A monadic approach to the McNeille completion of a lattice (Q1239175)

Nonstandard models of lattices serve as a tool to construct, for any given lattice \(L\), its MacNeille (or normal) completion \(\hat L\). \(\hat L\) is described as a partially ordered space of monads -- computed in an arbitrary highly saturated nonstandard model \({}^*L\) of \(L\) -- of filters of intervals in \(L\). We are led to a natural condition \((P)\) on \(L\) which roughly states that lattice operations in the monad space are ``close'' to those in \(L\). If \((P)\) holds in \(L\), \(\hat L\) and \(L\) are shown to satisfy the same set of lattice identities. As a special case, we obtain a new characterization (cf. [\textit{N. Funayama}, Proc. Imp. Acad. Tokyo 20, 1--2 (1944; Zbl 0063.01484)]) of those distributive lattices having distributive MacNeille completion: For \(L\) distributive, \(\hat L\) is distributive if and only if \((P)\) holds in \(L\), and \((P)\) takes a particularly simple form in that case. As an example, every distributive relatively complemented lattice satisfies \((P)\). For a general development of lattice extensions in nonstandard terms, the reader may wish to compare the present paper with the author's forthcoming one [Houston J. Math. 3, 423--439 (1977; Zbl 0376.06007)], where the above-mentioned results are considerably generalized.