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.
The constructive theory of the real numbers is the prototypical example where the pseudo-order formulation becomes crucial.
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.
A pseudo-order is a binary relation satisfying the three conditions: There are common constructive reformulations making use of contrapositions and the valid equivalences
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 on a pseudo-ordered set is given by
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.
Such a logical implication can classically be reversed, and then this condition exactly expresses trichotomy.
Constructively, the validity of the double-negation exactly means that there cannot be a refutation of any of the disjunctions in the classical claim
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): For any nontrivial interval given by some
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 numbners.