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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 strict weak order that is trichotomous is called a strict total order. [14] The total preorder which is the inverse of its complement is in this case a total order . For a strict weak order < {\displaystyle \,<\,} another associated reflexive relation is its reflexive closure , a (non-strict) partial order ≤ . {\displaystyle \,\leq .}
If the preordered set (,) also happens to be a partially ordered set (or more generally, if the restriction (,) is a partially ordered set) then is a maximal element of if and only if contains no element strictly greater than ; explicitly, this means that there does not exist any element such that and .
A set equipped with a total order is a totally ordered set; [5] the terms simply ordered set, [2] linearly ordered set, [3] [5] toset [6] and loset [7] [8] are also used. The term chain is sometimes defined as a synonym of totally ordered set, [5] but generally refers to a totally ordered subset of a given partially ordered set.
In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words, H is a maximal element of the partially ordered set of subgroups of G that are not equal to G. Maximal subgroups are of interest because of their direct connection with primitive permutation ...
In a slight abuse of terminology, the term is sometimes also used to refer not to such a relation, but to its corresponding partially ordered set. Partially ordered set. A partially ordered set (,), or poset for short, is a set together with a partial order on . Poset. A partially ordered set. Preorder. A preorder is a binary relation that is ...
NEW YORK (Reuters) -A New York judge partially lifted a gag order on Donald Trump on Tuesday following the Republican presidential candidate's conviction on criminal charges stemming from an ...
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.