Mp×q

Mp×q denotes the direct product of two mathematical structures, typically labeled M_p and M_q. The notation is used across various categories, including groups, rings, modules, vector spaces, and topological spaces, to indicate that the single object consists of ordered pairs whose first and second components come from M_p and M_q, respectively.

Construction and operation: If M_p and M_q are objects in a given category, their direct product Mp×q is the object consisting of all pairs (a, b) with a ∈ M_p and b ∈ M_q. The structure on Mp×q is defined componentwise: the operation in each coordinate is the same as in the corresponding factor. For example, if both factors are groups, the product operation is (a, b)·(a′, b′) = (a·a′, b·b′). If the factors are rings, addition and multiplication are defined coordinatewise, and for vector spaces the scalar multiplication acts on each coordinate.

Notation and universal property: There are natural projection maps π_p: Mp×q → M_p and π_q: Mp×q → M_q sending (a, b) to a and b, respectively. Mp×q satisfies the universal property: for any object X with maps to M_p and M_q, there exists a unique map from X to Mp×q making the diagram commute.

Special cases and properties: If M_p and M_q are finite groups, |Mp×q| = |M_p|·|M_q|. When both factors are cyclic of orders p and q, Mp×q is cyclic of order pq if p and q are coprime. The direct product is distinct from the direct sum in contexts with infinite indexing or extra structure.

Applications: Mp×q is used to construct new objects from known ones, analyze decompositions, and study how properties of factors influence the whole.