Heyting algebra

In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. Heyting algebras arise as models of intuitionistic logic, a logic in which the law of excluded middle does not in general hold. Complete Heyting algebras are a central object of study in pointless topology. == Formal definitions == A Heyting algebra ''H'' is a bounded lattice such that for all ''a'' and ''b'' in ''H'' there is a greatest element ''x'' of ''H'' such that $a\; \backslash wedge\; x\; \backslash le\; b$. This element is called the relative pseudo-complement of ''a'' with respect to ''b'', and is denoted $a\; \backslash Rightarrow\; b$ (or $a\; \backslash rightarrow\; b$). An equivalent definition can be given by considering the mappings $f\_a:\; H\; \backslash to\; H$ defined by $f\_a(x)=a\backslash wedge\; x$, for some fixed ''a'' in ''H''. A bounded lattice ''H'' is a Heyting algebra iff all mappings $f\_a$ are the lower adjoint of a monotone Galois connection. In this case the respective upper adjoints $g\_a$ are given by $g\_a(x)=\; a\; \backslash Rightarrow\; x$, where $\backslash Rightarrow$ is defined as above. A complete Heyting algebra is a Heyting algebra that is a complete lattice. In any Heyting algebra, one can define the pseudo-complement $\backslash lnot\; x$ of some element ''x'' by setting $\backslash lnot\; x\; =\; x\; \backslash Rightarrow\; 0$, where 0 is the least element of the Heyting algebra. An element ''x'' of a Heyting algebra is called regular if $x=\backslash lnot\backslash lnot\; x$. An element ''x'' is regular if and only if $x=\backslash lnot\; y$ for some element ''y'' of the Heyting algebra. == Properties == Heyting algebras are always distributivity. This is sometimes stated as an axiom, but in fact it follows from the existence of relative pseudo-complements. The reason is that, being the lower adjoint of a Galois connection, $\backslash wedge$ limit-preserving function (order theory) all existing suprema. Distributivity in turn is just the preservation of binary suprema by $\backslash wedge$. Furthermore, by a similar argument, the following infinite distributive law holds in any complete Heyting algebra: $x\backslash wedge\backslash bigvee\; Y\; =\; \backslash bigvee\; \backslash \{x\backslash wedge\; y\; :\; y\; \backslash in\; Y\backslash \}$ for any element ''x'' in ''H'' and any subset ''Y'' of ''H''. Not every Heyting algebra satisfies the two De Morgan laws. However, the following statements are equivalent for all Heyting algebras ''H'': # ''H'' satisfies both De Morgan laws. # $\backslash lnot(x\; \backslash wedge\; y)=\backslash lnot\; x\; \backslash vee\; \backslash lnot\; y$, for all ''x'', ''y'' in ''H''. # $\backslash lnot\; x\; \backslash vee\; \backslash lnot\backslash lnot\; x\; =\; 1$, for all ''x'' in ''H''. # $\backslash lnot\backslash lnot\; (x\; \backslash vee\; y)\; =\; \backslash lnot\backslash lnot\; x\; \backslash vee\; \backslash lnot\backslash lnot\; y$, for all ''x'', ''y'' in ''H''. The pseudo-complement of an element ''x'' of ''H'' is the supremum of the set $\backslash \{\; y\; :\; y\; \backslash wedge\; x\; =\; 0\backslash \}$ and it belongs to this set (i.e. $x\; \backslash wedge\; \backslash lnot\; x\; =\; 0$ holds). Boolean algebras are exactly those Heyting algebras in which $x\; =\; \backslash lnot\backslash lnot\; x$ for all ''x'', or, equivalently, in which $x\backslash vee\backslash lnot\; x=1$ for all ''x''. In this case, the element $a\; \backslash Rightarrow\; b$ is equal to $\backslash lnot\; a\; \backslash vee\; b$. In any Heyting algebra, the least and greatest elements 0 and 1 are regular. The regular elements of any Heyting algebra constitute a Boolean algebra. Unless all elements of the Heyting algebra are regular, this Boolean algebra will not be a sublattice of the Heyting algebra, because its join operation will be different. == Examples == * Every total order that is a bounded lattice is also a Heyting algebra, where $\backslash lnot\; 0\; =\; 1$ and $\backslash lnot\; a\; =\; 0$ for all ''a'' other than 0. * Every topology provides a complete Heyting algebra in the form of its open set lattice. In this case, the element $A\; \backslash Rightarrow\; B$ is the interior (topology) of the union of $A\_c$ and ''B'', where $A\_c$ denotes the complement of the open set ''A''. 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. It is defined to be the set of all propositional logic formulae, ordered via logical entailment: for any two formulae ''F'' and ''G'' we have $F\; \backslash le\; G$ iff $F\; \backslash models\; G$. At this stage $\backslash le$ is merely a preorder that induces a partial order which is the desired Heyting algebra. ==Heyting algebras as applied to intuitionistic logic== Arend Heyting (1898-1980) was himself interested in clarifying the foundational status of intuitionistic logic, in introducing this type of structures. The case of Peirce's law illustrates the semantic role of Heyting algebras. No simple proof is known that Peirce's law cannot be deduced from the basic laws of intuitionistic logic. A Heyting algebra, from the logical standpoint, is essentially a generalization of the usual system of truth values. Amongst other properties, the largest element, called in logic $\backslash top$, is analogous to 'true'. The usual two-valued logic system is the simplest example of a Heyting algebra, one in which the elements of the algebra are $\backslash top$ (true) and $\backslash bot$ (false). That is, in abstract terms, the two-element Boolean algebra is also a Heyting algebra. ''Classically valid'' formulas are those formulas that have a value of $\backslash top$ in this Boolean algebra under any possible assignment of true and false to the formula's variables — that is, they are formulas which are tautologies in the usual truth-table sense. ''Intuitionistically valid'' formulas are those formulas that have a value of $\backslash top$ in ''any'' Heyting algebra under any assignment of values to the formula's variables. One can construct a Heyting algebra in which the value of Peirce's law is not always $\backslash top$. From what has just been said, this does show that it cannot be derived. See Curry-Howard isomorphism for the general context, of what this implies in type theory. ==See also== * list of Boolean algebra topics == References == * F. Borceux, ''Handbook of Categorical Algebra 3'', In ''Encyclopedia of Mathematics and its Applications'', Vol. 53, Cambridge University Press, 1994. * G. Gierz, K.H. Hoffmann, K. Keinel, 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. Order theory

Heyting algebra

This definition looks identical to that of the concepf of Boolean algebra: a Boolean algebra is a complemented distributive lattice, provided one includes boundedness in the definition of lattice. What is the difference, if any, supposed to be? User:Michael Hardy 03:27, 9 Nov 2003 (UTC) Unclear, certainly. Heyting algebras are not in general Boolean algebras. The point is that negation is defined as not x = 'x implies bottom', and this doesn't in general satisfy not not x = x. I think relatively complemented mis-states the definition, which is presumably meant to be that x implies y always exists , as a supremum. User:Charles Matthews 08:31, 9 Nov 2003 (UTC)

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