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.
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.
In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise; Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound.
In order-theoretic mathematics, a graded partially ordered set is said to have the Sperner property (and hence is called a Sperner poset), if no antichain within it is larger than the largest rank level (one of the sets of elements of the same rank) in the poset. [1]
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions).
Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which all subsets
The term complete partial order, abbreviated cpo, has several possible meanings depending on context. A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. (A subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the subset.)
The Hasse diagram of the power set of three elements, partially ordered by inclusion. In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial ...