Some articles on poset, posets:
Ranked Poset
... In mathematics, a ranked partially ordered set - or poset - may be either a graded poset, or a poset that has the property that for every element x, all maximal chains among those with x as ... from the first in that it requires all minimal elements to have the same rank for posets with a least element, however, the two requirements are equivalent ... The third definition is even more strict in that it excludes posets with infinite chains and also requires all maximal elements to have the same rank ...
... In mathematics, a ranked partially ordered set - or poset - may be either a graded poset, or a poset that has the property that for every element x, all maximal chains among those with x as ... from the first in that it requires all minimal elements to have the same rank for posets with a least element, however, the two requirements are equivalent ... The third definition is even more strict in that it excludes posets with infinite chains and also requires all maximal elements to have the same rank ...
Boolean-valued Model - Relationship To Forcing
... In one form, forcing "adds to the universe" a generic subset of a poset, the poset being designed to impose interesting properties on the newly-added object ... The wrinkle is that (for interesting posets) it can be proved that there simply is no such generic subset of the poset ... forcing A forcing relation is defined between elements p of the poset and formulas φ of the forcing language ...
... In one form, forcing "adds to the universe" a generic subset of a poset, the poset being designed to impose interesting properties on the newly-added object ... The wrinkle is that (for interesting posets) it can be proved that there simply is no such generic subset of the poset ... forcing A forcing relation is defined between elements p of the poset and formulas φ of the forcing language ...
Locally Finite Poset
... In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval consists of finitely many elements ... Given a locally finite poset P we can define its incidence algebra ... In theoretical physics a locally finite poset is also called a causal set and has been used as a model for spacetime ...
... In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval consists of finitely many elements ... Given a locally finite poset P we can define its incidence algebra ... In theoretical physics a locally finite poset is also called a causal set and has been used as a model for spacetime ...
Chain Complete - Properties
... Zorn's lemma states that, if a poset has an upper bound for every chain, then it has a maximal element ... Thus, it applies to chain complete posets, but is more general in that it allows chains that have upper bounds but do not have least upper bounds ... Chain complete posets also obey the Bourbaki–Witt theorem, a fixed point theorem stating that, if f is a function from the poset to itself with the property that, for all x, f(x) ≥ x, then f has a fixed point ...
... Zorn's lemma states that, if a poset has an upper bound for every chain, then it has a maximal element ... Thus, it applies to chain complete posets, but is more general in that it allows chains that have upper bounds but do not have least upper bounds ... Chain complete posets also obey the Bourbaki–Witt theorem, a fixed point theorem stating that, if f is a function from the poset to itself with the property that, for all x, f(x) ≥ x, then f has a fixed point ...
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 ...
... 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 ...
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