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)
Famous quotes containing the words important, properties and/or results:
“The presence of a grandparent confirms that parents were, indeed, little once, too, and that people who are little can grow to be big, can become parents, and one day even have grandchildren of their own. So often we think of grandparents as belonging to the past; but in this important way, grandparents, for young children, belong to the future.”
—Fred Rogers (20th century)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“The peace conference must not adjourn without the establishment of some ordered system of international government, backed by power enough to give authority to its decrees. ... Unless a league something like this results at our peace conference, we shall merely drop back into armed hostility and international anarchy. The war will have been fought in vain ...”
—Virginia Crocheron Gildersleeve (1877–1965)