# Brouwer lattice

Brouwer structure, Brouwer algebra

A distributive lattice in which for each pair of elements there exists an element, called the pseudo-difference (frequently denoted by ), which is the smallest element possessing the property . An equivalent description of a Brouwer lattice is as a variety of universal algebras (cf. Universal algebra) with three binary operations , and , which satisfies certain axioms. The term "Brouwer algebra" was introduced in recognition of the connection between Brouwer lattices and Brouwer's intuitionistic logic. Instead of Brouwer lattices the so-called pseudo-Boolean algebras are often employed, the theory of which is dual to that of Brouwer lattices. Any Brouwer lattice can be converted to a pseudo-Boolean algebra by the introduction of a new order , and of new unions and intersections according to the formulas

and the operation of relative pseudo-complementation which corresponds to the pseudo-difference . Conversely, any pseudo-Boolean algebra can be regarded as a Brouwer lattice. The term "Brouwer lattice" is sometimes used to denote a pseudo-Boolean algebra (see, for instance, [2]).

#### References

 [1] J.C.C. McKinsey, A. Tarski, "The algebra of topology" Ann. of Math. (2) , 45 : 1 (1944) pp. 141–191 [2] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)