Important Properties and Results
The above definition of limit preservation is quite strong. 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.
Based on the fact that some limits can be expressed in terms of others, one can derive connections between the preservation properties. For example, a function f preserves directed suprema if and only if it preserves the suprema of all ideals. Furthermore, a mapping f from a poset in which every non-empty finite supremum exists (a so-called sup-semilattice) preserves arbitrary suprema if and only if it preserves both directed and finite (possibly empty) suprema.
However, it is not true that a function that preserves all suprema would also preserve all infima or vice versa.
Read more about this topic: Limit-preserving Function (order Theory)
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