Topic summary
Order relation
Extracted from the Wikipedia article Order theory.
Functions between orders
Besides being compatible with the mere order relations, functions between posets may also behave well with respect to special elements and constructions. For example, when talking about posets with least element, it may seem reasonable to consider only monotonic functions that preserve this element, i.e. which map least elements to least elements. If binary infima ∧ exist, then a reasonable property might be to require that f(x ∧ y) = f(x) ∧ f(y), for all x and y. All of these properties, and indeed many more, may be compiled under the label of limit-preserving functions.