latticewith

Latticewith is a term used in some mathematical and computer science contexts to denote a lattice that is equipped with an additional binary operation, commonly referred to as “with.” The idea behind latticewith is to study how an operation that combines two lattice elements interacts with the underlying order and lattice operations, such as join and meet. The term is not universally standardized and may be defined differently across sources, but the central notion is a lattice augmented by a second, well-behaved operation.

Formally, a latticewith consists of a lattice (L, ≤, ∨, ∧) together with a binary operation with: L × L → L. In typical formulations, the with operation is required to be commutative and associative, and it is monotone with respect to the lattice order. In some accounts, the operation is also said to distribute over joins (and sometimes over meets), meaning a with (b ∨ c) = (a with b) ∨ (a with c) and similarly for the meet. Depending on the source, additional axioms or variations may be imposed to suit particular applications or to emphasize connections with semilattices, monoids, or lattice-ordered structures.

Examples often cited include taking the power set lattice with union as the join and with as set-union as well (a degenerate but illustrative case), or using with to model aggregation of attributes in formal concept analysis or knowledge representation. In practice, latticewith serves as a flexible framework for examining how an aggregation-like operation coexists with lattice order.

See also: lattice, semilattice, formal concept analysis, lattice-ordered structures, data fusion.