Editing Heyting algebra (section)

===Heyting algebra of formulas equivalent with respect to a theory ''T''===
Given a set of formulas ''T'' in the variables {''A''<sub>''i''</sub>}, viewed as axioms, the same construction could have been carried out with respect to a relation ''F''≼''G'' defined on ''L'' to mean that ''G'' is a provable consequence of ''F'' and the set of axioms ''T''. Let us denote by ''H''<sub>''T''</sub> the Heyting algebra so obtained. Then ''H''<sub>''T''</sub> satisfies the same universal property as ''H''<sub>0</sub> above, but with respect to Heyting algebras ''H'' and families of elements 〈''a''<sub>''i''</sub>〉 satisfying the property that ''J''(〈''a''<sub>''i''</sub>〉)=1 for any axiom ''J''(〈''A''<sub>''i''</sub>〉) in ''T''. (Let us note that ''H''<sub>''T''</sub>, taken with the family of its elements 〈[''A''<sub>''i''</sub>]〉, itself satisfies this property.) The existence and uniqueness of the morphism is proved the same way as for ''H''<sub>0</sub>, except that one must modify the metaimplication {{nowrap|1 ⇒ 2}} in "[[#Provable identities|Provable identities]]" so that 1 reads "provably true ''from T''," and 2 reads "any elements ''a''<sub>1</sub>, ''a''<sub>2</sub>,..., ''a''<sub>''n''</sub> in ''H'' ''satisfying the formulas of T''."

The Heyting algebra ''H''<sub>''T''</sub> that we have just defined can be viewed as a quotient of the free Heyting algebra ''H''<sub>0</sub> on the same set of variables, by applying the universal property of ''H''<sub>0</sub> with respect to ''H''<sub>''T''</sub>, and the family of its elements 〈[''A''<sub>''i''</sub>]〉.

Every Heyting algebra is isomorphic to one of the form ''H''<sub>''T''</sub>. To see this, let ''H'' be any Heyting algebra, and let 〈''a''<sub>''i''</sub>: ''i''∈I〉 be a family of elements generating ''H'' (for example, any surjective family). Now consider the set ''T'' of formulas ''J''(〈''A''<sub>''i''</sub>〉) in the variables 〈''A''<sub>''i''</sub>: ''i''∈I〉 such that ''J''(〈''a''<sub>''i''</sub>〉)=1. Then we obtain a morphism ''f'':''H''<sub>''T''</sub>→''H'' by the universal property of ''H''<sub>''T''</sub>, which is clearly surjective. It is not difficult to show that ''f'' is injective.