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''Brouwer structure, Brouwer algebra''

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  $  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====
<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====
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]]].

====References====
<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>