TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-01

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A cartesian closed preorder has relative implications

Statement

Let $P$ be a preorder regarded as a category. If $P$ is cartesian closed, then for every $a, b \in P$ there is an element $b^{a} \in P$ such that for every $x \in P$ ,

$x \leq b^{a} ⟺ x \land a \leq b,$

where $\land$ is the binary product in the preorder.

Facts & Assumptions

Given: A cartesian closed preorder $P$ and elements $a, b, x \in P$ .

[L1]

A preorder can be regarded as a category with at most one morphism between two objects (A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps).

[L2]

In a cartesian closed category, the product functor $- \times a$ has right adjoint $(-)^{a}$ (Cartesian closed category).

[L3]

Binary products satisfy the usual universal property; in a preorder they are meets (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

Proof

technique · direct

1.1

By [L1], the inequality $u \leq v$ is the same as the existence of a morphism $u \to v$ in the associated thin category. By [L3], the product $x \times a$ is exactly the meet $x \land a$ .

givenL1L3

2.1

Let $b^{a}$ denote the exponential object given by [L2]. The adjunction $- \times a ⊣ (-)^{a}$ gives $Hom (x \land a, b) ≅ Hom (x, b^{a})$ . Since each hom-set in a preorder is either empty or a singleton, this bijection says precisely that $x \land a \leq b$ iff $x \leq b^{a}$ .

step 1.1L2algebra

3.1

Therefore every cartesian closed preorder has the stated relative implication operation.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. Mac Lane, Categories for the Working Mathematician, 2nd ed., IV.6 (standard reference, not scraped)
Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.4.10 (standard reference, not scraped)