@partition :poset

Copyright

2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.

poset__Scope

The most basic concepts in dedekind.order live here: species classified by the \emph{relation} they carry (reflexivity, transitivity, antisymmetry, directedness, comparability). Lattice structures (meet/join/distributive) live in :lattice; density and completeness profiles (Archimedean, Dedekind-complete, ...) live in :completeness.

poset__Structural_Synthesis

• Pre-Order: Reflexive and Transitive.
• Directed Set: A Pre-Order with a Join-Semilattice (Convergence).
• Directed Poset: A Directed Set with Antisymmetry (Identity).
• Total Order: A Poset with no "Parallelism" (The Chain).

Wikipedia: Order theory, Directed set, Partial order, Strict weak ordering

Note

"כל עגול אין לו ראשית, רק כרצון איש ואיש." — אברהם אבן עזרא, Hebrew Wikiquote. [Trans: "A circle has no beginning, except as each person chooses."]