MATH.22:5.2 - Remove total comparability and inspect a proposed new operation

A partial order is reflexive, antisymmetric and transitive. A total order additionally compares every pair: x<=y or y<=x. Suppose the work needs to retain pairs for which neither direction holds.

Remove total comparability. On pairs of natural numbers, define (a,b)<=(c,d) when a<=c and b<=d. The three partial-order laws follow coordinate by coordinate. The pairs (1,0) and (0,1) are incomparable. The ordinary order on natural numbers supplies a model with total comparability; the pair order supplies a model without it.

Now ask for a least common upper bound of two objects. In the pair order, it is the coordinatewise maximum. It is above both inputs, and any common upper bound is above each coordinatewise maximum. This proves both the construction and its leastness.

The partial-order axioms alone do not supply that operation. Take four distinct objects a,b,u,v. Besides reflexive comparisons, require a<=u, a<=v, b<=u and b<=v, and no others. This is a partial order. Both u and v are upper bounds of a and b, but neither is below the other; a and b are not upper bounds. There is no least upper bound of the pair.

Consequently, writing “x join y” for every pair in an arbitrary partial order adds an existence requirement unless a construction has already supplied it. The work can select an order where joins exist, add the join assumption and accept its narrower class of models, or undertake a construction that enlarges the objects. Choosing one of u or v alone supplies an upper bound, not the missing least one.

This separates two changes that a generic instruction to “relax the order” would hide: allowing incomparability and requiring a combining operation. A model of alternatives can use the resulting order; any decision to prefer one alternative remains a separate choice.