nLab
positive element

Idea

In classical mathematics, an element of a poset is positive if and only if it not a bottom element.

A more subtle definition is needed in constructive mathematics; analogously to how “nonempty sets” may need to be replaced by inhabited sets. In mathematics that is both constructive and predicative, positivity can be axiomatised but not defined.

Definitions

Let $L$ be a preordered set, and let $x$ be an element of $L$.

Definition

(in classical mathematics, and in predicative mathematics with classical logic)

An element $x\in L$ is positive if it is not a bottom element, equivalently if there exists an element $y\in L$ such that $x\le y$ is false: $\neg (x\le \bigvee \varnothing )$.

Definition

(in classical mathematics, and in impredicative constructive mathematics)

An element $x\in L$ is positive if whenever $x$ is bounded above by a join of some subset $A$ of $L$, $A$ is inhabited: $\forall A;x\le \bigvee A\Rightarrow \exists u\in A$.

Definition

(in predicative constructive mathematics)

A positivity predicate on $L$ is a predicate $\diamond x$, pronounced “$x$ is positive”, such that

• If $x$ is bounded above by a join of a subset and $x$ is positive, then some element of that subset is positive: $\diamond x\wedge x\le \bigvee A⊢\exists u\in A,\diamond u$.
• If $x$ is bounded above by a join on the assumption that $x$ is positive, then $x$ really is so bounded: $\diamond x\Rightarrow x\le \bigvee A⊢x\le \bigvee A$.

Any two of these definitions can be shown to be equivalent in the union of the corresponding foundational systems. Specifically:

• In predicative mathematics with classical logic, one can prove that every preorder has a unique positivity predicate, which must match Definition 1.

• In impredicative constructive mathematics, one can prove that every preorder has a unique positivity predicate, which must match Definition 2.

• In classical mathematics (with classical logic and impredicativity), all of the definitions are equivalent.

Examples

Every power set has a positivity predicate: $x$ is positive iff $x$ is inhabited. (Of course, there are few power sets in predicative mathematics, but often it is enough to think of the power set as a proper class.)

The positive predicate on a locale plays a role in the definition of an overt space. Locale theory is often considered constructively but impredicatively; in predicative constructive mathematics, a positivity predicate is used in the corresponding theory of formal topology.

Every atom of $L$ is positive, and indeed an atom is precisely a minimal positive element.

Properties

Although classically trivial, a key property of positivity in the constructive context is this:

$$\bigvee A=\bigvee A^{+},$$\bigvee A = \bigvee A^+ ,

where $A^{+}$ is the set of positive elements of $A$.