Heyting Algebra - Examples

Examples

  • Every Boolean algebra is a Heyting algebra, with given by .
  • Every totally ordered set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise.
  • The simplest Heyting algebra that is not already a Boolean algebra is the totally ordered set {0, ½, 1} with defined as above, yielding the operations:
b
a 		0 ½ 1
0 0 0 0
½ 0 ½ ½
1 0 ½ 1
b
a 		0 ½ 1
0 0 ½ 1
½ ½ ½ 1
1 1 1 1
b
a 		0 ½ 1
0 1 1 1
½ 0 1 1
1 0 ½ 1
a
0 1
½ 0
1 0

Notice that ½∨¬½ = ½∨(½ → 0) = ½∨0 = ½ falsifies the law of excluded middle.

  • Every topology provides a complete Heyting algebra in the form of its open set lattice. In this case, the element is the interior of the union of and, where denotes the complement of the open set . Not all complete Heyting algebras are of this form. These issues are studied in pointless topology, where complete Heyting algebras are also called frames or locales.
  • The Lindenbaum algebra of propositional intuitionistic logic is a Heyting algebra.
  • The global elements of the subobject classifier of an elementary topos form a Heyting algebra; it is the Heyting algebra of truth values of the intuitionistic higher-order logic induced by the topos.

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