完全海廷代数

数学特别是序理论中,完全海廷代数是作为完全海廷代数。完全海廷代数是三个不同范畴对象,它们是范畴CHey,locales的范畴Loc,它的对偶frames的范畴Frm

定义

考虑是完全格偏序集合P, ≤)。则P是完全海廷代数,如果任何下列等价条件中的一个成立:

  • P是海廷代数,就是说运算 ( x - )有一个右伴随(也叫做(单调)伽罗瓦连接的下伴随),对于每个P的元素x
  • 对于所有P的元素x和所有P的子集S,下列无限分配律成立:
  • P是分配格,就是说对于所有P中的x, yz,有着
并且P是交连续性的,就是说交运算 ( x - )对于所有P中的x斯科特连续性的。

例子

完全海廷代数引发自带有无限析取的(直觉)逻辑的林登鲍姆-塔斯基代数

引用

  • P. T. Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1982. (ISBN 0-521-23893-5)
Still a great resource on locales and complete Heyting algebras.
  • G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains, In Encyclopedia of Mathematics and its Applications, Vol. 93, Cambridge University Press, 2003. ISBN 0-521-80338-1
Includes the characterization in terms of meet continuity.
  • Francis Borceux: Handbook of Categorical Algebra III, volume 52 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.
Surprisingly extensive resource on locales and Heyting algebras. Takes a more categorical viewpoint.
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