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The term partial order usually refers to the reflexive partial order relations, referred to in this article as non-strict partial orders. However some authors use the term for the other common type of partial order relations, the irreflexive partial order relations, also called strict partial orders.
This "finer-than" relation on the set of partitions of X is a partial order (so the notation "≤" is appropriate). Each set of elements has a least upper bound (their "join") and a greatest lower bound (their "meet"), so that it forms a lattice , and more specifically (for partitions of a finite set) it is a geometric and supersolvable lattice.
Like partial orders and equivalence relations, preorders (on a nonempty set) are never asymmetric. A preorder can be visualized as a directed graph , with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices.
Of particular importance are relations that satisfy certain combinations of properties. A partial order is a relation that is reflexive, antisymmetric, and transitive, [3] an equivalence relation is a relation that is reflexive, symmetric, and transitive, [4] a function is a relation that is right-unique and left-total (see below). [5] [6]
The identity relation = on any set is also a partial order in which every two distinct elements are incomparable. It is also the only relation that is both a partial order and an equivalence relation because it satisfies both the antisymmetry property of partial orders and the symmetry property of equivalence relations. Many advanced properties ...
A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preorders are sometimes also called preference relations . The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves ...
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 order by means of the Hasse diagram .
In mathematics, a partially ordered space [1] (or pospace) is a topological space equipped with a closed partial order, i.e. a partial order whose graph {(,)} is a closed subset of . From pospaces, one can define dimaps , i.e. continuous maps between pospaces which preserve the order relation.