COVERS IN GENERALIZED FRAMES

Jan Paseka Department of Mathematics, Faculty of Science, Masaryk University Janáčkovo nám. 2a, 662 95 Brno, Czech Republic

0. Introduction

The aim of this paper is to present a candidate for the so called ’partial frame’ (see [10]) i.e. a meet-semilattice in which certain distinguished sets have joins distributing over finite meets. The basic reference for the present text is the classic book by Johnstone [5]. All basic definitions and conventions are taken from either in [5] or in [10]. A familiarity with [10] is assumed since we omit some of the proofs that we obtained as straightforward modifications of the corresponding proofs in that paper. For the motivating examples see the papers [1], [3] and [9]. For the theory of monads and basic facts concerning categories see [8].

The paper is divided in two parts. Section 1 introduces the notion of a partial frame. The monadicity of partial frames is established and a characterization of projective partial frames is given.

Section 2 contains the basic constructions of limits and colimits of partial frames. Especially, we give an explicit construction of a sum of partial frames.

Section 3 solves the problem of the preservation of some covering properties w.r. to sums of partial frames.

For F, G sets, F a finite subset of G we write F ⊆⊆ G. Meet denotes the category of meetsemilattices. Let F be a function with the domain Meet and the codomain Set such that F(L) ⊆ P(L) (the system of all subsets of L). If F ∈ F(L), we shall speak about a set of a type F or an F-set. If X = ↓(F) for F ∈ F(L) then we call X to be an F-generated set. The system of all nonempty F-generated sets will be denoted by F_{l}(L). Recall that we have a meet-semilattice

Typeset by AMS-TEX 2 JAN PASEKA

morphism ↓ from L to F_{l}(L) assigning to each element of L the principal ideal generated by it. For a subset m of L we define P_{F}(m)= {F ∈ F(L) : F ⊆ m}.

1. Partial frames

In the sequel, let F be function with the domain Meet and the codomain Set satisfying the following conditions:

(A1) L ∈ Meet, t ∈ L implies {t} ∈ F(L), ∅ ∈ F(L) ⊆ P(L),

(A2) L ∈ Meet, F, G ∈ F(L) implies F ∧ G ∈ F(L),

(A3) L ∈ Meet, F ∈ F(L), f ∈ F, f = ⋁ F_{f} for some F_{f} ∈ F(L) implies ⋃ F_{f} ∈ F(L), (A4) L, L′ ∈ Meet, h : L → L′ being a meet-semilattice morphism implies h(F(L)) =

F(h(L)) ⊆ F(L′), (A5) L ∈ Meet, ∅ ≠ F ∈ F(F_{l}(L)) implies ⋃{f : f ∈ F } ∈ F_{l}(L). Applying (A2) and (A3) we have that F_{l}(L) is a meet-semilattice closed under F-joins and by

(A4), we have that F ∈ F(L) implies ↓(F) ∈ F_{l}(L) (↓ is a meet-semilattice morphism). Recall that the conditions (A1) , (A2) , (A3) , and (A4) are very close to the conditions (ii), (iii), (iv) and (v) in [4]. 1.1 Definition. We shall say that a meet-semilattice L ∈ Meet is a partial F-frame (shortly a partial frame) over Meet if every set of the type F has a join and which satisfies the F-distributive law

(FD)  a ∧ ⋁ S = ⋁{a ∧ s : s ∈ S}

for all a ∈ L, S ∈ F(L). Recall that, for each meet-semilattice L ∈ Meet, the meet-semilattice F_{l}(L) is actually a partial frame over S. A morphism f : L → L′ of partial frames (F-morphism) is a meet-semilattice morphism preserving F joins. The category of partial frames over Meet is denoted by S_{F}. It is easy to check (by applying (A4)) that F_{l} is a functorial construction.

A F-cover C of a partial frame L is an F-subset of L such that its join is the top element of L. We denote by C_{F}(L) the set of all F-covers of L.

An element a of a partial frame L is said to be dense if, for any x ∈ L, x ∧ a = 0 implies x = 0. A dense cover of L is an F-subset D ⊆ L such that ⋁ U is dense in L. The system of all dense

PARTIAL FRAMES 3

covers of the locale L is denoted by D(L). For any a ∈ L, we put a^{∗} = {x ∈ L : x ∧ a = 0}. Then a is dense if and only if a^{∗} = {0}.

A partial subframe L of a partial frame L′ is a subset of L′ such that the inclusion is a morphism of partial frames i.e. a meet-subsemilattice L such that F ⊆ L, F ∈ F(L) implies F ∈ F(L′), ⋁ F ∈ L. A congruence on a partial frame L is an equivalence relation θ on L such that aθb, cθd ⟹ a∧cθb∧d and F = {f_{i}}, G = {g_{i}} ∈ F(L), f_{i}θg_{i} ⟹ ⋁ F θ ⋁ G. Since the intersection of congruences is a congruence the set CL of all congruences on L is a complete lattice. Moreover CL is a frame. To see this we put, for all congruences θ, ϕ, θ → ϕ = {(a, b) ∈ L × L : (a, b) ∧ (c, d) ∈ ϕ for all (c, d) ∈ θ}. Evidently, θ → ϕ is a congruence on L and α ⊆ θ → ϕ if and only if α ∩ θ ⊆ ϕ for any congruence α i.e. CL is a frame. Examples.

Let L ∈ Meet and put N (L) = {{a} : a ∈ L} ∪ {∅}.

(1) Put F(L)= N (L). Then F_{l}(L) is isomorphic to L and S_{F}= Meet.

(2) Put F(L)= {F : F ⊆⊆ L}. Then F_{l}(L) is a distributive lattice and S_{F} = DLat.

(3) Put F(L)= {F : F ⊆ L, cardF ∈ κ}, κ being a regular cardinal. Then F_{l}(L) is a κ-frame

(see [10]) and S_{F} = κ − Frm.

(4) Put F(L)= P(L). Then F_{l}(L) is a frame and S_{F} = Frm.

Suppose F, G satisfying (A1)-(A5) such that L ∈ S implies F(L) ⊆ G(L). As in [10], there is a forgetful functor U_{F,G} from S_{G} to S_{F}. We say that F is well behaved wrt. G and write F ⊑ G if the following conditions are satisfied:

(W1) for all L ∈ S_{F} there is an idempotent mapping : G(L) → G(L) such that {↓(⋁ F) : F ≺ G, F ∈ F(L)} = {↓(⋁ F) : F ≺ Ĝ, F ∈ F(L)} = ↓(Ĝ) for all G ∈ G(L),

(W2) L, L′ ∈ S_{F}, h : L → L′ being an F-morphism implies h(Ĝ_{L}) = { \atop h(G)}_{L′} for all G ∈ G(L).

Particularly, ↓(Ĝ) = { \atop ↓(G)}.

In the following, we shall always assume that F ⊑ G. Recall that, in the case of κ-frames, we may put Ĝ = {(⋁ F) : F ⊆ G, card(F) ∈ λ} for all G ⊆ L, card(G) ∈ κ.

We define an FG-ideal of a partial F-frame to be the G-generated set closed under F-joins. We

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denote by Q_{F,G}(L) the system of all FG-ideals. It is easy to check that Q_{F,G}(L) is a G-frame, ↓ : L → Q_{F,G}(L) is an F-morphism (using (A5) and the well behavedness) and that Q_{F,G} is a functorial construction (factorizing through G_{l}(L)). Moreover, Q_{F,G} preserves injections and surjections (look for principal ideals). The reader should compare our Q_{F,G}-construction with the F_{κ}^{λ}-construction in [10]. 1.2 Theorem. U_{F,G} has as a left adjoint Q_{F,G} from S_{F} to S_{G} with adjunction morphisms η_{F,G} : id → U_{F,G}Q_{F,G} and ϵ_{F,G} : Q_{F,G}U_{F,G} → id. Proof. First, we put η_{F,G}L(x) = ↓({x}) and ϵ_{F,G}K([F]) = ⋁ F for all L ∈ S_{F} and all K ∈ S_{G}. It is easy to see that both prescriptions are correct and the corresponding mappings are morphisms in S_{F}, S_{G} resp. To check the naturality condition is a routine verification.

Now, let f : Q_{F,G}(L) → K be a morphism in S_{G}, L ∈ S_{F}, K ∈ S_{G}. To conclude that the required adjunction holds, we have to show that there is a unique ϕ : L → K such that f = ϵ_{F,G}K ∘Q_{F,G}(ϕ). We put, for a ∈ L, ϕ(a) = f(↓({a})). This determines ϕ uniquely and since ϕ = f ∘ ↓ it is a morphism in S_{F}. □ 1.3 Corollary. The triple T = (U_{F,G}Q_{F,G}, η, U_{F,G}ϵQ_{F,G}) is a monad on S_{F}. 1.4 Corollary. S_{F} is monadic. Proof. This follows from the fact that Meet is a finitary algebraic category (this implies monadicity over Set) and every S_{F} is monadic over Meet by 1.3 (Meet is well behaved wrt. each S_{F}). □ 1.5 Definition. Let L be a partial G-frame, a, b ∈ L. We say that a is FG-below b and write a≪_{F,G}b if, for any G ∈ G(L), b ≤ ⋁ G there is g ∈ Ĝ such that a ≤ h. We say that an element c ∈ L is FG-compact if c≪_{F,G}c. Recall that a join of an F-set of FG-compact elements is an FG-compact element (using (A3)). The set of all FG-compact elements is denoted K_{FG}(L). L is called FG-coherent if each element of L is a join of a G-subset G, G = Ĝ, G ⊆ K_{FG}(L) and K_{FG}(L) is closed under finitary meets (including the empty meet). For a FG-coherent partial frame L, K_{FG}(L) is an F-frame.

We say that L is stably FG-contimuous if, for each a ∈ L, there is a subset k(a) ⊆ {x : x≪_{F,G}a}, k(a) ∈ Q_{F,G} such that a = ⋁{x : x ∈ k(a)}, 1 is FG-compact and a ∈ k(b), c ∈ k(d)

PARTIAL FRAMES 5

implies a ∧ c ∈ k(b ∧ d).

Recall that, for a stably FG-contimuous partial G-frame L, we have k(a) = {x : x≪_{F,G}a}. One can verify that k : L → Q_{F,G}(L) is a G-morphism and that ⋁ ∘k = id_{L}.

As in frames, where we speak about coherent and stably continuous frames, FG-coherence implies stable FG-continuity. Moreover, the functor Q_{F,G}: S_{F} → S_{G} induces an equivalence between S_{F} and the category FGCohS_{G} of all FG-coherent partial G-frames and those G-morphisms which preserve FG-compact elements. It is easy to check (see 1.6) that K_{FG} is functorial, going from FG-coherent partial G-frames into partial F-frames. The corresponding natural isomorphisms are ↓ : M → K_{FG}(Q_{F,G}(M)), ⋁ : Q_{F,G}(K_{FG}(L)) → L, M ∈ S_{F}, L ∈ FGCohS_{G}.

Note also that our definitions differ formally from that in [10], but one easily sees that they coincide for κ-frames.

As in [10] for κ-frames, one may prove the following.

1.6 Proposition. A partial G-frame L is F-free i.e. isomorphic to Q_{F,G}(M) for some partial F-frame M if and only if it is FG-coherent. Moreover, we have :

(1) for any partial F-frame M, Q_{F,G}(L) is FG-coherent and K_{FG}(Q_{F,G}(M)) = ↓(M) = {↓(x) : x ∈ M},

(2) for any FG-coherent partial G-frame L, we have an isomorphismus of partial G-frames ⋁ : Q_{F,G}(K_{FG}(L)) → L.

Proof. (1) Let ↓(G) ∈ K_{FG}(Q_{F,G}(M)), G = Ĝ ∈ G(M). Then by (A5) ↓(G) ⊆ ↓(g) for some g ∈ G i.e. K_{FG}(Q_{F,G}(M)) ⊆ ↓(M). The reverse inclusion is evident. The rest from the fact that every element of Q_{F,G}(M) is a ’good’ G-join of principal ideals i.e. it has the form ⋁{↓(g) : g ∈ G}, G = Ĝ ∈ G(L).

(2) The proof copies 1.4(b) from [10]. Recall only that, for for any FG-coherent partial Gframe L, one may easily check that a = ⋁ Ĝ, Ĝ ∈ G(L), Ĝ ⊆ K_{FG}(L) implies K_{FG}(L) ∩ ↓(Ĝ) = K_{FG}(L) ∩ ↓(a) and k(a) = ↓(Ĝ). □

Following Madden [10], we define an F-section for a G-morphism q : K → L to be an Fmorphism s : L → K such that q ∘ s = id_{L}. As in [10], a G-morphism with an F-section is

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surjective and the F-section injective. The leading example of a G-morphism with an F-section are our ϵ_{F,G}L = ⋁_{L} and η_{F,G}L = ↓ from the theorem 1.2. 1.7 Definition. A G-frame P is said to be F-projective if, whenever we are given a diagram of the form

\begin{aligned}& & & P \\ & & f & ↓ \\ K & ⟶^{q} & & L \end{aligned}

with q having an F-section, there exists a G-morphism g : P → K such that q ∘ g = f. 1.8 Proposition. A G-frame P is F-projective if and only if it is a retract of an F-free G-frame. 1.9 Lemma. A retract of a stably FG-continuous G-frame is stably FG-continuous. Proof. Let f : L → K, g : K → L be G-morphisms such that f ∘ g = id_{K}, L be stably FGcontinuous. Recall that a≪_{F,G}g(b) implies f(a)≪_{F,G}b. Namely, Ĝ ∈ G(K), b ≤ ⋁ Ĝ ⟹ g(Ĝ) ∈ G(L), g(b) ≤ ⋁ g(Ĝ) ⟹ there is by (W2) h ∈ Ĝ such that a ≤ g(h) ⟹ there is h ∈ Ĝ such that f(a) ≤ h. The rest follows immediately the corresponding part of the proof of 2.4 [10]. □ 1.10 Theorem. A G-frame P is F-projective if and only if it is a stably FG-continuous frame. Proof. One proves the proposition in the same manner as Madden did for κ-frames using 1.9. We note only that one has to use (A3) first time to count that an F-join of elements FG-below some element is FG-below it and second time to verify the existence of a G-morphism needed for projectiveness. □ 1.11 Corollary. The categories FGCohS_{G} of FG-coherent G-frames and FGStContS_{G} of stably FG-continuous G-frames are both coreflective in the category S_{G} of G-frames, with the coreflection functor Q_{F,G} and the coreflection map ϵ_{F,G}L = ⋁_{L} : Q_{F,G}(L) → L. Proof. The proof is a reformulating of Prop. 10 in [2]. Namely, by the same arguments it is sufficient to prove that any G-morphism h : M → L, M stably FG-continuous, uniquely factors through ⋁ : Q_{F,G}(L) → L. Let us consider the commutative diagram

Q_{F,G}(L) \underrightarrow{⋁_{L}} L \overleftarrow{⋁_{L}} Q_{F,G}(L)
↑^{Q_{F,G}(h)} ↑^{h} ↑^{Q_{F,G}(h)}
Q_{F,G}(M) \underrightarrow{⋁_{M}} M ⟶^{k} Q_{F,G}(M)

where k is given by k(a) = {x : x≪_{F,G}a}. The rest follows [2]. □

2. Limits and colimits in partial frames

First of all we note that, when constructing limits in partial frames, they are the same as for meet-semilatticess (in fact sets). This implies completeness of S_{F}. To prove cocompletness it is enough to investigate the coequalizers and sums. By (A4), the coequalizers in partial quantales coincide with coequalizers in meet-semilattices. A sum of partial frames can be constructed by taking their sum in Meet, lifting it in partial frames and factorizing the result through the least congruence such that the superposition of all injections and the lifting map will be morphisms of partial frames. In the sequel we will discuss this construction more explicitly.

There are different ways how to construct sums of frames such as in the papers [4], [5] or [6]. For partial frames seems not to be so elegant description. However, we will give an analogous description to that of [6].

Let L_{γ}, γ ∈ Γ be a family of partial frames and write B for the sum in Meet of L_{γ}. B consists in fact of formal reduced meets ⋀_{γ∈Γ_{0}} x_{γ}, Γ_{0} = {γ_{i} : i = 1, …, n}, over the disjoint union ⨿L_{γ} of the system L_{γ}. Evidently, if x is an nonzero element of B ’containing’ some element x_{γ} of L_{γ} then we may write x = x_{γ} ∧ ⋀ X_{p}, here X_{p} = {x_{δ} : δ ≠ γ, δ ∈ Γ_{0}} ∪ {1_{δ} : δ ≠ γ, δ ∉ Γ_{0}} is a subset of ⨿L_{γ}.

We shall call an F-connector a system c = (m, (R_{γ}^{↑}, R_{γ}^{↓}, γ ∈ Γ), m_{1}, m_{2}), where m ⊆ B, m_{1}, m_{2} ∈ F_{l}(B), m_{1}, m_{2} ⊆ m, R_{γ}^{↓}⊆ P_{F}(m) × m, R_{γ}^{↓}⊆ m×P_{F}(m) such that the following conditions hold:

Let k ⊆ m. Whenever

(1)  (m_{1} ⊆ k) & [(nR_{γ}^{↑}x) & (n ⊆ k) ⟹ x ∈ k] & [(xR_{γ}^{↓}n) & (x ∈ k) ⟹ n ⊆ k]

or

(2)  (m_{2} ⊆ k) & [(nR_{γ}^{↑}x) & (x ∈ k) ⟹ n ⊆ k] & [(xR_{γ}^{↓}n) & (n ⊆ k) ⟹ x ∈ k]

holds, it is k = m.

Elements m_{1}, m_{2} ∈ F_{l}(B) are said to be C_{F}-equivalent (m_{1} ∼_{C_{F}} m_{2}), if either m_{1} = ↓(0) = m_{2}

8 JAN PASEKA

or there exists an F-connector c = (m, (R_{γ}^{↑}, R_{γ}^{↓}, γ ∈ Γ),m_{1}, m_{2}) such that it holds:

(xR_{γ}^{↑}m′) or (m′R_{γ}^{↓}x) ⟹ (x ≠ 0)

&(x_{γ} = ⋁ m′_{γ}) & (x = x_{γ} ∧ ⋀ X_{p}, m′ = m′_{γ} ∧ ⋀ X_{p}) . (3) 2.1 Proposition. “ ∼_{C_{F}} ” is an equivalence relation on F_{l}(B). Proof. Let m ⊆ B. We put xS_{γ}^{↓}m′, m′S_{γ}^{↑}x if and only if (x ≠ 0)&(x_{γ} = ⋁ m′_{γ}) & (x = x_{γ} ∧ ⋀ X_{p}) & (m′ = m′_{γ} ∧ ⋀ X_{p}) for each x ∈ m, m′ ∈ P_{F}(m). Such relations S_{γ}^{↑}, S_{γ}^{↓} will be called standard, an F-connector with standard relations S_{γ}^{↑} and S_{γ}^{↓} is said to be standard.

Let m_{1} ∈ F_{l}(B). We put m = m_{1} = m_{2} and let c be the corresponding standard F-connector. Then m_{1}∼_{C_{F}} m_{1} i.e. ∼_{C_{F}} is reflexive.

Let m_{1}∼_{C_{F}} m_{2}, c be the corresponding F-connector. Clearly, if we interchange m_{1} with m_{2} and define the “new” R_{γ}^{↑} as an inverse relation to the “old” R_{γ}^{↓} and similarly for the “new” R_{γ}^{↓}, we obtain an F-connector c′ i.e. m_{2}∼_{C_{F}} m_{1}. This yields the symmetry of ∼_{C_{F}}.

Let m_{1}∼_{C_{F}} m_{2}, m_{2}∼_{C_{F}} m_{3}. Then there exist F-connectors

a = (m(a), (R_{γ}^{↑}(a), R_{γ}^{↓}(a), γ ∈ Γ), m_{1}, m_{2}) , b = (m(b), (R_{γ}^{↑}(b), R_{γ}^{↓}(b), γ ∈ Γ), m_{2}, m_{3}) such that (3) holds. We put m = m(a) ∪ m(b), R_{γ}^{↑}=R_{γ}^{↑}(a)∪R_{γ}^{↑}(b)⊆ S_{γ}^{↑}(m), R_{γ}^{↓}=R_{γ}^{↓}(a)∪R_{γ}^{↓}(b)⊆ S_{γ}^{↓}(m) and c = (m, (R_{γ}^{↑}, R_{γ}^{↓}, γ ∈ Γ), m_{1}, m_{3}). Then c is clearly an F-connector satisfying (3).

We shall check the condition (1).

Let (m_{1} ⊆ k) & [(nR_{γ}^{↑}x) & (n ⊆ k) ⟹ x ∈ k] & [(xR_{γ}^{↓}n) & (x ∈ k) ⟹ n ⊆ k]. We put k(a) = k ∩ m(a) and k(b) = k ∩ m(b). Then (m_{1} ⊆ k(a)) & [(nR_{γ}^{↑}(a) x) & (n ⊆ k(a)) ⟹ x ∈ k(a)] & [(xR_{γ}^{↓}(a) n) & (x ∈ k(a)) ⟹ n ⊆ k(a)]. Then k(a) = m(a), so we have m_{2} ⊆ k. Now, let (m_{2} ⊆ k(b)) & [(nR_{γ}^{↑}(b) x) & (n ⊆ k(b)) ⟹ x ∈ k(b)] & [(xR_{γ}^{↓}(b) n) & (x ∈ k(b)) ⟹ n ⊆ k(b)]. Again, k(b) = m(b) i.e. k = m.

The same argument may be used for checking (2). Since the condition (3) is trivially preserved by the union of relations, we are ready. □ 2.2 Proposition. “ ∼_{C_{F}} ” is an F-congruence on F_{l}(B) i.e. F_{l}(B)/_{∼_{C_{F}}} is a partial frame. Proof. Let m_{1}(a)∼_{C_{F}} m_{2}(a), m_{1}(b)∼_{C_{F}} m_{2}(b). Then there exist F-connectors a = (m(a), (R_{γ}^{↑}(a), R_{γ}^{↓}(a), γ ∈ Γ), m_{1}(a), m_{2}(a)) , b = (m(b), (R_{γ}^{↑}(b), R_{γ}^{↓}(b), γ ∈ Γ), m_{1}(b), m_{2}(b))

PARTIAL FRAMES 9

satisfying (3).

Put m = m(a) ∩ m(b), m_{1} = m_{1}(a) ∩ m_{1}(b), m_{2} = m_{2}(a) ∩ m_{2}(b), R_{γ}^{↓}= {(x ∧ y, {h ∧ y : h ∈ m′}) : ( xR_{γ}^{↓}(a) m′ & y ∈ m(b) ) or ( xR_{γ}^{↓}(b) m′ & y ∈ m(a) )}, R_{γ}^{↑}= {({h ∧ y : h ∈ m′}, x ∧ y) : ( m′R_{γ}^{↑}(a) x & y ∈ m(b) ) or ( m′R_{γ}^{↑}(b) x & y ∈ m(a) )}. Then m_{1}, m_{2} ∈ F_{l}(B) and we have to check that c = (m, (R_{γ}^{↑}, R_{γ}^{↓}, γ ∈ Γ), m_{1}, m_{2}) is an F-connector satisfying (3).

Checking (1), let (m_{1} ⊆ k) & [(nR_{γ}^{↑}x) & (n ⊆ k) ⟹ x ∈ k] & [(xR_{γ}^{↓}n) & (x ∈ k) ⟹ n ⊆ k]. We put k(a) = {z ∈ m(a) : {z∧h : h ∈ m_{1}(b)} ⊆ k}. Then (m_{1}(a) ⊆ k(a)) & [ (nR_{γ}^{↑}(a) x) & (n ⊆ k(a)) ⟹ x ∈ k(a)] & [(xR_{γ}^{↓}(a) n) & (x ∈ k(a)) ⟹ n ⊆ k(a)], which is easy to compute. Then k(a) = m(a) i.e. m(a) ∩ m_{1}(b) ⊆ k. Again, we put k(b) = {z ∈ m(b) : {z ∧ h : h ∈ m_{1}(a)} ⊆ k}. Then (m_{1}(b) ⊆ k(b)) & [ (nR_{γ}^{↑}(b) x) & (n ⊆ k(b)) ⟹ x ∈ k(b)] & [(xR_{γ}^{↓}(b) n) & (x ∈ k(b)) ⟹ n ⊆ k(b)]. Then k(b) = m(b) i.e. m = m(a) ∩ m(b) ⊆ k.

In the same manner we may check the condition (2). That (3) holds is evident. Alltogether, we proved that ∼_{C_{F}} actually preserves finite meets.

Let m_{1}(a_{i})∼_{C_{F}} m_{2}(a_{i}), i ∈ I, {m_{1}(a_{i})}, {m_{2}(a_{i})} ∈ F(F_{l}(B)). Then there exist F-connectors

a_{i} = (m(a_{i}), (R_{γ}^{↑}(a_{i}), R_{γ}^{↓}(a_{i}), γ ∈ Γ), m_{1}(a_{i}), m_{2}(a_{i})) , i ∈ I

satisfying (3). We put a = (⋃ m(a_{i}), (⋃ R_{γ}^{↑}(a_{i}), ⋃ R_{γ}^{↓}(a_{i}), γ ∈ Γ), ⋃ m_{1}(a_{i}), ⋃ m_{2}(a_{i})). Using (A3), we obtain that ⋃ m_{1}(a_{i}), ⋃ m_{2}(a_{i}) ∈ F_{l}(B).

Now, we will check (1). Let (⋃ m_{1}(a_{i}) ⊆ k) & [(n ⋃ R_{γ}^{↑}(a_{i})x) & (n ⊆ k) ⟹ x ∈ k] & [(x ⋃ R_{γ}^{↓}(a_{i})n) & (x ∈ k) ⟹ n ⊆ k]. We put k(a_{i}) = k ∩ m(a_{i}) for i ∈ I. Then (m_{1}(a_{i}) ⊆ k(a_{i})) & [(nR_{γ}^{↑}(a_{i}) x) & (n ⊆ k(a_{i})) ⟹ x ∈ k(a_{i})] & [(xR_{γ}^{↓}(a_{i}) n) & (x ∈ k(a_{i})) ⟹ n ⊆ k(a_{i})]. Clearly k(a_{i}) = m(a_{i}), so we have k = m.

By the symmetry argument we may check the condition (2). That (3) holds is trivial. This gives us that ∼_{C_{F}} preserves F-joins. □

We shall denote by [m] the class of m ∈ F_{l}(B) in the factor partial frame F_{l}(B)/_{∼_{C_{F}}}. For any γ ∈ Γ, we define a mapping τ_{γ}:L_{γ}⟶B to be the injection from L_{γ} to B in Meet. 2.3 Proposition. Given a system f_{γ} : L_{γ}⟶ C of morphisms in S_{F}, C being a partial frame, there exists a unique morphism f : F_{l}(B)/_{∼_{C_{F}}} ⟶ C of partial frames such that the following condition holds:

(4)  f([↓(x)] = ⋀{f_{γ}(x_{γ}) : γ ∈ Γ_{0}} for any x ∈ B.

Proof. As in [6], we have only to check that f, which is uniquely determined by the prescription f([u]) = ⋁{f([↓(x)]) : x ∈ u}, is correctly defined (the join exists since u is F-generated). In fact we will go first to F_{l}(B) and then factorize. Put g(u) = ⋁{f([↓(x)]) : x ∈ u} for all u ∈ F_{l}(B). It is evident that g is an F-morphism. Now, let m_{1}∼_{C_{F}} m_{2}. Then there exists an F-connector a = (m, (R_{γ}^{↑}, R_{γ}^{↓}, γ ∈ Γ), m_{1}, m_{2}) such that (3) holds. Put k = {x ∈ m : f([↓(x)] ≤ g(m_{1})}. It is easy to check that k satisfies the condition (1) i.e. k = m. Then g(m_{2}) ≤ g(m_{1}). By a symmetry argument we have that g(m_{1}) ≤ g(m_{2}) i.e. f([m_{1}]) = g(m_{1}) = g(m_{2}) = f([m_{2}]). □ 2.4 Theorem. The factor partial frame F_{l}(B)/_{∼_{C_{F}}} has F-joins and finite meets given by the prescriptions

⋁{[m_{i} : i ∈ I} = [⋁{n_{i} : i ∈ I}]

[m] ∧ [n] = [{x ∧ y : x ∈ m, y ∈ n}] (5)

for all m, n ∈ F_{l}(B), {m_{i}} ∈ F(F_{l}(B)).

Defining mappings i_{γ} : L_{γ}⟶, F_{l}(B)/_{∼_{C_{F}}} γ ∈ Γ, by i_{γ}(x) = [↓(τ_{γ}(x))], x ∈ L_{γ}, we have partial frame morphisms such that F_{l}(B)/_{∼_{C_{F}}} is a sum ⨁_{γ∈Γ} L_{γ} of L_{γ}, γ ∈ Γ, with injections i_{γ}. Proof. By the preceding we know that F_{l}(B)/_{∼_{C_{F}}} is a partial frame, with F-joins and finite meets given by (5). Again by (5), i_{γ} are morphisms of partial frames. Given a system f_{γ} : L_{γ} ⟶ C of partial frame morphisms , C being a partial frame, there exists by the preceding proposition a unique morphism f : F_{l}(B)/_{∼_{C_{F}}} ⟶ C of partial frames such that (4) holds i.e. f · i_{γ} = f_{γ}. □ 2.6 Lemma. Let x, y ∈ B. Then

(6)  [↓(x)] ≤ [↓(y)] ⟺ x ≤ y.

Proof. At first, we check that for x, y ∈ B we have:

(7)  [↓(x)] = [↓(y)] ⟺ x = y.

PARTIAL FRAMES 11

Let [↓(x)] = [↓(y)]. Then we have ↓(x)∼_{C_{F}} ↓(y) i.e. there is an F-connector a = (m, (R_{γ}^{↑}, R_{γ}^{↓}, γ ∈ Γ), ↓(x), ↓(y)) such that (3) holds. Clearly, ↓(x) satisfies (1) and ↓(y) satisfies (2) i.e. ↓(x) = m = ↓(y).

(6) then follows from (7) as in [6]. □

Recall that this says that i_{γ} preserves arbitrary meets which exist in L_{γ}.

For any X ⊆ B we define s(X) := {[↓(x)] : x ∈ X} i.e. we have defined a map s : 2^{B} → 2^{L}.

3. Covers in partial frames

In the following we shall always assume that, for any partial frame K, F(K) is a lower set in the usual ordering i.e. any subset of an F-set is an F-set and moreover, that any finite subset of K is an F-set.

Consider a system S_{γ} of covers of L_{γ}, γ ∈ Γ, such that S_{γ} = {1} except for, at most, finitely many γ ∈ Γ. Then the system ∏_{γ∈Γ} S_{γ}={⋀ i_{γ}(x_{γ}) : x_{γ} ∈ S_{γ}, γ ∈ Γ} is a cover of L by (5).

Recall the following important lemma which is a generalization of a lemma for frames due to K from [7]. 3.1 Lemma. Let T = {[↓(t)] : t ∈ T′}, T′ ⊆⊆ B finite, be a cover of L. Then there is a system S_{γ} of covers of L_{γ}, γ ∈ Γ, such that S_{γ} = {1} except for, at most, finitely many γ ∈ Γ and {⋀ i_{γ}(x_{γ}) : x_{γ} ∈ S_{γ}, γ ∈ Γ} refines T. Proof. We shall slightly modify the proof of K. We can easily find a finite system {K_{t} : t ∈ T′} of finite subsets K_{t} ⊆⊆ Γ and elements a(α, t), α ∈ K_{t}, t ∈ T′ such that

T = { ⋀_{α∈K_{t}} i_{α}(a(α, t)) : t ∈ T′}.

We put H := ⋃_{t∈T′} K_{t}. We shall denote by C_{β}, β ∈ H the finite meet semi-lattice generated in L_{β} by the finite subset {a(β, t) : β ∈ K_{t}, t ∈ T′}.

We define

k := {z ∈ B : (∀β ∈ H)(∃B_{β} ⊆⊆ C_{β})(⋁ i_{β}(B_{β}) ≥ [↓(z)]) and ⋀_{β∈H} i_{β}(B_{β}) < T }.

Since T is a cover of L there is an F-connector c = (m, (R_{γ}^{↑}, R_{γ}^{↓}, γ ∈ Γ), m_{1}, m_{2}), where m = m_{2} = B, m_{1} = ⋃_{t∈T′} ↓(t) ∈ F_{l}(B).

12 JAN PASEKA

Evidently, k ⊆ m, k is a lower set. It is easy to see that to prove 1 ∈ k = m = B (this is in fact our proposition) we have only to check the following implication

(nR_{γ}^{↑}x) & (n ⊆ k) ⟹ x ∈ k.

It is enough to do it for a standard relation S_{γ}^{↑}. Let (x ≠ 0)&(x_{γ} = ⋁ m′_{γ}) & (x = x_{γ} ∧ ⋀ X_{p}) & (m′ = m′_{γ} ∧ ⋀ X_{p}) and m′ ⊆ k, m′_{γ} ∈ F(L_{γ})

for some γ ∈ Γ. We have to prove that x ∈ k. If γ ∉ H then C_{γ} = {1} and the rest is evident. Let γ ∈ H. For any y ∈ m′_{γ} we put

K_{y} = {B^{y} ⊆⊆ C_{γ} : ⋁ i_{γ}(B^{y}) ≥ i_{γ}(y) ∧ [↓(⋀ X_{p})]}

and

K = {K_{y} : y ∈ m′_{γ}}.

Evidently, the set K ⊆ 2^{C_{γ}} is finite. We shall define for any E ∈ K an element w_{E} = ⋁{y ∈ m′_{γ} : K_{y} = E}.

Then there are sets B_{β}^{E} ⊆⊆ C_{β}, β ∈ H such that

⋁ i_{β}(B_{β}^{E}) ≥ i_{γ}(w_{E}) ∧ [↓(⋀ X_{p})] and ⋀_{β∈H} i_{β}(B_{β}^{E}) < T.

We put

B_{β} = \{ \begin{matrix}⋃_{E∈K} B_{γ}^{E} & if β = γ \\ ⋀_{E∈K} B_{β}^{E} & otherwise. \end{matrix}

We have then for any β ∈ H

⋁ i_{β}(B_{β}) ≥ ⋁_{E∈K} i_{γ}(w_{E}) ∧ [↓(⋀ X_{p})] = ⋁_{y∈m′_{γ}} i_{γ}(y) ∧ [↓(⋀ X_{p})]

and

⋀_{β∈H} i_{β}(B_{β}) = ⋃_{E∈K} (i_{γ}(B_{γ}^{E}) ∧ ⋀_{β∈H−{γ}} i_{β}(B_{β})) < ⋃_{E∈K} ⋀_{β∈H} i_{γ}(B_{β}^{E}) < T. □

In the following we shall generalize our results for frames obtained in [11]. We shall show that under some restricted assumptions certain covering properties of partial frames are preserved in their sums. Recall that Axiom of Choice is prerequisite of our considerations (except in the case of compact partial frames).

PARTIAL FRAMES 13

Definition 3.2. Let L be a partial frame, C, D ⊆ L. We shall say that D finitely regularly (strongly) refines C if, for each d ∈ D, there exists F ⊆⊆ C (c ∈ C) such that d ◁ ⋁F i.e. there is e ∈ L such that e ∨ ⋁F = 1 and e ∧ d = 0 (d ◁ c). We shall write D ◁ _{f}C (D ◁ C). Clearly, ◁ ⊆ ◁ _{f}. Putting σ(C) = ⋃{X ⊆ L : X ◁ _{f}C} we have that σ(C) ◁ _{f}C.

Recall that the relations ◁ _{f}, ◁ are preserved by morphisms of partial frames and that the following holds:

(i) E < D ◁ _{f}C < F implies E ◁ _{f}F, (ii) E < D ◁ C < F implies E ◁ F, (iii) D_{i} ◁ _{f}C_{i}, i ∈ I, I ≠ ∅ implies ⋃ D_{i} ◁ _{f} ⋃ C_{i}, (iv) D_{i} ◁ C_{i}, i ∈ I, I ≠ ∅ implies ⋃ D_{i} ◁ ⋃ C_{i},

(v) B ◁ _{f}A, D ◁ _{f}C implies B ∧ D ◁ _{f}A ∧ C, (vi) B ◁ A, D ◁ C implies B ∧ D ◁ A ∧ C.

We shall say that a partial frame L is finitely (cover) regular if each cover of L has a finitely regular (strong) refinement which is a cover as well. A partial frame L is said to be regular if any its element a is a join of some F-subset of the set σ(a) = {x ∈ L : x ◁ a}. Regular partial frames are cover regular and cover regular partial frames are finitely regular.

Moreover, it is easy to prove that any paracompact (compact) partial frame is finitely regular. In the sequel, let B, E be functions with the domain S_{F} and the codomain Set satisfying the

following conditions:

(E0) L ∈ S_{F} implies 1 ∈ B(L), B(L) ⊆ L is closed under finite meets,

(E1) L ∈ S_{F}, t ∈ B(L) implies {t} ∈ E(L), E(L) ⊆ 2^{B(L)} ∩ F(L),

(E2) L ∈ S_{F}, t ∈ B(L), E ∈ E(L) implies {t} ∧ E ∈ E(L),

(E3) L ∈ S_{F}, E ∈ E(L) and for each f ∈ E(L) there is E_{f} ∈ E(L) such that f = ⋁E_{f} implies ⋃{E_{f} : f ∈ E} ∈ E(L),

(E4) L, K ∈ S_{F}, E ∈ E(L), H ∈ E(K) implies i_{L}(E) ∧ i_{K}(H) ∈ E(L ⊕ K).

If E ∈ E(L), we shall speak about a set of a type E. It is sometimes convenient to suppose the following strengthening of the condition (E1):

(E1’) L ∈ S_{F}, C ⊆⊆ B(L) implies C ∈ E(L), E(L) ⊆ 2^{B(L)} ∩ F(L).

14 JAN PASEKA

Definition 3.3.. We shall say that a partial frame L is an E-partial frame (ED-partial frame) if C ∈ C(L) implies there is D ∈ C(L) ∩ E(L) (D ∈ D(L) ∩ E(L)) such that D < C i.e. every cover of L has a (dense) refinement of the type E.

A morphism f : K → L of partial frames is said to be E-good if f(E(K)) ⊆ E(L).

Remark 3.4. In the sequel, let L_{γ}, γ ∈ Γ, be a family of finitely regular E-partial frames, K ⊆ B, \{↓[(u)] : u ∈ K\} ∈ F(L), ⋁s(K) = 1, \overline{K} = \{x ∈ B : x ∈ M′ ⊆⊆ B, ⋁s(M′) = 1 implies there is a subset Q′_{x} of B, s(Q′_{x}) < s(K ∪ (M′ − {x})), s(Q′_{x}) ◁ _{f}s(K ∪ (M′ − {x})), s(Q′_{x}) ∈ E(L) ∩ C(L)}.

Lemma 3.5. Let S_{γ}, γ ∈ Γ be a system of covers of L_{γ}, γ ∈ Γ, S_{γ} = {1} for all but finitely many γ ∈ Γ. Then there are T_{γ}, γ ∈ Γ, T_{γ} < S_{γ}, T_{γ} ◁ _{f}S_{γ}, T_{γ} ∈ C(L_{γ}) ∩ E(L_{γ}), γ ∈ Γ such that T_{γ} = {1} for all but finitely many γ ∈ Γ.

Proof. Let Γ_{0} ⊆ Γ be finite such that γ ∉ Γ_{0} implies S_{γ} = {1}. For each γ ∈ Γ_{0} there is T_{γ}, T_{γ} < S_{γ}, T_{γ} ◁ _{f}S_{γ}, T_{γ} ∈ C(L_{γ}) ∩ E(L_{γ}), for each γ ∉ Γ_{0} we put T_{γ} = {1} i.e. T_{γ} < S_{γ}, T_{γ} ◁ _{f}S_{γ}, T_{γ} ∈ C(L_{γ}) ∩ E(L_{γ}) by (E0), (E1). □

Lemma 3.6. Let t ∈ B, s(t) ∈ B(L), A ⊆⊆ B, W ⊆⊆ \overline{K}, (K ∪ A) ∩ W = ∅, s(t) ◁ _{f}s(A ∪ W). Then there is T ⊆ B such that s(T) < s(K ∪ A), s(T) ◁ _{f}s(K ∪ A), s(T) ∈ E(L), ⋁s(T) = s(t).

Proof. The proof will be done by induction according to card W. If card W = 0 we are ready – we put T = {t}. Let card W = n > 0 and the lemma holds for all k < n. Then there is an element w ∈ W such that ⋁ s(D(t) ∪ A ∪ (W − {w}) ∪ {w}) = 1, where D(t) ⊆⊆ B such that ⋁ s(D(t)) ∧ s(t) = 0 and ⋁ s(D(t)) ∨ ⋁ s(A ∪ W) = 1. Since w ∈ \overline{K} ,D(t) ∪ A ∪ (W − \{w\}) is finite there is Q_{w} ⊆ B, s(Q_{w}) < s(K ∪ (D(t) ∪ A ∪ (W − {w})), s(Q_{w}) ◁ _{f}s(K ∪ D(t) ∪ A ∪ (W − {w}), s(Q_{w}) ∈ E(L) ∩ C(L)}.

We put S = {t} ∧ Q_{w}. Then s(S) < s(K ∪ A ∪ (W − {w})). Let z ∈ S. Then z = t ∧ y for some y ∈ Q_{w}. We have 1 = s(y)^{∗} ∨ s(K′ ∪ D(t) ∪ A ∪ (W − {w})), K′ ⊆⊆ K. This implies 1 = s(z)^{∗} ∨ s(K′ ∪ A ∪ (W − {w})) i.e. s(z) ◁ _{f}s(K ∪ A ∪ (W − {w})), ⋁s(S) = s(t), s(S) ∈ E(L) by (E2). By induction, for each z ∈ S there is T_{z} ⊆ B such that s(T_{z}) < s(K ∪ A), s(T_{z}) ◁ _{f}s(K ∪ A), ⋁s(T_{z}) = s(z), s(T_{z}) ∈ E(L).

PARTIAL FRAMES 15

We put T = ⋃{T_{z} : z ∈ S}. Then by (E3) s(T) ∈ E(L), ⋁s(T) = ⋁{T_{z} : z ∈ S} = ⋁s(S) = s(t), s(T) ◁ _{f}s(K ∪ A). □ Proposition 3.7. It holds \overline{K} = B. Proof. Evidently, K ⊆ \overline{K}. Namely, if k ∈ K, k ∈ M′ ⊆⊆ B, ⋁s(M′) = 1 then by 3.1 there are finite covers S_{γ}, γ ∈ Γ of L_{γ}, γ ∈ Γ, S_{γ} = {1} for all but finitely many γ ∈ Γ, 𝕏S_{γ} < s(M′). Then by 3.5 there are T_{γ}, γ ∈ Γ, T_{γ} < S_{γ}, T_{γ} ◁ _{f}S_{γ}, T_{γ} ∈ C(L_{γ}) ∩ E(L_{γ}), γ ∈ Γ such that T_{γ} = {1} for all but finitely many γ ∈ Γ. We put Q_{k} = 𝕏T_{γ}. Then Q_{k} = s(Q′_{k}), Q′_{k} ⊆ B, and by (E4) and that M′ ⊆⊆ K ∪ (M′ − {k}) we have that Q_{k} < s(K ∪ (M′ − {k})), Q_{k} ◁ _{f}s(K ∪ (M′ − {k})), Q_{k} ∈ E(L) ∩ C(L).

Since K is a cover of L there is an F-connector c = (m, (R_{γ}^{↑}, R_{γ}^{↓}, γ ∈ Γ), m_{1}, m_{2}), where m = m_{2} = B, m_{1} = ⋃_{u∈K} ↓(u) ∈ F_{l}(B).

Evidently, \overline{K} ⊆ m, \overline{K} is a lower set. Namely, let x ∈ \overline{K}, z ≤ x, z ∈ B, z ∈ M′ ⊆⊆ B, ⋁s(M′) = 1. Then ⋁ (s ((M′ − {z}) ∪ {x})) = 1 and it implies there is a subset Q′_{x} of B, s(Q′_{x}) < s(K ∪ (M′ − {z})), s(Q′_{x}) ◁ _{f}s(K ∪ (M′ − {z})), s(Q′_{x}) ∈ E(L) ∩ C(L)}. We may put Q′_{z} = Q′_{x}. Then z ∈ \overline{K}.

It is easy to see that to prove 1 ∈ \overline{K} = m = B we have only to check the following implication as in 3.1

(nR_{γ_{0}}^{↑}x) & (n ⊆ \overline{K}) ⟹ x ∈ k,

γ_{0} ∈ Γ arbitrary, fixed.

It is enough to do it for a standard relation S_{γ_{0}}^{↑}.

Let (x ≠ 0)&(x_{γ_{0}} = ⋁ W_{γ_{0}}) & (x = x_{γ_{0}} ∧ ⋀ X_{p}) & (W = W_{γ_{0}} ∧ ⋀ X_{p}) and W ⊆ \overline{K}, W_{γ_{0}} ∈ F(L_{γ_{0}}). We have to prove that x ∈ \overline{K}.

Let x ∈ M′ ⊆⊆ B, ⋁s(M′) = 1. Again by 3.1 we have that there are finite covers S_{γ}, γ ∈ Γ of L_{γ}, γ ∈ Γ, S_{γ} = {1} for all but finitely many γ ∈ Γ and 𝕏S_{γ} < s(M′). We put S′_{γ_{0}} = {y ∈ S_{γ_{0}} : y≰x_{γ_{0}}} ∪ W_{γ_{0}}, S′_{γ} = S_{γ} for γ ≠ γ_{0}. By 3.5 there are T_{γ}, γ ∈ Γ, T_{γ} < S′_{γ}, T_{γ} ◁ _{f}S′_{γ}, T_{γ} ∈ C(L_{γ}) ∩ E(L_{γ}), γ ∈ Γ such that T_{γ} = {1} for all but finitely many γ ∈ Γ. We put T = 𝕏T_{γ}, S′ = 𝕏S′_{γ}. Then T < S′ < s((M′ − {x}) ∪ W), ⋁T = 1 and clearly T ◁ _{f}s(S′), T ∈ E(L) by (E4) .

Evidently, for each s(t) ∈ T there are

A_{t} ⊆⊆ (M′ − {x} ∪ (W ∩ K) ⊆ (M′ − {x} ∪ K,
W_{t} ⊆⊆ (W − K) ∩ (B − (M′ − \{x\})) ⊆ \overline{K}

such that (A_{t} ∪ K) ∩ W_{t} = ∅ and s(t) ◁ _{f}s(A_{t} ∪ W_{t}). By the preceding lemma, there are sets T_{t} ⊆ B such that s(T_{t}) < s(K ∪ A_{t}), s(T_{t}) ◁ _{f}s(K ∪ A_{t}), s(T_{t}) ∈ E(L), ⋁s(T_{t}) = s(t). We put Q′_{x} = ⋃{T_{t} : t ∈ T }. Then Q′_{x} ⊆ B, s(Q′_{x}) < s(K ∪ (M′ − {x})), s(Q′_{x}) ◁ _{f}s(K ∪ (M′ − {x})), s(Q′_{x}) ∈ E(L) ∩ C(L)\} by (E3). We have that x ∈ \overline{K}. □ Theorem 3.8 General cover theorem. Let L_{γ}, γ ∈ Γ be a family of finitely regular E-partial frames. Then the sum L = ⨁_{γ∈Γ} L_{γ} is a finitely regular E-partial frame. Proof. Let C ∈ C(L). Then there is a basic refinement s(K), K ⊆ B of C. By 3.7. 1 ∈ \overline{K} i.e. putting M′ = {1} there is a subset Q of B, s(Q) < s(K), s(Q) ◁ _{f}s(K), s(Q) ∈ E(L) ∩ C(L)}. □ Corollary 3.9. Let L_{γ}, γ ∈ Γ be a family of finitely regular partial framess. Then the sum L = ⨁_{γ∈Γ} is a finitely regular partial frame. Proof. We put, for any partial frame K, B(K) = K, E(K) = F(K). Clearly, the conditions (E0)- (E4) are satisfied. □ Theorem 3.10. Let L_{γ}, γ ∈ Γ be a family of compact partial frames. Then the sum L is a compact partial frame. Proof. We put B(L) = L, F(L) = Fin(L) = {A : A ⊆⊆ L}. One can easily check that (E0)-(E4 are satisfied. □ Theorem 3.11. Let L_{γ}, γ ∈ Γ be a family of paracompact partial frames. Then the sum L is paracompact. Proof. We put B(L) = L, F(L) = LFin(L) = {A : A ∈ F(L), A is locally finite}. Since every paracompact locale is finitely regular the rest follows 3.10. □

Recall that a partial frame is κ-compact, κ being a a regular cardinal, if every cover has subcover of a cardinality < κ. Proceeding in the same vein as above we have

PARTIAL FRAMES 17

Theorem 3.12.. Let L_{γ}, γ ∈ Γ be a family of κ-compact finitely regular partial frames. Then the product L is κ-compact. Proof. Put B(L) = L, E(L) = {F : F ∈ F(L), cardF ∈ κ}. □ Theorem 3.13. Let L_{γ}, γ ∈ Γ be a family of supercompact partial frames. Then the sum L is supercompact. Proof. We put B(L) = {1}, E(L) = {{1}}. □ Remark 3.14. From now, let L_{γ}, γ ∈ Γ, be a family of finitely regular ED-partial frames, K ⊆ B, {↓[(u)] : u ∈ K} ∈ F(L), ⋁s(K) = 1, K̂ = {x ∈ B : x ∈ M′ ⊆⊆ B, ⋁s(M′) = 1 implies there is a subset Q′_{x} of B, s(Q′_{x}) < s(K ∪ (M′ − {x})), s(Q′_{x}) ◁ _{f}s(K ∪ (M′ − {x})), s(Q′_{x}) ∈ E(L) ∩ D(L)}. Lemma 3.15. Let S_{γ}, γ ∈ Γ be a system of covers of L_{γ}, γ ∈ Γ, S_{γ} = {1} for all but finitely many γ ∈ Γ. Then there are T_{γ}, γ ∈ Γ, T_{γ} < S_{γ}, T_{γ} ◁ _{f}S_{γ}, T_{γ} ∈ D(L_{γ}) ∩ E(L_{γ}), γ ∈ Γ such that T_{γ} = {1} for all but finitely many γ ∈ Γ. Proof. It is evident. □ Lemma 3.16. Let t ∈ B, s(t) ∈ B(L), A ⊆⊆ B, W ⊆⊆ K̂, (K ∪ A) ∩ W = ∅, s(t) ◁ _{f}s(A ∪ W). Then there is T ⊆ B such that s(T) < s(K ∪ A), s(T) ◁ _{f}s(K ∪ A), s(T) ∈ E(L), (⋁s(T))^{∗} = s(t)^{∗}. Proof. The proof follows the same arguments as the proof of lemma 2.2 in [12] Proposition 3.17. It holds K̂ = B. Proof. It is a straightforward modification of 3.7. □

Now, we have Theorem 3.18 General dense cover theorem. Let L_{γ}, γ ∈ Γ be a family of finitely regular ED-partial frames. Then the sum L = ⨁_{γ∈Γ} L_{γ} is a finitely regular ED-partial frame. Theorem 3.19. Let L_{γ}, γ ∈ Γ be a family of finitely regular almost paracompact (almost κcompact) partial frames. Then the sum L = ⨁_{γ∈Γ} is a finitely regular almost paracompact (almost κ-compact) partial frame.

18 JAN PASEKA

Proof. Recall that a partial frame is almost paracompact (almost κ-compact) if every its cover has a locally finite dense refinement (a refinement of a cardinality < κ.) We put B(L) = L, F(L) = {A : A ⊆⊆ L A is locally finite} (F(L) = {A : A ⊆⊆ L cardA ∈ κ}). One can easily check that (E0)-(E4) are satisfied. The rest is evident. □

Acknowledgment.

The first and second sections of the paper was written while the author was visiting the Technische Hochschule Darmstadt under the auspices of the DAAD Foundation. Excellent conditions provided by the Arbeitsgruppe Allgemeine Algebra are gratefully acknowledged. The last section was written under the assistence by a grant of Grantová agentura ČR, grant number 201/93/0950.

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