LEMdomain

LEMdomain is a concept in theoretical computer science and mathematical logic that denotes a class of informational domains whose internal logic validates the Law of Excluded Middle (LEM). The term combines 'LEM' for the law with 'domain', indicating a semantic space in which statements possess a definitive true or false value. In practice, LEMdomains are modeled as Boolean-algebra-like structures or as special lattice domains equipped with a classical truth valuation.

Definition and properties: A LEMdomain is typically a complemented distributive lattice or Boolean algebra with a least element and a greatest element. For every element p, there exists a complement ¬p such that p ∨ ¬p is the top element and p ∧ ¬p is bottom. This enforces decidability of propositions, aligning with classical logic. Monotone maps between LEMdomains preserve joins and meets, supporting semantic interpretation of programs and proofs.

Examples and relation: The two-element Boolean algebra {true, false} is the canonical LEMdomain. The power set P(S), ordered by inclusion, also forms a LEMdomain with complement as set-theoretic complement. In constructive or intuitionistic settings, LEMdomains do not generally apply. In domain theory, LEMdomains contrast with generally non-Boolean domains that model constructive logics. They are related to Boolean algebras, lattice-ordered sets, and toposes.

Applications: Used in formal semantics of programming languages that assume classical logic, in automated theorem proving, and in type systems where decidable truth values are required. Limitations include reduced expressiveness for constructive mathematics and potential mismatch with models of computation that rely on partial information.