** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** Antichain principle: Every partially ordered set has a maximal antichain.

* Incidence algebras of locally finite partially ordered sets are unitary associative algebras considered in combinatorics.

The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set.

In mathematics, especially in order theory, the cofinality cf ( A ) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.

The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image.

* The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal ( and must be contained in every other cofinal subset ).

* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.

Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.

This is partially ordered under inclusion and the subsets with m elements are maximal.

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.

Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets ( contrast partially ordered sets which need not be directed ).

Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open ( X ) under inclusion.

Like every partially ordered set, Open ( X ) forms a small category by adding a single arrow U → V if and only if.

* In mathematics, a certain kind of subset of a partially ordered set.

** Filter ( mathematics ), a special subset of a partially ordered set

In 1971, American novelist Frank Yerby published The Man From Dahomey, a historical novel set partially in Dahomey.

During the same year, British author George MacDonald Fraser published Flash for Freedom !, the third novel in the Flashman series that was set partially in Dahomey.

This was, at least partially, a reference to a scene in the original A Nightmare on Elm Street where the character Nancy Thompson ( portrayed by Heather Langenkamp ), watches the original Evil Dead on a television set in her room.

Scullin was partially influenced by the precedent set by the Government of the Irish Free State, which always insisted upon the Governor-General of the Irish Free State being an Irishman.

It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

The modern mathematical understanding is to describe such a structural sequence in terms of an " abstract " polygon which is a partially ordered set ( poset ) of elements.

* Graded poset, a partially ordered set equipped with a rank function, sometimes called a ranked poset

A set with a binary relation R on its elements that is reflexive ( for all a in the set, aRa ), antisymmetric ( if aRb and bRa, then a = b ) and transitive ( if aRb and bRc, then aRc ) is described as a partially ordered set or poset.

The term poset as an abbreviation for partially ordered set was coined by Garrett Birkhoff in the second edition of his influential book Lattice Theory.

In mathematical order theory, an ideal is a special subset of a partially ordered set ( poset ).

In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set ( poset ).

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set ( poset ) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually.

In set theory, a tree is a partially ordered set ( poset ) ( T, <) such that for each t ∈ T, the set

Note also that the term bounded poset is sometimes used to refer to a partially ordered set which has both a least and a greatest element.

A graded partially ordered set is said to have the Sperner property when one of its largest antichains is formed by a set of elements that all have the same rank ; Sperner's theorem states that the poset of all subsets of a finite set, partially ordered by set inclusion, has the Sperner property.

By analogy with the Dedekind – MacNeille completion of a partially ordered set, every partially ordered set can be extended uniquely to a minimal chain-complete poset.

Formally, P ( with <) will be a ( strict ) partially ordered set, or poset.

Chomp is a special case of a poset game where the partially ordered set on which the game is played is a product of total orders with the minimal element ( poisonous block ) removed.

* Bounded poset a partially ordered set that has both a greatest element and a least element

* Zigzag poset, another term for a partially ordered set in mathematics commonly called a fence ( mathematics )

Formally, let P be a poset ( partially ordered set ), and let F be a filter on P ; that is, F is a subset of P such that:

A power set, Partial order | partially ordered by Inclusion ( set theory ) | inclusion, with rank defined as number of elements, forms a graded poset.

In mathematics, in the branch of combinatorics, a graded poset, sometimes called a ranked poset ( but see the article for an alternative meaning ), is a partially ordered set ( poset ) P equipped with a rank function ρ from P to N compatible with the ordering ( so ρ ( x )< ρ ( y ) whenever x < y ) such that whenever y covers x, then.