Absorption law

The absorption laws state that in a lattice, joining an element with its meet with another element, or meeting it with their join, leaves it unchanged.

Lattice (order theory): A partially ordered set in which every pair of elements has a greatest lower bound and a least upper bound. Its meet and join operations satisfy the absorption identities.
Lattice identities: Equations that hold for meet and join in every lattice. Absorption is one of the defining equations in this family.
Boolean algebra: A complemented distributive lattice with least and greatest elements. Its meet and join correspond to conjunction and disjunction, where absorption simplifies formulas.
Distributive law: An identity describing how one binary operation distributes over another. Distribution expands or factors expressions; absorption eliminates a nested expression.
Meet (order theory): The greatest lower bound of two elements in a partially ordered set. Absorption combines a meet with a join involving the same element.
Commutativity: The property that an operation gives the same result when its operands are exchanged. Meet and join are commutative, allowing absorption equations to be written in equivalent forms.
Propositional logic: A formal system for reasoning with propositions and truth-functional connectives. The laws simplify expressions such as a conjunction with a disjunction containing that proposition.
De Morgan's laws: Identities relating negation of conjunctions or meets to disjunctions or joins of negations. These laws require complementation, while absorption holds in every lattice.
Join (order theory): The least upper bound of two elements in a partially ordered set. Absorption combines a join with a meet involving the same element.
Associativity: The property that regrouping operands does not change the result of an operation. Associativity lets lattice expressions be regrouped alongside absorption simplifications.

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Consensus theoremRelated: It supplies a basic simplification law used to derive the consensus identity.