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Down-set. See lower set. Dual. For a poset (P, ≤), the dual order P d = (P, ≥) is defined by setting x ≥ y if and only if y ≤ x. The dual order of P is sometimes denoted by P op, and is also called opposite or converse order. Any order theoretic notion induces a dual notion, defined by applying the original statement to the order dual ...
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.
List or describe a set of sentences in the language L σ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in L σ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields.
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.
The set of integers and the set of even integers have the same order type, because the mapping is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there ...
Also note that the empty set usually has upper bounds (if the poset is non-empty) and thus a bounded-complete poset has a least element. One may also consider the subsets of a poset which are totally ordered, i.e. the chains. If all chains have a supremum, the order is called chain complete. Again, this concept is rarely needed in the dual form.
Functions are set-theoretical name sets as special cases of binary relations. A fuzzy set is a named set (U, m, [0,1]) where U is a set, [0,1] is a unit interval and m is a membership function. A graph G is a named set (V, E, V) where V is the set of vertices (nodes) of G and E is the set of edges of G.