On profinite completions and canonical extensions

We show that if a variety V of monotone lattice expansions is finitely generated, then profinite completions agree with canonical extensions on V . The converse holds for varieties of finite type. It is a matter of folklore that the profinite completion of a Boolean algebra B is given by the power set of the Stone space of B, or in the terminology of Jónsson and Tarski [5], by the canonical extension of B. Similarly, the profinite completion of a distributive lattice D is given by the lattice of upsets of the Priestley space of D, or equivalently, by the canonical extension of D [4]. In [1], Bezhanishvili et. al., give a description of the profinite completion of a Heyting algebra in terms of its dual space. As a consequence of this, they obtain that for a variety V of Heyting algebras, profinite completions coincide with canonical extensions for all members of V if, and only if, V is finitely generated. It is our purpose here to show that this result holds in a wider setting. Theorem. For any variety V of monotone lattice expansions, the first condition below implies the second, and for varieties of finite type the conditions are equivalent. (1) V is finitely generated. (2) Profinite completions coincide with canonical extensions on V . Before proving the theorem, we recall a few basics. As defined by Gehrke and Harding [2], the canonical completion of a bounded lattice L is a pair (e, C) where (i) e : L → C is a bounded lattice embedding, (ii) C is a complete lattice, (iii) each element of C is a join of meets and a meet of joins of elements of the image e[L] of L, and (iv) if F, I are a filter and ideal of L, then ∧ e[F ] ≤ ∨ e[I] implies F ∩ I = ∅. Suppose L is a bounded lattice and f is an n-ary operation on L that in each coordinate either preserves or reverses order. We call f a monotone operation on L, and call a bounded lattice L with a family of monotone operations a monotone Presented by I. Hodkinson. Received May 14, 2005; accepted in final form September 8, 2005. 2000 Mathematics Subject Classification: 06B23; 08B25, 06E25.