Limit-preserving Function (order Theory)

Limit-preserving Function (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.

The purpose of this article is to clarify the definition of these basic concepts, which is necessary since the literature is not always consistent at this point, and to give general results and explanations on these issues.

Read more about Limit-preserving Function (order Theory):  Background and Motivation, Formal Definition, Special Cases, Important Properties and Results

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Limit-preserving Function (order Theory) - Important Properties and Results
... Indeed,every function that preserves at least the suprema or infima of two-element chains,i.e ... of sets of two comparable elements,is necessarily monotone ... Hence,all the special preservation properties stated above induce monotonicity ...

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