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The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviation is vacuously true. However, its opposite poset has deviation 1. Let k be an algebraically closed field and consider the poset of ideals of the polynomial ring k[x] in one variable. Since the deviation of this poset is the ...
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
For two elements a, b of a partially ordered set P, the interval [a,b] is the subset {x in P | a ≤ x ≤ b} of P. If a ≤ b does not hold the interval will be empty. Interval finite poset. A partially ordered set P is interval finite if every interval of the form {x in P | x ≤ a} is a finite set. [2] Inverse. See converse. Irreflexive.
Forcing with this poset collapses λ down to κ. Levy collapsing: If κ is regular and λ is inaccessible, then P is the set of functions p on subsets of λ × κ with domain of size less than κ and p(α, ξ) < α for every (α, ξ) in the domain of p. This poset collapses all cardinals less than λ onto κ, but keeps λ as the successor to κ.
1. A tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation < 2. A tree is a collection of finite sequences such that every prefix of a sequence in the collection also belongs to the collection. 3. A cardinal κ has the tree property if there are no κ-Aronszajn trees tuple
Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value. [1] [2] Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram.
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.
Thus, an equivalent definition of the dimension of a poset P is "the least cardinality of a realizer of P." It can be shown that any nonempty family R of linear extensions is a realizer of a finite partially ordered set P if and only if, for every critical pair ( x , y ) of P , y < i x for some order < i in R .