View source for Brouwer lattice

''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

<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>

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]]]).

====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 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]]].

====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>