Pseudo-order on arvau

Wikipedia / Pseudo-order

In constructive mathematics, pseudo-order is a name given to certain binary relations appropriate for modeling continuous orderings.

In classical mathematics, its axioms constitute a formulation of a strict total order (also called linear order), which in that context can also be defined in other, equivalent ways.

Examples

The constructive theory of the real numbers is the prototypical example where the pseudo-order formulation becomes crucial. A real number is less than another if there exists (one can construct) a rational number greater than the former and less than the latter. In other words, here x < y holds if there exists a rational number z such that x < z < y. Notably, for the continuum in a constructive context, the usual trichotomy law does not hold, i.e. it is not automatically provable. The axioms in the characterization of orders like this are thus weaker (when working using just constructive logic) than alternative axioms of a strict total order, which are often employed in the classical context.

Definition

A pseudo-order is a binary relation satisfying the three conditions:

,
,
For all , , and , if then either or ,

Auxiliary notation

There are common constructive reformulations making use of contrapositions and the valid equivalences of two elements defines a reflexive partial order. In these terms, the first condition reads

and it really just expresses the asymmetry. It implies irreflexivity, as familiar from the classical theory.

Classical equivalents to trichotomy

The second condition exactly expresses the anti-symmetry of the associated partial order,

With the above two reformulations, the negation signs may be hidden in the definition of a pseudo-order.

A natural apartness relation. With it, the second condition exactly states that this relation is tight,

Together with the first axiom, this means equality can be expressed as negation of apartness. Note that the negation of equality is in general merely the double-negation of apartness.

Now the disjunctive syllogism. Such a logical implication can classically be reversed, and then this condition exactly expresses trichotomy. As such, it is also a formulation of connectedness.

Discussion

Asymmetry

The non-contradiction principle, whether or not this proposition represents a decidable problem.

, the double-negated strong connectedness" thus constitutes a (non-strict) total order.

Co-transitivity

(the partial order) is transitive. So that property is called co-transitivity. Using the asymmetry condition, one quickly derives the theorem that a pseudo-order is actually transitive as well. Transitivity is common axiom in the classical definition of a linear order.

The condition also called is comparison (as well as weak linearity is either above the lower bound or below the upper bound. Since this is an implication of a disjunction, it ties to the trichotomy law as well. And indeed, having a pseudo-order on a Dedekind-MacNeille-complete poset implies the principle of excluded middle. This impacts the discussion of completeness in the constructive theory of the real numbers.

Relation to other properties

This section assumes classical logic. At least then, following properties can be proven:

If R is a co-transitive relation, then

R is also quasitransitive;
R satisfies axiom 3 of semiorders;
incomparability w.r.t. R is a transitive relation; and
R is connex iff it is reflexive.

Sufficient conditions for a co-transitive relation R to be transitive also are:

R is left Euclidean;
R is right Euclidean;
R is antisymmetric.

A semi-connex relation R is also co-transitive if it is symmetric, left or right Euclidean, transitive, or quasitransitive. If incomparability w.r.t. R is a transitive relation, then R is co-transitive if it is symmetric, left or right Euclidean, or transitive.