We prove that there exist profinite Heyting algebras that are not isomorphic to the profinite completion of any Heyting algebra. This resolves an open problem from 2009. More generally, we characterize those varieties of Heyting algebras in which profinite algebras are isomorphic to profinite completions. It turns out that there exists largest such. We give different characterizations of this variety and show that it is finitely axiomatizable and locally finite. From this it follows that it is decidable whether in a finitely axiomatizable variety of Heyting algebras all profinite members are profinite completions. In addition, we introduce and characterize representable varieties of Heyting algebras, thus drawing connection to the classical problem of representing posets as prime spectra.

我们证明存在不与任何 Heyting 代数的超完备性同构的超限 Heyting 代数。这解决了 2009 年的一个悬而未决的问题。更一般地说，我们描述了 Heyting 代数的变种，其中超限代数与超限完成同构。事实证明，存在最大的这样的。我们给出了这种多样性的不同特征，并表明它是有限公理化的和局部有限的。由此推导出，在 Heyting 代数的有限公理化变体中，是否所有超限成员都是超限完成是可判定的。此外，我们介绍和表征了 Heyting 代数的可表示变体，从而与将偏序集表示为素谱的经典问题联系起来。