Poset

Any such poset has a dual poset.

Completion If a complete lattice is freely generated from a given poset used in place of the set of generators considered above, then one speaks of a completion of the poset.

In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals.

Mathematical representation main Mathematically, in its most general form, a hierarchy is a partially ordered set or poset. citation The system in this case is the entire poset, which is constituted of elements.

Note that this concept of boundedness has nothing to do with finite size, and that a subset S of a bounded poset P with as order the restriction of the order on P is not necessarily a bounded poset.

An abstract polyhedron is a certain kind of partially ordered set (poset) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements (faces, edges, etc.) of a polyhedron.

Any geometric polyhedron is then said to be a "realization" in real space of the abstract poset.

A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.

A subset of a poset in which no two distinct elements are comparable is called an antichain (e.g. the set of singletons in the top-right figure).

Definition A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if every strictly ascending sequence of elements eventually terminates.

For this process, elements of the poset are mapped to (Dedekind-) cuts, which can then be mapped to the underlying posets of arbitrary complete lattices in much the same way as done for sets and free complete (semi-) lattices above.

If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element for it.

In category theory Every poset (and every preorder ) may be considered as a category in which every hom-set has at most one element.

Let (I, ≤) be a directed poset (not all authors require I to be directed).

Let J be a directed poset (considered as a small category by adding arrows i → j if and only if i ≤ j) and let F : J op → C be a diagram.

More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

Motivation Intuitively, a filter on a partially ordered set (poset) contains those elements that are large enough to satisfy some criterion.

One distinguished member of this algebra is that poset's "Möbius function".

Such a poset may be "realised" as a geometrical polyhedron having the same topological structure.

The aforementioned result that free complete lattices do not exist entails that an according free construction from a poset is not possible either.