QuasiQuasiorder

A quasiquasiorder is a binary relation on a set that is reflexive and transitive. It is a generalization of a partial order, which is a reflexive, antisymmetric, and transitive relation. In a quasiquasiorder, antisymmetry is not required, meaning that if aRb and bRa, it is not necessarily true that a=b. The term "quasiquasiorder" is not standard in mathematical literature, and the properties of reflexivity and transitivity are often referred to simply as a preorder or a quasiorder. A preorder is a relation that is reflexive and transitive. Therefore, a quasiquasiorder is essentially a preorder.

The concept arises in various areas of mathematics and computer science, particularly in contexts where the strict equality of elements is not the primary concern, but rather a notion of "at least as good as" or "precedes". For instance, in theoretical computer science, preorders are used to describe computational models where states can be related by a notion of refinement or implication. If a relation R is a quasiquasiorder on a set S, then for any elements a, b, and c in S, the following properties hold: reflexivity (aRa) and transitivity (if aRb and bRc, then aRc). The absence of antisymmetry allows for distinct elements to be related in both directions. This can be visualized as a directed graph where edges represent the relation, with loops for reflexivity and paths for transitivity, but without the restriction that two nodes connected by edges in both directions must be the same node.