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)

Comments

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