Canonical Extensions and Profinite Completions of Semilattices and Lattices

Abstract

Canonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class ${{\boldsymbol{\mathcal{S}}}}_{\wedge}$ of (unital) meet semilattices. Any ${\mathbf S}\in {{\boldsymbol{\mathcal{S}}}}_{\wedge}$ embeds into the algebraic closure system Filt(Filt(S)). This iterated filter completion, denoted Filt²(S), is a compact and ${\textstyle{\bigvee}\,}{\textstyle{\bigwedge}\,}$-dense extension of S. The complete meet-subsemilattice S ^δ of Filt²(S) consisting of those elements which satisfy the condition of ${\textstyle{\bigwedge}\,}{\textstyle{\bigvee}\,}$-density is shown to provide a realisation of the canonical extension of S. The easy validation of the construction is independent of the theory of Galois connections. Canonical extensions of bounded lattices are brought within this framework by considering semilattice reducts. Any S in ${{\boldsymbol{\mathcal{S}}}}_{\wedge}$ has a profinite completion, ${\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})$. Via the duality theory available for semilattices, ${\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})$ can be identified with Filt²(S), or, if an abstract approach is adopted, with ${\mathbb F_{\sqcup}}({\mathbb F_{\sqcap}}({\mathbf S}))$, the free join completion of the free meet completion of S. Lifting of semilattice morphisms can be considered in any of these settings. This leads, inter alia, to a very transparent proof that a homomorphism between bounded lattices lifts to a complete lattice homomorphism between the canonical extensions. Finally, we demonstrate, with examples, that the profinite completion of S, for ${\mathbf S} \in {{\boldsymbol{\mathcal{S}}}}_{\wedge}$, need not be a canonical extension. This contrasts with the situation for the variety of bounded distributive lattices, within which profinite completion and canonical extension coincide.