Where does join collide?

In order and lattice theory, the join of two elements of a partially ordered set is their least upper bound: the smallest element that is greater than or equal to both, written a ∨ b, and dual to the meet (their greatest lower bound). Birkhoff's Lattice theory developed the algebra of posets in which every pair of elements has a join and a meet, grounding structures from Boolean algebras to concept lattices. The referent is a binary operation on a partial order that returns a supremum when one exists, a purely algebraic notion with nothing to do with combining database tables or gluing topological spaces.[1]

Unit

The least upper bound a ∨ b of two elements of a poset

Recovery means

Not applicable

Measured by

Defined as the supremum of two elements under the order relation, where such a least upper bound exists

In the relational model of data, a join is an operation that combines rows from two relations (tables) by matching values on one or more shared attributes, producing a new relation whose tuples pair records satisfying the join condition. Codd's 1970 A relational model of data for large shared data banks set out the relational algebra in which the join and its natural, equi-, and theta- variants recombine data that normalization has split across relations. The referent is a table-combining operator over finite relations, defined by a predicate on attributes and evaluated into result rows, unrelated to the lattice-theoretic least upper bound or the topological gluing of spaces that share its name.[2]

Unit

The result relation formed by matching rows across two tables

Recovery means

Not applicable

Measured by

Characterised by the join predicate, the cardinality of the result relation, and the query plan that evaluates it

In topology, the join X ∗ Y of two spaces is the space of all line segments connecting each point of X to each point of Y, formally a quotient of X × Y × [0,1] that collapses the two ends; for instance, the join of spheres S^m ∗ S^n is the sphere S^{m+n+1}. Hatcher's Algebraic topology treats the join among the standard constructions (alongside products, wedges, and smash products) used to build and compare spaces. The referent is a geometric way of combining two spaces into a higher-dimensional one, sharing only the English word with the relational operator and the lattice supremum.[3]

Unit

The space X ∗ Y built from all segments joining X to Y

Recovery means

Not applicable

Measured by

Constructed as the quotient of X × Y × [0,1] collapsing the ends; its homotopy type tracked through its cell structure