Heyting-Algebra

In mathematics, Heyting algebras are special partial orders; simultaneously, the notion of Heyting algebra is a generalization of the concept of Boolean algebra. Heyting algebras arise as models of intuitionistic logic, a logic in which the law of excluded middle does not apply in general. Complete Heyting algebras are a central subject of the point -free topology.

The Heyting algebra is named after Arend Heyting.

Formal definition

A Heyting algebra is a limited association with the property that there is for all, and in a greatest element in with

This element is called and written the relative Pseudokomplement of respect.

An equivalent definition can be given by the following figures:

For fixed in. A limited association is a Heyting algebra if and only if all the pictures are Linksadjungierte a Galoisverbindung. In this case, the respective Rechtsadjungierte is given by being defined as above.

A complete Heyting algebra is a Heyting algebra, which is a complete lattice.

In any Heyting algebra can be the Pseudokomplement an element defined by, where 0 is the smallest element of the Heyting algebra. It is true, and is also the largest element with this property. However, in general does not apply, so that only a Pseudokomplement and is not a real complement.

Examples

• Each total order, which is a limited association, is also a Heyting algebra in which different for all a 0.

• Every Boolean algebra is a Heyting algebra, with defined as.

• The simplest Heyting algebra, which already is not a Boolean algebra, is the linearly ordered set {0, ½, 1} with the following operations:

• The association of open sets of a topological space is a complete Heyting algebra. In this case, the interior of the association of B, wherein the complement of the open set A is. Not all complete Heyting algebras are generated in this way. Related issues are studied in the point- free topology in which complete Heyting algebras and frames or locales are called.

• The Lindenbaum algebra of intuitionistic propositional logic is a Heyting algebra. It is defined as the set of all propositional formulas, ordered by the logical entailment relation: for two formulas F and G is precisely when. However, it is only a quasi-ordering, which induces a partial order, which is then the desired 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 induced from the topos of intuitionistic logic of higher level.

Properties

Heyting algebras are always distributive, that is, an association of A with a binary operation is a Heyting algebra if and only if:

The distributive law is sometimes postulated as an axiom, but it follows from the existence of relative Pseudokomplemente. The reason for this is that preserved as the lower adjoint of a Galois connection all existing suprema. But Distributivity is nothing more than the preservation of binary Suprema by.

With a similar argument, the following can be infinitäres distributive show in complete Heyting algebras:

For all x in H and subsets Y of H.

Not every Heyting algebra satisfies the two De Morgan 's laws. However, the following statements are in any Heyting algebra H are equivalent:

The Pseudokomplement an element x in H is the supremum of the set, and it belongs to this amount (ie it applies ).

An element x of a Heyting algebra H is called regular if one of the following equivalent conditions holds:

A Heyting algebra H is a Boolean algebra if and only if one of the following equivalent conditions holds:

In this case the element is the same.

In any Heyting algebra are the smallest and the largest element, 0 and 1 regular.

The regular elements of a Heyting algebra form a Boolean algebra. If not all elements of the Heyting algebra are regular, this Boolean algebra is not a sublattice of the Heyting algebra, because the supremum operations are different.

Is a Heyting algebra, it may be that the corresponding dual association is also a Heyting algebra. If this is so, one can form an item in the Pseudokomplement and interpret this as an element of. It is then always valid. If applies, is.

In contrast to many -valued logics share Heyting algebras with Boolean algebras the following property: If the negation has a fixed point ( ie one ), then the Heyting algebra is trivial: it consists of only one element.

Importance for intuitionistic logic

Arend Heytings motivation to introduce this term was to clarify the meaning of intuitionistic logic for the foundations of mathematics. The Peircean law illustrates the role that Heyting algebras for the semantics play intuitionistic logic. There is no known simple proof which shows that the Peircean law can not be derived using the proof rules for intuitionistic logic.

A Heyting algebra is seen from the logical point of view, a generalization of the usual amount of truth values ​​. Among other corresponds to the greatest element 1 of a Heyting algebra of the truth value "true"; the smallest element 0 corresponds to "false". The usual two-valued logic is the simplest example of a Heyting algebra - it consists only of these two elements. Abstract told the two -element Boolean algebra is also ( as every Boolean algebra ) is a Heyting algebra.

Classically valid formulas are those which under any assignment of the propositional variables in the divalent Boolean algebra value of 1 ( "true") revealed that the usual propositional tautologies. ( Equivalently, can be considered in all Boolean algebras, all layouts.) Intuitionistic valid formulas, however, are those which give the value 1 for all Heyting algebras and all assignments. In the above three-element Heyting algebra is the value of Peirce's law is not always 1: if we prove P and Q with ½ 0, then the value is not 1, but only ½. According to what we said above, this means that the Peircean law is not valid intuitionistically - classic but it is already valid.