# Brouwer lattice

Brouwer structure, Brouwer algebra

A distributive lattice in which for each pair of elements $a, b$ there exists an element, called the pseudo-difference (frequently denoted by $a {} _ {-} ^ {*} b$), which is the smallest element $c$ possessing the property $b+c \geq a$. An equivalent description of a Brouwer lattice is as a variety of universal algebras (cf. Universal algebra) with three binary operations $\cup$, $\cap$ 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 $(a \leq ^ \prime b) \iff (b \leq a)$, and of new unions and intersections according to the formulas

$$(a \cup ^ \prime b) \iff \ (a \cap b),\ \ (a \cap ^ \prime b) \iff \ (a \cup b)$$

and the operation of relative pseudo-complementation $a \iff b$ which corresponds to the pseudo-difference $a {} _ {-} ^ {*} b$. 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)