täiskomplementaarsed

Täiskomplementaarsed is a term used in mathematics, particularly in lattice theory and Boolean algebra, to describe structures in which complements exist for all elements relative to the bounds of the system. In standard terminology these structures are called complemented lattices, and when every element has a unique complement, they are often linked to Boolean algebras in the presence of additional laws such as distributivity.

Definition and basic idea: Let L be a bounded lattice with least element 0 and greatest element 1. An element a ∈ L has a complement b ∈ L if a ∨ b = 1 and a ∧ b = 0. If every element of L has at least one complement, L is called complemented. If every element has a unique complement, L is sometimes described as uniquely complemented. The terminology "täiskomplementaarsed" reflects the idea that the complement operation is defined for all elements of the structure.

Relationship to Boolean algebras: A lattice that is both complemented and distributive satisfies the axioms of a Boolean algebra. In a Boolean algebra, the complement operation is an involution (the complement of the complement is the original element), and it corresponds to the logical NOT operation in propositional logic or the set-theoretic complement in a power set lattice.

Examples and applications: The classic example is the Boolean algebra of all subsets of a set X, with union, intersection, and set-theoretic complement relative to X. More generally, power set lattices form Boolean algebras and hence are uniquely complemented. Applications appear in logic, set theory, digital circuit design, and information theory.

Notes: In Estonian mathematical writing, täiskomplementaarsed may be used to describe structures with full complementarity properties. The precise English equivalents are complemented lattice, uniquely complemented lattice, or Boolean algebra, depending on the presence of distributivity and other axioms.

See also: Boolean algebra, complemented lattice, orthocomplemented lattice, lattice theory, propositional logic.