Complemented Lattice

A complemented lattice is an algebraic structure  such that  is a bounded lattice and for each element , the element  is a complement of , meaning that it satisfies

1. 

2. .

A related notion is that of a lattice with complements. Such a structure is a bounded lattice  such that for each , there is  such that  and .

One difference between these notions is that the class of complemented lattices forms a variety, whilst the class of lattices with complements does not. (The class of lattices with complements is a subclass of the variety of lattices, but it is not a subvariety of the class of lattices.) Every lattice with complements is a reduct of a complemented lattice, by the axiom of choice. To see this, let  be a lattice with complements. For each , let  denote the set of complements of . Because  is a lattice with complements, for each ,  is nonempty, so by the axiom of choice, we may choose from each collection  a distinguished complement  for . This defines a function  which is a complementation operation, meaning that it satisfies the properties stated above for the complementation operation of a complemented lattice. Augmenting the bounded lattice  with this operation yields a complemented lattice,  of which the original lattice with complements  is a reduct.