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''Brouwer structure, Brouwer algebra''
 
''Brouwer structure, Brouwer algebra''
  
A [[Distributive lattice|distributive lattice]] in which for each pair of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176601.png" /> there exists an element, called the pseudo-difference (frequently denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176602.png" />), which is the smallest element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176603.png" /> possessing the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176604.png" />. An equivalent description of a Brouwer lattice is as a variety of universal algebras (cf. [[Universal algebra|Universal algebra]]) with three binary operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176607.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176608.png" />, and of new unions and intersections according to the formulas
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b0176609.png" /></td> </tr></table>
+
$$
 +
(a \cup  ^  \prime  b)  \iff \
 +
(a \cap b),\ \
 +
(a \cap  ^  \prime  b)  \iff \
 +
(a \cup b)
 +
$$
  
and the operation of relative pseudo-complementation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b01766010.png" /> which corresponds to the pseudo-difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017660/b01766011.png" />. 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]]]).
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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>
 
 
  
 
====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
How to Cite This Entry:
Brouwer lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brouwer_lattice&oldid=43167
This article was adapted from an original article by V.A. Yankov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article