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A lattice P is transferable for a class of lattices \(\mathcal {K}\) if whenever P can be embedded into the ideal lattice \(\mathfrak {I}K\) of some \(K\in \mathcal {K}\), then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice \(\mathfrak {I}K\) with the MacNeille completion \(\overline{K}\). Basic properties of MacNeille transferability are developed. Particular attention is paid to MacNeille transferability in the class of Heyting algebras where it relates to stable classes of Heyting algebras, and hence to stable intermediate logics.

MacNeille transferability and stable classes of Heyting algebras