Formal Definition
Consider two partially ordered sets P and Q, and a function f from P to Q. Furthermore, let S be a subset of P that has a least upper bound s. Then f preserves the supremum of S if the set f(S) = {f(x) | x in S} has a least upper bound in Q which is equal to f(s), i.e.
- f(sup S) = sup f(S)
Note that this definition consists of two requirements: the supremum of the set f(S) exists and it is equal to f(s). This corresponds to the abovementioned parallel to category theory, but is not always required in the literature. In fact, in some cases one weakens the definition to require only existing suprema to be equal to f(s). However, Wikipedia works with the common notion given above and states the other condition explicitly if required.
From the fundamental definition given above, one can derive a broad range of useful properties. A function f between posets P and Q is said to preserve finite, non-empty, directed, or arbitrary suprema if it preserves the suprema of all finite, non-empty, directed, or arbitrary sets, respectively. The preservation of non-empty finite suprema can also be defined by the identity f(x v y) = f(x) v f(y), holding for all elements x and y, where we assume v to be a total function on both orders.
In a dual way, one defines properties for the preservation of infima.
The "opposite" condition to preservation of limits is called reflection. Consider a function f as above and a subset S of P, such that sup f(S) exists in Q and is equal to f(s) for some element s of P. Then f reflects the supremum of S if sup S exists and is equal to s. As already demonstrated for preservation, one obtains many additional properties by considering certain classes of sets S and by dualizing the definition to infima.
Read more about this topic: Limit-preserving Function (order Theory)
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