A Heyting algebra is supplemented if each element a has a dual pseudo-complement \(a^+\), and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigate a new type of completion of Heyting algebras arising in the context of algebraic proof theory, the so-called hyper-MacNeille completion. We show that the hyper-MacNeille completion of a Heyting algebra is the MacNeille completion of its centrally supplemented extension. This provides an algebraic description of the hyper-MacNeille completion of a Heyting algebra, allows development of further properties of the hyper-MacNeille completion, and provides new examples of varieties of Heyting algebras that are closed under hyper-MacNeille completions. In particular, connections between the centrally supplemented extension and Boolean products allow us to show that any finitely generated variety of Heyting algebras is closed under hyper-MacNeille completions.

如果每个元素a具有双重伪补码\（a ^ + \），则补充Heyting代数，如果Heyting代数被补充并且每个补充都是中心的，则Heyting代数是中心的补充。我们证明，每个Heyting代数在与原始Heyting代数相同的各种变体中都有一个中央补充扩展。我们使用此工具来研究在代数证明理论的背景下出现的新型Heyting代数完成，即所谓的超MacNeille完成。我们表明，Heyting代数的超MacNeille完成是其中央补充扩展的MacNeille完成。这提供了Heyting代数的超MacNeille完备的代数描述，允许开发hyper-MacNeille完备的进一步的性质，并提供了在Hyper-MacNeille完备条件下封闭的Heyting代数的新示例。尤其是，