limit preserving (order theory)

In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima. Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of the set. Depending on the type of sets for which a function satisfies this property, it may preserve finite, directed, non-empty, or just arbitrary suprema or infima. Each of these requirements appears naturally and frequently in many areas of order theory and there are various important relationships among these concepts and other notions such as monotonicity. If the implication of limit preservation is inverted, such that the existence of limits in the range of a function implies the existence of limits in the domain, then one obtains functions that are limit-reflecting.