U×UF

U×UF is a symbolic expression that combines two terms U and UF with the Cartesian product symbol ×. The meaning of the expression depends on the definitions of U and UF and varies across fields and specific texts.

In mathematics, the most common interpretation is as the Cartesian product of two sets. If U and UF are sets, then U×UF denotes the set of all ordered pairs (u, uf) where u is an element of U and uf is an element of UF. When U and UF carry additional structure—for example, if they are groups, rings, or topological spaces—U×UF can denote the direct product, equipped with componentwise operations and, in the topological case, the product topology.

Examples: If U = {a, b} and UF = {0, 1}, then U×UF = {(a,0), (a,1), (b,0), (b,1)}. If U and UF are groups, the direct product U×UF forms a new group whose operation is performed separately on each coordinate. If they are topological spaces, the product inherits the product topology.

Variants and related notations: Some fields substitute other symbols for product notions, such as ⊗ for tensor products or different symbols for semidirect or free products, depending on context. In computer science or data modeling, U and UF might represent types, universes, or domains, with × indicating a combination or pairing of elements from the two structures.

See also: Cartesian product, direct product, product topology, tensor product.

Notes: There is no single universal object named U×UF; its exact meaning is determined by the definitions and structures assigned to U and UF in the given work.