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Difference between revisions of "Brouwer lattice"

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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 [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].
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In Western literature pseudo-Boolean algebras are more commonly called Heyting algebras. Complete Heyting algebras (often called frames or [[locale]]s) 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 [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].
  
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.T. Johnstone,  "Stone spaces" , Cambridge Univ. Press  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Simmons,  "A framework for topology" , ''Logic colloquium '77'' , ''Studies in logic and foundations of math.'' , '''96''' , North-Holland  (1978)  pp. 239–251</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.T. Johnstone,  "Stone spaces" , Cambridge Univ. Press  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Simmons,  "A framework for topology" , ''Logic colloquium '77'' , ''Studies in logic and foundations of math.'' , '''96''' , North-Holland  (1978)  pp. 239–251</TD></TR></table>

Revision as of 08:03, 16 April 2018

Brouwer structure, Brouwer algebra

A distributive lattice in which for each pair of elements there exists an element, called the pseudo-difference (frequently denoted by ), which is the smallest element possessing the property . An equivalent description of a Brouwer lattice is as a variety of universal algebras (cf. Universal algebra) with three binary operations , 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 , and of new unions and intersections according to the formulas

and the operation of relative pseudo-complementation which corresponds to the pseudo-difference . 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