Search results
Results From The WOW.Com Content Network
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 a partially ordered set (,), the irreflexive kernel of is denoted as < and is defined by < if and . For arbitrary members x , y ∈ P , {\displaystyle x,y\in P,} exactly one of the following cases applies:
In the first case, the graph is the undirected Hasse diagram of the partially ordered set, and in the second case, the graph is simply the underlying (undirected) graph of the partially ordered set. However, if T is a tree of height > ω, then the Hasse diagram definition does not work.
A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear. By checking these properties, one immediately sees that the well-known orders on natural numbers , integers , rational numbers and reals are all orders in the above sense.
The directed preordered set (,) is partially ordered if and only if has exactly one element. All pairs of elements from R {\displaystyle R} are comparable and every element of R {\displaystyle R} is a greatest element (and thus also a maximal element) of ( R , ≤ ) . {\displaystyle (R,\leq ).}
A partially ordered group G is called integrally closed if for all elements a and b of G, if a n ≤ b for all natural n then a ≤ 1. [1]This property is somewhat stronger than the fact that a partially ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent. [2]
A partial order of dimension 4 (shown as a Hasse diagram) and four total orderings that form a realizer for this partial order.. In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.
A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set S is the set of its downwardly closed subsets ordered by inclusion. S is embedded in this (complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x.