crosslattice

Crosslattice is a term used in several disciplines to denote a lattice-like algebraic structure that is augmented with an additional binary operation designed to model the interaction or combination of two elements. There is no universally accepted definition, and the precise axioms vary by author and context.

In order-theoretic treatments, a crosslattice consists of a lattice (L, ≤, ∨, ∧) together with a binary operation ⊗: L × L → L. Typical requirements include monotonicity in each argument (a ≤ a' and b ≤ b' imply a ⊗ b ≤ a' ⊗ b'), along with axioms that relate ⊗ to ∨ and ∧, sometimes in a cross-distributive form such as a ⊗ (b ∨ c) ≤ (a ⊗ b) ∨ (a ⊗ c) and (a ∨ b) ⊗ c ≤ (a ⊗ c) ∨ (b ⊗ c). Some variants require commutativity or associative properties for ⊗, while others use weaker conditions.

Crosslattices relate to other ordered algebraic structures such as bilattices, lattices with operators, and various product or tensor-type constructions. They are discussed in theoretical contexts that aim to model combined attributes, multi-criteria reasoning, information fusion, or logics that seek to capture interactions between elements in a structured way.

Constructions often derive from product lattices with coordinate-wise operations or from defining ⊗ in terms of existing lattice operations (for example via expressions involving ∨, ∧, and complemented forms in certain classes of lattices). Researchers study representations, morphisms, and congruences of crosslattices, as well as potential applications in formal concept analysis and knowledge representation.