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Graded poset
Graded poset A locally finite graded poset consists of a poset P along with a function . Equivalently, a poset P is graded if it admits a ... the rank of minimal elements. In general poset theory, the rank of a minimal ...
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Bounded complete poset
Bounded complete poset In the mathematical field of order theory ... theory. Note also that the term bounded poset is sometimes used to refer to a ... important to distinguish between a bounded complete poset and a bounded cpo. For a typical example of a bounded complete poset, consider the set of all finite decimal ...
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Glossary of order theory
... intersection of open sets is open. Algebraic poset . A poset is algebraic if it has a base ... compact elements. Antichain. An antichain is a poset in which no two elements are comparable ... is not symmetric. An atom in a poset P with least element 0, is an ... that are unequal to 0. A atomic poset P with least element 0 is ...
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Boolean-valued model
... the universe" a generic subset of a poset, the poset being designed to impose interesting properties on ... is no such generic subset of the poset. There are three usual ways of dealing ... is defined between elements p of the poset and formulas φ of the forcing language ... the desired purpose, and that contains the poset. Then there do exist filters on ...
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Completeness (order theory)
... just the least element of the whole poset. Other common names for the least element ... join for v are most common. A poset in which only non-empty finite suprema ... once. If all directed subsets of a poset have a supremum, then the order is ... of a dcpo is the filtered complete poset. Dcpos with a least element ("pointed dcpos ... a least upper bound, then the respective poset is called bounded complete. The term ...
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Order theory
... it is called a partially ordered set, poset, or just an ordered set if the ... the vertices are the elements of the poset and the ordering relation is indicated by ... given by the least element of a poset. For example, 1 is the least element ... Now there are also elements of a poset that are special with respect to some ... bounds. Given a subset S of some poset P, an upper bound of S ...
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Theory of the order (translated from Spanish)
... is called set partially ordered, or, shortly, poset. The term ordered set sometimes also it ... grafos where vértices is the elements of poset and the order relation is indicated by ... basic example is given by minimum of poset. For example, 0 is the minimum of ... of divisibilidad. There are also elements of poset that they are special with respect to ... level. Given a subgroup S of certain poset P, a superior level of S ...
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Domain theory
... Hence, one defines a base of a poset P as being a subset B of ... a directed set with supremum x. The poset P is a continuous poset if it has some base. Especially, P ... a base of compact elements. Such a poset is called algebraic . From the viewpoint of ... some cases, however, the base for a poset is countable. In this case, one ...
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Forcing (mathematics)
... forcing expounded here. Forcing posets A forcing poset is an ordered triple (P, ≤, 1 ... his co-authors.) Associated with a forcing poset P are the P-names. P-names ... G. A good example of a forcing poset is (Bor(I) , ⊆ , I ), where I ... consistency result is that given a forcing poset P, we may assume that there exists ... analysis. Cohen forcing The simplest nontrivial forcing poset is ( Fin(ω,2) , ⊇ , 0 ), ...
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Lattice (order)
... lattice is a partially ordered set (or poset), in which all nonempty finite subsets have ... explained below. Lattices as posets Consider a poset (L, ≤). L is a lattice if ... a bounded lattice. One can convert any poset into a bounded lattice by adding a ... restrict to the compact elements of a poset for obtaining these directed sets, then the poset is even algebraic . Both concepts can ...
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