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Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects).
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A simplicial model category is a simplicial category with a model structure that is compatible with the simplicial structure. [3] Given any category C and a model category M, under certain extra hypothesis the category of functors Fun (C, M) (also called C-diagrams in M) is also a model category.
If and is a pair of adjoint functors, then and is also a pair of adjoint functors. The functor category D C {\displaystyle D^{C}} has all the formal properties of an exponential object ; in particular the functors from E × C → D {\displaystyle E\times C\to D} stand in a natural one-to-one correspondence with the functors from E ...
A category C consists of two classes, one of objects and the other of morphisms.There are two objects that are associated to every morphism, the source and the target.A morphism f from X to Y is a morphism with source X and target Y; it is commonly written as f : X → Y or X Y the latter form being better suited for commutative diagrams.
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets ), orderings in which some pairs are comparable and others might not be
For example, the membership axiom produces a class that may contain elements that are not ordered pairs, while the intersection contains only the ordered pairs of . The circular permutation and transposition axioms do not imply the existence of unique classes because they specify only the 3‑tuples of class B . {\displaystyle B.}