Difference between revisions of "Brouwer lattice"
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''Brouwer structure, Brouwer algebra'' | ''Brouwer structure, Brouwer algebra'' | ||
| − | A [[Distributive lattice|distributive lattice]] in which for each pair of elements | + | A [[Distributive lattice|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|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|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 | + | 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, [[#References|[2]]]). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.C.C. McKinsey, A. Tarski, "The algebra of topology" ''Ann. of Math. (2)'' , '''45''' : 1 (1944) pp. 141–191</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1967)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.C.C. McKinsey, A. Tarski, "The algebra of topology" ''Ann. of Math. (2)'' , '''45''' : 1 (1944) pp. 141–191</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1967)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
Latest revision as of 06:29, 30 May 2020
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 |
Brouwer lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brouwer_lattice&oldid=46165