joinpreserving

Joinpreserving is a term used in order theory to describe a function between join-based partially ordered sets that preserves joins. Formally, a function f between join-semilattices (or lattices) L and M is joinpreserving if for all elements a and b in L, f(a ∨ b) = f(a) ∨ f(b), where ∨ denotes the join operation (least upper bound). This property implies that f is monotone: if a ≤ b, then f(a) ≤ f(b), since b = a ∨ b and f(b) = f(a ∨ b) = f(a) ∨ f(b).

Variants and related notions: If the domain and codomain are complete lattices, a stronger version called complete join-preserving (or a complete join-homomorphism) requires preservation of arbitrary joins, not just binary ones. A map that preserves finite joins is often called a join-semilattice homomorphism. A joinpreserving map need not preserve meets (greatest lower bounds).

Examples: The identity map on any join-semilattice is joinpreserving. Projections from a product lattice L × M to L or to M preserve joins. The map f(x) = x ∨ c, for a fixed c in L, is joinpreserving in any lattice L. Conversely, a map can be joinpreserving without preserving meets, illustrating that joinpreservation is weaker than full lattice homomorphism.

Context and significance: Joinpreserving maps form morphisms in the category of join-semilattices and, in the complete setting, in the category of complete lattices. They are central in domain theory, lattice theory, and various areas of computer science where structure-preserving maps between ordered objects are studied.