This does the standard thing of treating preorders as the default generalization of partial orders. But an (arguably) more natural, and more useful, generalization of partial orders is acyclicity.

Unfortunately acyclicity isn't called an "order" so people assume it's something unrelated. But "orders" are just second-order properties that binary relations can fulfill, and acyclicity is also such a property.

Acyclicity is a generalization of strict (irreflexive) partial orders, just like strict partial orders are a generalization of strict total (linear) orders. Every strict partial order relation is acyclic, but not every acyclic relation is a strict partial order.

A strict partial order is a binary relation that is both acyclic and transitive, i.e. a strict partial order is the transitive closure of an acyclic relation.

Binary relations of any kind can be represented as sets of pairs, or as directed graphs. If the binary relation in the directed graph is acyclic, that graph is called a "directed acyclic graph", or DAG. In a DAG the transitive closure (strict partial order) is called the reachability relation.

Examples of common acyclic relations that are not strict partial orders: x∈y (set membership), x causes y, x is a parent of y.