From Objects to Diagrams for Ranges of Functors

Abstract

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Cevian operations on distributive lattices

“…For each x ∈ X, the map ρ x def = π X x ⊗ A is a surjective lattice homomorphism from B onto A ∂x (cf. 2.6.7, 3.1.2, and 3.1.3 in [11]). In particular, if x ∈ X (1) , then ∂x ∈ {1, 2, 3}, thus A ∂x ∼ = {0, 1}, and we may pick b Since all B x are finite, they are finitely presented within S, thus we can apply the Armature Lemma [11, Lemma 3.2.2] to those data, with the B x in place of the required S x and the identity of B in place of χ.…”

“…This means that X = (a, u) | a ∈ P , u : U → ω 2 with a = U with componentwise ordering (≤ on the first component, extension ordering on the second one), and ∂(a, u) = a whenever (a, u) ∈ X. It follows from (the proof of) [11,Lemma 3.5.5] that X is lower finite and that furthermore, it is, together with the map ∂, a principal ω-lifter of P (cf. Definition 6.1).…”

“…For a morphism ϕ : A → B between finitely presented P -scaled Boolean algebras and an atom b of B, we define b ϕ as the unique atom of A such that b ≤ ϕ(b ϕ ). Then the product morphism ϕ ⊗ S def = σ |b|B |b ϕ |A | b ∈ At B goes from A⊗ S to B⊗ S. This defines a functor from the finitely presented members of Bool P to S. Since every member of Bool P is a small directed colimit of a direct system of finitely presented objects, it follows, using [11,Proposition 1.4.2], that this functor can be uniquely extended, up to isomorphism, to a functor from Bool P to S that preserves all small directed colimits. This functor will also be denoted by − ⊗ S. For a P -scaled Boolean algebra A, the object A ⊗ S will be called a condensate of S.…”

“…Now define X as the poset denoted by P ω 2 in the proof of [11,Lemma 3.5.5], together with the canonical isotone map ∂ : X → P . This means that X = (a, u) | a ∈ P , u : U → ω 2 with a = U with componentwise ordering (≤ on the first component, extension ordering on the second one), and ∂(a, u) = a whenever (a, u) ∈ X.…”

Compact Intersection Property and description of congruence lattices

“…Our results correspond to the two main approaches to the problem of describing Con K. The approach based on lifting of diagrams has been recently greatly developed by P. Gillibert. (See [6] or [7].) The description based on topological representation has been investigated by M. Ploščica ([12], [13], [14]).…”