# 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)

In Western literature pseudo-Boolean algebras are more commonly called Heyting algebras. Complete Heyting algebras (often called frames or locales) have been extensively studied on account of their connections with topology: the lattice of open sets of any topological space is a locale, and locales can in some respects be considered as generalized topological spaces. See [a1], [a2], [a3].

#### References

 [a1] M.P. Fourman, D.S. Scott, "Sheaves and logic" M.P. Fourman (ed.) C.J. Mulvey (ed.) D.S. Scott (ed.) , Applications of sheaves , Lect. notes in math. , 753 , Springer (1979) pp. 302–401 [a2] P.T. Johnstone, "Stone spaces" , Cambridge Univ. Press (1983) [a3] H. Simmons, "A framework for topology" , Logic colloquium '77 , Studies in logic and foundations of math. , 96 , North-Holland (1978) pp. 239–251
How to Cite This Entry:
Brouwer lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brouwer_lattice&oldid=46165
This article was adapted from an original article by V.A. Yankov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article