Another note on precompact uniform frames

Journal Pre-proof Another note on precompact uniform frames

P. Bhattacharjee, I. Naidoo

PII:

S0166-8641(19)30429-8

DOI:

https://doi.org/10.1016/j.topol.2019.107023

Reference:

TOPOL 107023

To appear in:

Topology and its Applications

Received date:

31 January 2019

Revised date:

12 April 2019

Accepted date:

16 April 2019

Please cite this article as: P. Bhattacharjee, I. Naidoo, Another note on precompact uniform frames, Topol. Appl. (2019), 107023, doi: https://doi.org/10.1016/j.topol.2019.107023.

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ANOTHER NOTE ON PRECOMPACT UNIFORM FRAMES P. BHATTACHARJEE1,2 AND I. NAIDOO2

Abstract. In [17] pointfree compactness and precompactness are characterized by the convergence and clustering of classical filters, ultrafilters and Cauchy filters. Banaschewski in [2] introduced the notion of general filters as bounded meet semilattice homomorphisms. Together with Hong in [5, 6, 7] they described the notion of convergence of such filters as those that are cover preserving. In this paper, we revisit the notion of general filters in a frame and introduce the concepts of clustering general filters, maximal general filters and general ultrafilters. These concepts have hitherto not been considered for general filters. We use these concepts to characterize, among other things, almost compact frames, Boolean frames and precompact uniform frames.

Introduction Convergence of classical filters in pointfree topology has its roots dating back to the paper of Banaschewski and Pultr [9] in their study of the Samuel compactification and completion of uniform frames where a filter in a frame is said to converge whenever it contains a completely prime filter. This concept is clearly motivated by the topological concept of a convergent filter of a space. Motivated by Herrlich’s work on nearness spaces in [15], Hong in [16] presents an equivalent preferred cover approach to filter convergence that we will use in our article. Therein, a filter F in a frame L converges whenever it meets every cover of L and clusters if secF meets every cover of L. However, on an historical note, it is not well known and is certainly worth noting and elaborating that in 1992, the idea of clustering of filters, postulated in 1995 by Hong [16], appears subtly in the doctoral thesis of Dube [12] with the notion of a near subset, which translates to the notion of clustering in Hong’s sense. Dube in [12], whilst independently working on the concept of nearness for frames, defines particularly for a (regular) frame L with its fine nearness (Cov L is the set of all covers of the frame), A ⊆ L is near if for each B ∈ Cov L there exists b ∈ B such that b ∧ a = 0 for each a ∈ A. Thus, in the terminology of Hong, A is near provided that B ∩ secA = ∅ for each cover B. Furthermore, Dube [12] then shows that 2010 Mathematics Subject Classification. Primary: 06D22; Secondary: 54A20. Key words and phrases. frame, general filter, general ultrafilter, convergence, clustering, Boolean frame, uniform frame, precompact, (weakly) Cauchy filter. 1

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A is near provided that {a∗ : a ∈ A} ∈ / Cov L, that is

 a∈A

a∗ = 1,

which for a filter F is the equivalent formulation (using its pseudocomplements) of clustering as given by Hong [16, Proposition 1.3]. The above characterizations of convergence and clustering of filters underpins our study. Taking a cue from nearness spaces, Hong’s definition in [16] of a convergent filter in a frame is a weaker notion than that of Banaschewski and Pultr [9], but equivalent to it within the class of regular frames. Hong also defined what it means for a filter in a frame to be clustered. His definition is a natural extension of clustering in the field of spaces. With the advent of nearness frames by Banaschewski and Pultr [10] it soon became apparent that the classical filters in frames did not suffice to describe the notion of completeness. To address this deficiency, Banaschewski [2] introduced general filters on a frame as follows. Let L be a frame, and denote by 2 = {0, 1} the two-element frame. The characteristic function of a set K ⊆ L is the mapping χK : L → 2 defined by  1 if a ∈ K χK (a) = 0 if a ∈ / K. A subset F ⊆ L is a (proper) filter in L if and only if χF is a meet semilattice homomorphism. Replacing 2 with an arbitrary frame T , Banaschewski defined a T -valued filter on a frame L to be a 0-meet semilattice homomorphism φ : L → T . If T is unspecified, one speaks of a general filter on L. General filters have since been put to a good use in a number of papers (see, for instance Banaschewski and Hong [5, 6, 7]) to characterize compact frames and complete nearness frames, and also to give a Bourbaki-like Extension Theorem for frames. In each of these cases it was necessary to define convergence for general filters. This was done by realizing that a filter F ⊆ L converges precisely when χF takes covers to covers. That then motivated the definition that a general filter is convergent in case it takes covers to covers. The notion of clustering for general filters has not been considered before. It is one of the objectives of this paper to address that, and we do so in Section 2 where we use the notion to characterize almost compact frames (Proposition 2.5). In Section 3 we define a balanced general filter and a general ultrafilter, each of which has hitherto not been considered although their classical counterparts are well known. The latter is not defined to be a maximal general filter, but rather to be one satisfying a condition that generalizes the characterization that a filter F ⊆ L is an ultrafilter if and only if for every a ∈ L either a ∈ F or a∗ ∈ F . It turns out that ultrafilters are maximal (Proposition 3.6), but not every maximal filter is an ultrafilter (Example 3.8). A balanced general filter is one which sends no dense element to zero. This is motivated by the fact that a filter in a frame is balanced (see Dube [13]) in case it contains every dense element, and F is balanced if and only if χF sends no dense element to 0. Whereas a filter in a frame is an ultrafilter if and only if it is prime and balanced (Dube, [13, Proposition 4]), for general filters we have that every ultrafilter is prime and balanced (Proposition 3.4), but not conversely, as Example 3.5 shows. We end Section 3 with several characterizations of Boolean frames in terms of general filters and ultrafilters (Theorem 3.11).

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In Section 4 we define a stronger variant of clustering to generalize a similar one for classical filters. We then use this notion to characterize compact regular frames (Corollary 4.5). In the last section we consider uniform frames. In particular we focus on precompact uniform frames, those uniform frames in which the uniformity is generated by their finite uniform covers. Naidoo in [17] introduced the pointfree counterpart of classical weakly Cauchy filters and presented classical filter characterizations of precompact uniform frames. We revisit these characterizations given in [17], define weakly Cauchy general filters and provide equivalent formulations of precompactness via the convergence and clustering of general filters. 1. Preliminaries 1.1. Frames and their homomorphisms. Our reference for frames is Picado and Pultr [20], and our notation is fairly standard. We recall that a frame (L, ≤) is a bounded lattice which is complete that satisfies the following distributive law for each x ∈ L and any S ⊆ L,   (x ∧ s). x∧ S= s∈S

  We will denote the bottom of L by 0L = ∅ and the top by 1L = ∅. Where there is no confusion we will merely write 0 and 1 for the bottom and top element of the frame respectively. For frames L and M , a frame homomorphism is a map h : L → M which preserves finite meets and all joins. The resulting category of frames and their homomorphisms is denoted by Frm. All frames that we consider are non-trivial in the sense that 0 = 1. A frame homomorphism h : L −→ M has a right Galois adjoint denoted by h∗ : M −→ L which is a meet preserving map (that is not necessarily a frame homomorphism) such that h(x) ≤ y iff x ≤ h∗ (y). Explicitly, h∗ is given by  h∗ (y) = {x ∈ L : h(x) ≤ y}. h is a dense frame homomorphism in case x = 0L whenever h(x) = 0M . An equivalent characterization for h to be a dense frame homomorphism is that h∗ (0M ) = 0L . Another useful property of the right adjoint is that h is onto iff hh∗ = idM . For each element x in a frame L its pseudocomplement is the element  x∗ = {y ∈ L : y ∧ x = 0}. In any frame L, given any X ⊆ L we will use the notation adopted by Naidoo in [18] for the collection of pseudocomplements of X as X ¬ = {x∗ : x ∈ X}.  L is a regular frame if for each x ∈ L, x = {y ∈ L : y ≺ x} where y ≺ x (y rather below x) means that there is s ∈ L such that y ∧ s = 0 and s ∨ x = 1, equivalently y ≺ x iff y ∗ ∨ x = 1. We also write y ≺≺ x (y completely below x) which means that there is a scale {cα ∈ L : α ∈ Q ∩ [0, 1]} such that c0 = y, c1 = x and cα ≺ cβ whenever α < β. Those frames in which each element is a join of elements completely below it are called completely regular frames. We will also require the Booleanization of a frame L denoted by BL which is ∗∗ the collection of all regular elements ofL (those  x∗∗∈ L such that x = x ). We recall that the join in BL is given by S = ( S) , for any S ⊆ BL. Meets are

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calculated as in L. BL is a regular frame and we shall write bL : L → BL for the dense onto frame homomorphism bL (a) = a∗∗ . An element a ∈ L is dense if a∗ = 0. We write D(L) for the set of all dense elements of L and we note that a ∨ a∗ ∈ D(L) for each a ∈ L. An element x ∈ L is said to be complemented if x ∨ x∗ = 1. The set of all complemented elements of L is denoted by C(L) and L is a called a zero-dimensional frame if C(L) is a base for L.  Here B is a base for the frame L if for each x ∈ L there is T ⊆ B such that x = T. In what follows, A ⊆< ω L will mean that  A is a finite subset of the frame L. A cover of L is a subset A ⊆ L such that A = 1. We denote the collection of all covers of L by Cov L. Compact frames are of course those in which every cover has a finite subcover. A frame Lis called almost compact if for every C ∈ Cov L there ˇ [19] for more exists F ⊆< ω C such that F ∈ D(L). (See Paseka and Smarda details concerning almost compact frames.)  Anelement x ∈ L is a compact element if whenever S ⊆ L and x ≤ S then x ≤ T for some T ⊆< ω S. The set of all compact elements of L is given by KL and L is called an algebraic frame if KL is a base for L. Furthermore, L satisfies the finite intersection property (in short, FIP) on KL if KL is closed under finite meets. An algebraic frame that satisfies the FIP is called an M -frame (see Bhattacharjee [11]). Also, an element x ∈ L is a prime element or is meet-irreducible if x = 1 and a ∧ b ≤ x implies that either a ≤ x or b ≤ x for any a, b ∈ L. Spec(L) is the set of all prime elements of L and we call it the prime spectrum of L. We have already mentioned that 2 denotes the two-element frame. We denote the three-element chain by 3 = {0 < m < 1}, and by 4 = {0, c, c∗ , 1} the four-element Boolean algebra. 1.2. Filters in a frame and general filters on a frame. A set F ⊆ L is called a filter if it is an up-set in L such that 0 ∈ / F , 1 ∈ F , and a ∧ b ∈ F for all a, b ∈ F . Thus, “filter” always means a proper filter. In particular, a filter F is called: (a) (b) (c) (d) (e)

prime if x ∨ y ∈ F implies x ∈ F or y ∈ F for each x, y ∈ L;  completely prime if S ∈ F implies S ∩ F = ∅ for each S ⊆ L; balanced if it equals the intersection of ultrafilters containing it; Boolean if for each x ∈ F , there is y ∈ F ∩ C(L) such that y ≤ x; an ultrafilter if whenever G is a filter in L with F ⊆ G, then F = G.

A filter F ⊆ L is an ultrafilter if and only if, for each x ∈ L, either x ∈ F or x∗ ∈ F ; and a filter G is balanced if and only if D(L) ⊆ G (see Dube [13]). The next two quick results on prime and completely prime filters have not been considered before. Lemma 1.1. In any zero-dimensional frame every completely prime filter is Boolean. Proof. Suppose that L is a zero-dimensional frame and P is a completely prime  filter in L with x ∈ P . Then x = T for some T ⊆ C(L). Since P is completely prime, t ∈ P for some t ∈ T . Thus t ∈ P ∩ C(L) with t ≤ x. Hence, P is Boolean.  The following result shows that the principle filters ↑c = {x ∈ L : c ≤ x} of all compact elements do not distinguish with being prime or completely prime.

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Lemma 1.2. Let L be any frame and c ∈ KL. The filter ↑c is prime if and only if it is completely prime in L.  Proof.  Suppose that c ∈ KL and that ↑c is aprime filter in L. If A ∈ ↑c, then c≤ A. Using the compactness of c, c ≤ T for some T ⊆< ω A. Therefore, T ∈ ↑c. Since ↑c is prime and T is finite, t ∈ ↑c for some t ∈ T . Thus ↑c ∩ A = ∅, proving that ↑c is completely prime. Since every completely prime filter is prime, the equivalence then follows.  We record the following for zero-dimensional M -frames. Proposition 1.3. With choice, the only zero-dimensional M -frame in which ↑c is prime for each c ∈ KL is 2, the topology of the one-point space. Proof. Suppose that L is such an M -frame. We first show that for any c ∈ KL, c∗ ∈ Spec(L). To see this consider u, v ∈ KL with u ∧ v ≤ c∗ . Then since compact elements are complemented in zero-dimensional M -frames, c = c∗∗ ≤ (u ∧ v)∗ = u∗ ∨ v ∗ ; the latter is true since u and v are complemented. Then c ≤ u∗ or c ≤ v ∗ since ↑c is a prime filter. Thus, u ≤ c∗ or v ≤ c∗ , showing that c∗ is prime. With choice, algebraic frames are spatial so that L OX (the frame of open subsets) for some completely regular space X. But in OX the primes are complements of singletons, which then makes compact elements singletons. So, being algebraic this makes X discrete and hence an one-point space, for if p = q in X, then the compact {p, q} ∈ OX satisfies {p, q} = {p}∨{q}, violating the condition on compact elements.  The following definitions come from Hong [16]. Let F be a filter in L. The set sec F is defined by sec F = {a ∈ L | a ∧ x = 0 for all x ∈ F }. A filter F ⊆ L converges (or is convergent) if it meets every cover of L, and it clusters (or is clustered ) if sec F meets every cover of L. It is shown in [16] that F clusters if and only if



F ¬ = 1.

Furthermore, Dube [13, Proposition 7] makes the observation that for a filter F on a frame L,  sec F = {U : U is an ultrafilter and F ⊆ U }. By the above description, an alternative characterization of the clustering of a classical filter may be realized as follows. A filter F in a frame L clusters if and only if for each A ∈ Cov L there is an ultrafilter U ⊇ F such that A ∩ U = ∅. As already mentioned in the Introduction, a general filter on a frame L is any bounded meet semilattice ϕ : L → T from L into any arbitrary frame T . Following Banaschewski and Hong [7], we say a general filter ϕ : L → T is: (a) prime if ϕ is a lattice homomorphism; (b) completely prime  if ϕ preserves all joins iff ϕ is a frame homomorphism; (c) regular if ϕ(a) = {ϕ(x) | x ≺ a} for all a ∈ L; (d) convergent if ϕ[C] ∈ Cov T for every C ∈ Cov L; and (e) strongly convergent if h ≤ ϕ for some frame homomorphism h : L → T .

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Strong convergence implies convergence. It is not difficult to show that each of the adjectives above is “conservative”, in the sense that a filter F in L has the property of filters described by a given adjective if and only if the general filter χF : L → 2 has the corresponding property of general filters. We record two results on general filters that have classical antecedents. A classical filter F ⊆ L is called regular if for any x ∈ F there is a y ≺ x that also  belongs to F . A convergent regular classical filter F is completely prime. Indeed, if S∈F  for some S ⊆ L, then there is a y ∈ F such that y ≺ S. The set {y ∗ } ∪ S is a / F , it follows that F meets S, as required. cover of L, and so meets F . Since y ∗ ∈ This goes back to the work of Banaschewski and Pultr [10, Proposition 2]. The following proposition gives a generalized version of this fact. Proposition 1.4. A convergent regular general filter is a frame homomorphism. Proof. Let ϕ : L → T be such a general filter. We need to show that ϕ preserves all joins. For any A ⊆ L we clearly have   {ϕ(a) | a ∈ A} ≤ ϕ A ; so we need only verify the opposite inequality. Since ϕ is regular,  

  ϕ A = ϕ(x) | x ≺ A .  Consider any x ≺ A. Then {x∗ } ∪ A is a cover of L, so that, by convergence,  ϕ(x∗ ) ∨ {ϕ(a) | a ∈ A} = 1,  ∗ whence ϕ(x)   ≤ {ϕ(a) | a ∈ A} since ϕ(x) ∧ ϕ(x ) = 0. It follows therefore that ϕ( A) ≤ {ϕ(a) | a ∈ A}, and hence equality.  For the next result we consider h∗ the right adjoint of a frame homomorphism. An extension of a frame L is a frame M together with a dense onto frame homomorphism h : M → L. In [1], Banaschewski calls an extension h : M → L of L flat in case h∗ is a lattice homomorphism. Given an extension h : M → L and a general filter ϕ : M → T , the composite ϕh∗ : L → T is a general filter which we may call the image of ϕ under h. It has been observed by Banaschewski [1] that an extension h : M → L is flat precisely when the image h[P ] of every prime classical filter P in M is prime in L. This is extended to general filters. Proposition 1.5. For an extension h : M → L to be flat, it is necessary and sufficient that h maps prime general filters to prime general filters. Proof. If the extension is flat and ϕ : M → T is prime, then both ϕ and h∗ are lattice homomorphisms, which makes ϕh∗ : L → T prime. Conversely, assume that the condition holds, and suppose, by way of contradiction, that the extension is not flat. Then there exist elements a and b in M such that h∗ (a) ∨ h∗ (b) < h∗ (a ∨ b). By the dual version of Stone’s Separation Lemma, there / F . The is a classical prime filter F in M such that h∗ (a ∨ b) ∈ F but h∗ (a) ∨ h∗ (b) ∈ characteristic function χF : L → 2 is a general prime filter, and so, by hypothesis, χF h∗ : L → T is prime. Hence, in light of h∗ (a ∨ b) ∈ F , we have 1 = χF h∗ (a ∨ b) = χF h∗ (a) ∨ χF h∗ (b), which implies χF h∗ (a) = 1 or χF h∗ (b) = 1, whence h∗ (a) ∈ F or h∗ (b) ∈ F , neither / F . Therefore h is flat.  of which is possible because h∗ (a) ∨ h∗ (b) ∈

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2. Clustering of general filters Henceforth, we shall call filters in a frame “classical filters”, and filters on a frame (i.e. meet semilattice homomorphisms as above) “general filters”. We aim to introduce the concept of clustering for general filters in a “natural” way, so that for the 2-valued filters it reduces to clustering for classical filters. Convergence of general filters is defined in terms of a condition involving the behaviour of the filter with regard to covers of its domain; namely, that it should take covers to covers. Following this, we seek a condition in terms of covers of the domain to define clustering. In that regard let us observe the following simple characterization of clustered classical filters in terms of covers. Lemma 2.1. A classical filter F in a frame L is clustered if and only if for every cover C of L there exists c ∈ C such that c∗ ∈ / F. Proof. Assume that F clusters, and let C be a cover of L. If for each c ∈ C we had c∗ ∈ F , then c∗∗ ∈ F ¬ and we would have    1 = C ≤ {c∗∗ | c ∈ C} ≤ {z ∗ | z ∈ F }, which would contradict the fact that F clusters. Conversely, assume the condition holds. Let C be a cover of L. If F did not cluster, the set F ¬ = {x∗ | x ∈ F } would be a cover of L, so that, by the condition, / F . But this is impossible since F is an there would exist z ∈ F such that z ∗∗ ∈ up-set.  Given a general filter ϕ : L → T , we denote by ϕ← (1) the classical filter ϕ← (1) = {x ∈ L | ϕ(x) = 1}. To motivate the definition of clustering we shall adopt, firstly, we compare convergence of a general filter ϕ : L → T with that of the classical filter ϕ← (1). It is easy to check that =⇒ ϕ convergent; ϕ← (1) convergent that is, the convergence of a general filter is weaker than the convergence of its induced filter. The implication is not reversible since, for instance, the identity map ϕ : 4 → 4 is a convergent filter on 4 for which ϕ← (1) does not converge as it misses the cover {c, c∗ }. Definition 2.2. We say that a general filter ϕ : L → T clusters (or is clustered ) if for every cover C of L there exists c ∈ C such that ϕ(c∗ ) = 1. A simple calculation shows that for an arbitrary general filter ϕ, we have ϕ← (1) clustered

=⇒

ϕ clustered

This notion generalizes the classical one because, as already remarked, it follows immediately from Lemma 2.1 that a classical filter F ⊆ L clusters if and only if the general filter χF : L → 2 clusters. In fact, any general filter ϕ : L → T clusters precisely when the filter ϕ← (1) clusters. To seethe non-immediate implication, suppose ϕ clusters but ϕ← (1) does not. Then [ϕ← (1)]¬ = 1 so that {x∗ | ϕ(x) = 1} is a cover of L. Consequently, by the clustering of ϕ there is an a ∈ L such that ϕ(a) = 1 but ϕ(a∗∗ ) = 1. This is impossible since ϕ is an increasing map.

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As expected, convergence implies clustering for general filters. Proposition 2.3. A convergent general filter clusters. Proof. If a general filter ϕ : L → T does not cluster, then there is a cover C of L such that ϕ(c∗ ) = 1 for every c ∈ C. Since ϕ is a semilattice homomorphism, this implies 0 = ϕ(c) ∧ ϕ(c∗ ) = ϕ(c) for every c ∈ C, in which case ϕ[C] is not a cover of T , implying that ϕ does not converge.  We shall now show that, just as in the case of filters in a frame, almost compactness can also be characterized in terms of general filters. In Hong [16] and ˇ Paseka and Smarda [19] there is a characterization in terms of maximal filters in a frame. We start with a lemma. Since general filters are compared argument-wise, so that only filters with the same domain and codomain can be compared, there is no danger of ambiguity in talking about maximal general filters. Lemma 2.4. Every general filter on a frame is below a maximal one. Proof. Let ϕ : L → T be a general filter on L, and put 

S = {L −→ T |  is a general filter and ϕ ≤ }. Let C ⊆ S be a chain, and define a map ϕ : L → T by  ϕ(a) = {γ(a) | γ ∈ C}. We show that ϕ is a general filter on L. It is immediate that ϕ(0) = 0 and ϕ(1) = 1. It is also clear that ϕ preserves order. To show that it preserves binary meets, consider any a, b ∈ L. Then   ϕ(a) ∧ ϕ(b) = {γ(a) | γ ∈ C} ∧ {δ(b) | δ ∈ C}  = {γ(a) ∧ δ(b) | γ ∈ C, δ ∈ C} by the frame law. Since C is a chain, for any γ, δ ∈ C we have γ(a) ∧ δ(b) ≤ γ(a) ∧ γ(b) = γ(a ∧ b) or γ(a) ∧ δ(b) ≤ δ(a) ∧ δ(b) = δ(a ∧ b), so that ϕ(a) ∧ ϕ(b) = ≤



{γ(a) ∧ δ(b) | γ ∈ C, δ ∈ C}



{η(a ∧ b) | η ∈ C}

= ϕ(a ∧ b), and hence ϕ(a ∧ b) = ϕ(a) ∧ ϕ(b). Thus, ϕ is an upper bound for C. So, by Zorn’s Lemma, S has a maximal element, as required.  We remark that a classical filter F ⊆ L is an ultrafilter if and only if the general filter χF : L → 2 is a maximal 2-valued filter on L. Observe, as well, for use in the next proof that a general filter that is below a clustered one is clustered. Proposition 2.5. The following are equivalent for a frame L. (1) L is almost compact. (2) Every general filter on L clusters. (3) Every maximal general filter on L clusters.

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Proof. (1) ⇒ (2): Suppose that L is almost compact and let ϕ : L → T be a general filter on L. Let C be a cover of L. Since L is almost compact, there is a finite {c1 , . . . , cn } ⊆ C such that (c1 ∨ · · · ∨ cn )∗ = 0. Then c∗1 ∧ · · · ∧ c∗n = 0, hence ϕ(c∗1 ) ∧ · · · ∧ ϕ(c∗n ) = 0 since ϕ is a meet semilattice homomorphism. If ϕ(c∗i ) = 1 for each i = 1, 2, . . . , n, we have a contradiction. Thus, for some i = 1, 2, . . . , n, ϕ(c∗i ) = 1 which proves the implication. (2) ⇒ (3): This is trivial. (3) ⇒ (1): Let F ⊆ L be a classical filter, and consider the general filter χF : L → 2. By Lemma 2.4, there is a maximal filter τ : L → 2 with χF ≤ τ . By the present hypothesis, τ clusters, which implies χF clusters, and hence F clusters. It therefore follows from Hong [16, Corollary 1.4] that L is almost compact.  3. General ultrafilters and some applications We saw in the previous section that every general filter is below a maximal one. Now we aim to define general ultrafilters and balanced general filters taking a cue from the classical case (see, for instance, Dube [13]). In fact, our definition of a general ultrafilter will be based on the characterization that a classical filter F in a frame is an ultrafilter if and only if for any x ∈ L, either x ∈ F or x∗ ∈ F . We continue to follow our maxim that a property of general filters should not be strictly stronger than the classical notion it generalizes. Definition 3.1. We say a general filter ϕ : L → T is: (1) an ultrafilter if, for all x ∈ L, ϕ(x) ∨ ϕ(x∗ ) = 1; (2) balanced if ϕ(x) = 0 for any x ∈ D(L). Clearly, a classical filter is an ultrafilter if and only if its characteristic function is an ultrafilter, and it is balanced if and only if its characteristic function is balanced. It is also immediate that if ϕ← (1) is an ultrafilter, then ϕ is an ultrafilter. Equally clear is that if ϕ← (1) is balanced, then ϕ is balanced. We have simple examples to show that the converses fail. Example 3.2. Let ϕ : 4 → 4 be the identity map. Then ϕ is an ultrafilter on 4, but the classical filter ϕ← (1) is not an ultrafilter in 4 as it contains neither c nor c∗ . Example 3.3. Let φ : 3 → 3 be the identity map. The dense elements of 3 are m and 1, and neither is mapped to 0. So φ is balanced. On the other hand, m∈ / φ← (1), which implies that φ← (1) is not balanced. For classical filters there is a characterization which states that F is an ultrafilter if and only if it is prime and balanced (Dube, [13, Proposition 4]). For general filters we have only one implication in this regard. Theorem 3.4. Every general ultrafilter is prime and balanced. Proof. Let ϕ : L → T be a general ultrafilter. To show that ϕ is balanced, let x ∈ L be dense. Then x∗ = 0, so that ϕ(x∗ ) = 0. Since ϕ is an ultrafilter, this implies 1 = ϕ(x) ∨ ϕ(x∗ ) = ϕ(x) and hence ϕ(x) = 0. Therefore ϕ is balanced. To show that it is prime, consider any a, b ∈ L. Since ϕ is an ultrafilter, we have

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ϕ(z) ∨ ϕ(z ∗ ) = 1 for any z ∈ L. Thus,   ϕ(a ∨ b) = ϕ(a ∨ b) ∧ ϕ(a) ∨ ϕ(a∗ ) ∧ ϕ(b) ∨ ϕ(b∗ ) 



= ϕ(a ∨ b) ∧ ϕ(a) ∨ ϕ(a∗ ) ∧ ϕ(a ∨ b) ∧ ϕ(b) ∨ ϕ(b∗ )   = ϕ(a) ∨ ϕ(a∗ ∧ b) ∧ ϕ(b) ∨ ϕ(a ∧ b∗ )   ≤ ϕ(a) ∨ ϕ(b) ∧ ϕ(b) ∨ ϕ(a) =

ϕ(a) ∨ ϕ(b),

whence we deduce that ϕ(a) ∨ ϕ(b) = ϕ(a ∨ b) as the opposite inequality holds by virtue of ϕ being an increasing map. Therefore ϕ is a prime filter.  Example 3.5. The converse of the preceding result is not true. Consider the general filter φ : 3 → 3 given by the identity map. We observed above that φ is balanced. It is also prime because it is a frame map. Note though that φ is not an ultrafilter since φ(m) ∨ φ(m∗ ) = φ(m) ∨ φ(0) = m = 1. We shall now show that every general ultrafilter is maximal, and give an example to show that the converse fails. Proposition 3.6. Every general ultrafilter is maximal. Proof. Let ϕ : L → T be a general ultrafilter on L, and consider any general filter τ L −→ T on L with ϕ ≤ τ . For any x ∈ L, we have ϕ(x∗ ) ≤ τ (x∗ ). Since ϕ is an ultrafilter, 1 = ϕ(x) ∨ ϕ(x∗ ) ≤ ϕ(x) ∨ τ (x∗ ), which implies that τ (x) ≤ ϕ(x) since τ (x) ∧ τ (x∗ ) = 0. Thus τ ≤ ϕ, and hence equality. Therefore ϕ is maximal.  An argument similar to that employed in the proof of the implication (3) ⇒ (1) in Proposition 2.5 shows that if every general ultrafilter on L clusters, then L is almost compact. It follows from this and the fact that general ultrafilters are maximal that we have the following corollary. Corollary 3.7. A frame is almost compact if and only if every general ultrafilter on it clusters. Here is an example showing that not every maximal general filter is an ultrafilter. Example 3.8. Let L = OR, and consider the identity map idL : L → L. Being a general filter, there is, by Lemma 2.4, a maximal filter μ : L → L such that idL ≤ μ. We claim that μ is not an ultrafilter. To see this, put a = (−∞, 0) in L, so that a∗ = (0, ∞). If μ was an ultrafilter we would have μ(a) ∨ μ(a∗ ) = 1, whence μ(a) ∨ μ(a)∗ = 1, implying that μ(a) = 0 or μ(a) = 1 since L is a connected frame. Now we cannot have μ(a) = 0 as that would lead to the absurdity 0 = a = idL (a) ≤ μ(a) = 0. We can also not have μ(a) = 1 since that would imply μ(a∗ ) = 0, leading to a similar absurdity. We end this section with characterizations of Boolean frames in terms of general filters. We need a definition motivated by what Hong [16] calls Boolean filters in a frame. These are classical filters which are generated by their complemented elements.

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Definition 3.9. A general filter ϕ : L → T on L is Boolean if for every a ∈ L with ϕ(a) = 1, there is a complemented c ≤ a such that ϕ(c) = 0. We now show that Lemma 1.1 also carries over for general filters. Lemma 3.10. On a zero-dimensional frame every completely prime general filter is Boolean. Proof. Let L be a zero-dimensional frame and ϕ : L −→ T be a completely prime general filter. Suppose that x ∈ L and  ϕ(x) = 1. Since L is zero-dimensional we may find A ⊆C(L) such that x = A. Then ϕ being completely prime gives  ϕ(a). Consequently, ϕ(a) = 0 for some a ∈ A. Since a is 1 = ϕ(x) = ϕ( A) = a∈A

complemented and a ≤ x we are done.



In the proof that follows we shall use the fact that, in any frame, if x ≺ a, then x∗∗ ≤ a. We shall also use the dual version of Stone’s Separation Lemma which enables us to say if x < 1 in a frame, then there is a prime filter in the frame that misses x. Theorem 3.11. The following are equivalent for a frame L: (1) L is Boolean. (2) Every general filter on L is regular. (3) The general filter bL : L → BL is regular. (4) Every general filter on L is Boolean. (5) Every prime general filter on L is an ultrafilter. Proof. (1) ⇒ (2): Assume that L is Boolean, and  let ϕ : L → T be a general filter on L. For any a ∈ L, a ≺ a, so that ϕ(a) = {ϕ(x) | x ≺ a}, and hence ϕ is regular. (2) ⇒ (3): This is trivial. (3) ⇒ (1): Assume that bL : L → BL is regular. Then, for any a ∈ L we have   a ≤ a∗∗ = bL (a) = {bL (x) | x ≺ a} = {x∗∗ | x ≺ a} ≤ a, which says that a = a∗∗ . Therefore L is Boolean. (1) ⇔ (4): If L is Boolean, ϕ : L → T is a general filter on L, and a ∈ L is such that ϕ(a) = 1, then c = a is a complemented element of L such that c ≤ a and ϕ(c) = 0. Therefore ϕ is Boolean. Conversely, let 0 = b ∈ L. Consider the general filter χF : L → 2 on L, where F is the classical filter F = ↑b in L. By the present hypothesis, χF is Boolean. Since χF (b) = 1, there exists some complemented c ≤ b such that χF (c) = 0. This in turn implies that χF (c) = 1, that is, c ∈ ↑b, meaning that c ≥ b. Consequently, c = b and hence L is Boolean. (1) ⇔ (5): If L is Boolean and ϕ : L → T is a prime general filter on L, then for any a ∈ L we have a ∨ a∗ = 1, which, by primeness of ϕ, implies that 1 = ϕ(a ∨ a∗ ) = ϕ(a) ∨ ϕ(a∗ ), thus showing that ϕ is an ultrafilter. Conversely, suppose, by the way of contradiction, that L is not Boolean. Take a ∈ L such that a∨a∗ < 1. By the dual version of Stone’s Separation Lemma, there / F . The general filter χF : L → 2 is prime, is a prime filter F ⊆ L such that a ∨ a∗ ∈ so, by the present hypothesis, it is an ultrafilter. In consequence, χF (a) ∨ χF (a∗ ) =

P. BHATTACHARJEE1,2 AND I. NAIDOO2

12

1, which implies that a ∈ F or a∗ ∈ F , neither of which is possible since a ∨ a∗ ∈ / F. Therefore L is Boolean.  4. A stronger variant of clustering The possible paucity of points in frames gives rise to the following “deficiency” regarding clustering of classical filters. In any Boolean frame L with no atoms the classical filter ↑a, for any a = 0, clusters but is contained in no convergent filter. This does not accord with the situation in spaces. In trying to remedy this Dube and Naidoo in [14] define a classical filter F in a frame L to be strongly clustered as follows:  a filter F in a frame L strongly clusters if F ¬ ≤ p for some p ∈ Spec(L). We then have the following characterization of strong clustering for classical filters. Theorem 4.1. A filter F in a frame L strongly clusters if and only if there is a completely prime filter P ⊆ L such that P ⊆ sec F .  Proof. Suppose that F strongly clusters. Then {x∗ : x ∈ F } ≤ p for some p ∈ Spec L. Then P = {x ∈ L : x  p} is  completely prime. If x ∈ P and x ∧ y = 0 for some y ∈ F , then x ≤ y ∗ ≤ {x∗ : x ∈ F } ≤ p which is a contradiction. Thus for each x ∈ P , x ∧ y = 0 for each y ∈ F . Thus x ∈ secF so that P ⊆ secF  . Conversely, if Q ⊆  secF for some completely prime filter Q in L, then p = (L\Q) ∈ Spec L. If {x∗ : x ∈ F } ∈ Q, since Q is completely ∗ . Then x∗ ∈ secF which is a contradiction. Thus prime,  ∗ x ∈ Q for some x ∈ F / Q and hence {x∗ : x ∈ F } ≤ p. Hence F strongly clusters.  {x : x ∈ F } ∈ Dube and Naidoo in [14] then prove that a classical filter in a regular frame strongly clusters precisely when it is contained in a convergent classical filter. It is on the basis of that that we formulate the following definition. Definition 4.2. A general filter ϕ : L → T strongly clusters (or is strongly clustered ) if there is a convergent general filter τ : L → T such that ϕ ≤ τ . It is immediate from the definition that a convergent general filter is strongly clustered. Below we list some quick observations regarding how the strong clustering relates to other properties of filters. Recall from Banaschewski and Hong [7] that a general filter is said to be strongly convergent if there is a frame homomorphism below it. Of course, strong convergence implies convergence. Proposition 4.3. The following properties hold. (1) If a general filter strongly clusters, then it clusters. (2) A maximal general filter (and hence, an ultrafilter) strongly clusters if and only if it converges. (3) A prime general filter on a regular frame strongly clusters if and only if it strongly converges. Proof. (1) Let ϕ : L → T be a strongly clustered general filter. By definition, there is a convergent τ : L → T such that ϕ ≤ τ . Let C be cover of L. We cannot have ϕ(c∗ ) = 1 for every c ∈ C as that would imply  ∗    ϕ(c∗ ) ≤ τ (c∗ ) ≤ τ (c)∗ = τ [C] = 0. 1= c∈C

c∈C

c∈C

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13

We conclude therefore that ϕ is clustered. (2) This is immediate. (3) Since strong convergence implies convergence, which, in turn, implies strong clustering, only one implication needs to be shown. So suppose that ϕ : L → T is a strongly clustering prime filter on a regular frame L. Take a convergent τ : L → T such that ϕ ≤ τ . As shown in Banaschewski and Hong [5, Lemma 4], the map τ ◦ : L → T defined by  τ ◦ (a) = {τ (x) | x ≺ a} is a frame homomorphism. We show that τ ◦ ≤ ϕ. Let b ∈ L, and consider any x ≺ b. Then x∗ ∨ b = 1, which, by the primeness of ϕ, implies 1 = ϕ(x∗ ) ∨ ϕ(b) ≤ τ (x∗ ) ∨ ϕ(b), whence τ (x) ≤ ϕ(b), and consequently τ ◦ (b) ≤ ϕ(b), by the regularity of L. Therefore ϕ strongly converges.  Remark 4.4. Part of the proof of (3) in the foregoing proposition mimics the proof of Banaschewski, Hong and Pultr [8, Lemma 2.1]. We end with the following characterizations of compact regular frames in terms of strong clustering. Corollary 4.5. The following are equivalent for a regular frame L: (1) L is compact. (2) Every prime general filter on L strongly clusters. (3) Every general ultrafilter on L strongly clusters. Proof. (1) ⇒ (2): Since every cover of a compact frame admits a finite subcover, every prime general filter on a compact frame takes covers to covers, and is therefore strongly clustered. (2) ⇒ (3): This follows from the fact that every general ultrafilter is prime (Proposition 3.4). (3) ⇒ (1): If every general ultrafilter on L strongly clusters, then every general ultrafilter on L clusters, and so L is almost compact by Corollary 3.7. Since L is regular, this implies that L is compact.  5. Precompact Uniform frames revisited We have the following calculus for covers of a frame L. Let A, B ∈ Cov L and x ∈ L. Then we say that A refines B, expressed as A ≤ B, if for every a ∈ A there exists b ∈ B suchthat a ≤ b. We define Ax to be the largest element in A that meets x, that is Ax = {a ∈ A : a ∧ x = 0}. Also, A ∧ B = {a ∧ b : a ∈ A and b ∈ B}, AB = {Ab : b ∈ B} and we say that A star refines B, expressed as A ≤∗ B, if AA ≤ B. For any collection of covers U ⊆ Cov L, we define the relation
U on L as follows. For x, y ∈ L x
U y

iff

Ax ≤ y

for some

A∈U

and we say that x is uniformly below y if x
U y (or for brevity x
y). U is said  to be an admissible collection of covers if for each y ∈ L we have that y = {x ∈ L : x
y}. A nearness on L is an admissible filter U in (Cov L, ≤). The pair (L, U) is then called a nearness frame. The members in U are called uniform

14

P. BHATTACHARJEE1,2 AND I. NAIDOO2

covers. If L is a nearness frame, then we will write the nearness as UL. A frame homomorphism h : (L, UL) −→ (M, UM ) between nearness frames is called a uniform frame homomorphism if h preserves uniform covers. The resulting category of nearness frames and uniform frame homomorphisms is denoted by NFrm. If further for each B ∈ UL, A ≤∗ B for some A ∈ UL, then (L, UL) is called a uniform frame or L is uniformizable and UL is called the uniformity on L. The resulting category is denoted by UFrm. It is well known that a frame is uniformizable precisely when it is completely regular. Thus whenever we encounter uniform frames the underlying frame is completely regular and hence regular. Our general references for uniform and nearness frames are Banaschewski [2, 3], Banaschewski, Hong and Pultr [8, 9] or Banaschewski and Pultr [10]. Naidoo in [17] defines a classical filter F in a uniform frame (L, UL) to be weakly Cauchy if secF ∩ A = ∅ for each A ∈ UL. We also have essentially equivalent characterizations of weakly Cauchy filters as for clustering given in Lemma 2.1 and Hong [16, Proposition 1.3] with similar proofs. Proposition 5.1. The following are equivalent for a classical filter F in a uniform frame (L, UL). (1) F is weakly Cauchy. / F. (2) For each A ∈ UL, there is a ∈ A such that a∗ ∈ (3) F ¬ is not a uniform cover. Proof. (1) ⇒ (2): Suppose that F is a weakly Cauchy filter and A ∈ UL. Since F is weakly Cauchy, we may then find a ∈ A such that a ∧ x = 0 for each x ∈ F . Consequently, a∗ ∈ / F. (2) ⇒ (3): Suppose (2). If F ¬ = {x∗ | x ∈ F } is a uniform cover then by the / F ¬ . However, since a ≤ a∗∗ we have hypothesis we may find a ∈ F such that a∗∗ ∈ ∗∗ / UL. a ∈ F . We thus have a contradiction. Thus we must have that F ¬ ∈ (2) ⇒ (3): Suppose that F is not weakly Cauchy. We may then find a uniform cover A such that secF ∩ A = ∅. If a ∈ A then a ∈ / secF so that a ∧ x = 0 for some x ∈ F . Then a ≤ x∗ so that A ≤ F ¬ showing that F ¬ is a uniform cover, which is a contradiction. Thus F is a weakly Cauchy filter.  We now capture this notion for general filters in the category UFrm as follows. Definition 5.2. A general filter ϕ : L −→ T on a uniform frame is called weakly Cauchy or W-Cauchy if for each A ∈ UL there exists a ∈ A such that ϕ(a∗ ) = 1. We can now see from the above proposition that a classical filter F in a uniform frame L is weakly Cauchy iff the general filter χF : L −→ 2 is W-Cauchy. We recall from Banaschewski [2] that a general filter ϕ : L −→ T between uniform frames is (a) Cauchy if for eachA ∈ UL, ϕ[A] ∈ Cov T and (b) regular if ϕ(x) = {ϕ(y) : y
x} for each x ∈ L. The following result is proved similarly to that of Proposition 2.3 with the minor modification of considering uniform covers. Lemma 5.3. Every general Cauchy filter is W-Cauchy. Precompact uniform frames (those uniform frames whose uniform covers are generated by the finite uniform covers) have classical filter characterizations as

ANOTHER NOTE ON PRECOMPACT UNIFORM FRAMES

15

given in Naidoo [17]. We attempt below to formalize some these results using general filters. We note that precompact uniform frames are characterized by uniform almost compactness as given by Naidoo [17, Theorem 3.1] where L is uniformly almost compact provided that for each A ∈ UL there  is B ⊆< ω A for which B ∈ D(L). We will use the above equivalent characterization of precompactness in the proof of our next results. We remark that if a uniform frame L is not precompact we may find a particular filter in L, which is constructed as follows. L is not  Since ∗ A, ( B) =  0. Then precompact there is a A ∈ UL such that for each B ⊆ <ω  {( D)∗ : D ⊆< ω A} generates a proper classical filter in L. We shall denote this filter by F ∇ and will require it in the next results. Lemma 5.4. If every general ultrafilter on a uniform frame L is W-Cauchy, then L is precompact. Proof. Suppose that every general ultrafilter on L is W-Cauchy. If L is not precompact, then consider the proper classical filter F ∇ in L. We may find a classical ultrafilter U ⊇ F ∇ . Consider the general ultrafilter χU : L −→ 2. By the hypothesis χU is W-Cauchy so that there is a ∈ A such that χU (a∗ ) = 1. Hence χU (a∗ ) = 0. / F ∇ which is a contradication.  Since U ⊇ F ∇ we must have that a∗ ∈ We have the following extension to general filters of the result in Naidoo [17, Corollary 3.1]. Theorem 5.5. Every general ultrafilter on a uniform frame L is Cauchy iff L is precompact. Proof. Suppose that every general ultrafilter on L is Cauchy. If L is not precompact then again consider the proper classical filter F ∇ in L. We may find a classical ulthe general ultrafilter χU : L −→ 2. By the hypothesis trafilter U ⊇ F ∇ . Consider  χU is Cauchy so that χU [A] = 1. Thus for some a ∈ A, χU (a) = 0 whence a ∈ U . / F ∇. Since U ⊇ F ∇ , we arrive at the contradiction that a∗ ∈ Now suppose that L is precompact and ϕ : L −→ T is a general ultrafilter on L. Let A ∈ UL. We may then find a finite B ∈ UL that refines A. Then for each b ∈ B there is ab ∈ A such that b ≤ ab . Since ϕ is prime, we have      1=ϕ B = ϕ(b) ≤ ϕ(ab ) ≤ ϕ(a). b∈B

b∈B

Thus ϕ[A] ∈ Cov T whence ϕ is Cauchy.

a∈A



We now have the following result that relates prime general filters, W-cauchy and precompactness. Theorem 5.6. L is a precompact uniform frame iff every prime general filter on L is W-Cauchy. Proof. Suppose that L is precompact and ϕ : L −→ T is a prime general filter. If A ∈ UL, by  precompactness there is a finite uniform B ≤ A. Since ϕ is prime we  ∗ ) ≤ [ϕ(b)]∗ for each b ∈ B, we have 1= ϕ( B) = {ϕ(b) : b ∈ B}. Since   ϕ(b ∗ ∗ ∗ ϕ(b ) ≤ [ϕ(b)] = 0 so that ϕ(b ) = 0. Thus ϕ(b∗ ) = 1 for some have b∈B

b∈B

b∈B

P. BHATTACHARJEE1,2 AND I. NAIDOO2

16

b ∈ B. Consequently there is a ∈ A such that b ≤ a and thus a∗ ≤ b∗ whence ϕ(a∗ ) = 1. Hence ϕ is W-cauchy. Now suppose that every prime general filter is W-Cauchy. Since every general ultrafilter is prime by Theorem 3.4, every general ultrafilter is thus W-Cauchy. By the above Lemma 5.4 L is precompact.  Corollary 5.7. If every maximal general filter on L is W-Cauchy, then L is precompact. Proof. Let ϕ : L −→ T be any general filter. By Lemma 2.4 ϕ ≤ τ for some general maximal filter τ : L −→ T . By the hypothesis, τ is W-Cauchy. Let A ∈ UL. Then there is a ∈ A such that τ (a∗ ) = 1. Consequently, ϕ(a∗ ) = 1 so that ϕ is W-Cauchy. Thus every general filter on L is W-Cauchy. In particular, every general ultrafilter on L is W-Cauchy and hence, by Lemma 5.4 L is precompact.  Proposition 5.8. A uniform frame L is precompact iff for any general filter ϕ : L −→ T there is a W-Cauchy general filter τ : L −→ T such that ϕ ≤ τ . Proof. Suppose that L is precompact and ϕ : L −→ T is a general filter. We may find a maximal general filter τ : L −→ T such that ϕ ≤ τ by Lemma 2.4. Since L is precompact τ is W-Cauchy by Theorem 5.7. On the other hand suppose that ϕ : L −→ T is such that there is a W-Cauchy τ : L −→ T such that ϕ ≤ τ . If A ∈ UL, then there is a ∈ A such that τ (a∗ ) = 1. Consequently, ϕ(a∗ ) = 1 so that ϕ is W-Cauchy. Thus by Lemma 5.4 L is precompact.  We end with our last result which is a structured version of Proposition 2.5. Proposition 5.9. The following are equivalent for a uniform frame (L, UL). (1) L is precompact. (2) Every general filter on L is W-Cauchy. (3) Every maximal general filter on L is W-Cauchy. Proof. (1) ⇒ (2): exception of using stead. (2) ⇒ (3): This (3) ⇒ (1): This

The proof is similar to (1) ⇒ (2) of Proposition 2.5 with the uniform almost compactness and taking a uniform cover C inis trivial. is Corollary 5.7.



Acknowledgement. We thank the anonymous referee for the remarks and recommendations which have helped in the revision of the first version of this paper. Also, we are extremely grateful and indebted to our colleague and friend Professor Themba Dube at the University of South Africa (UNISA) for the numerous fruitful discussions on convergence in frames that materialized the completion of this paper. Bhattacharjee thanks the Topology Research Group and the Research Directorate at UNISA for financial support that enabled her to visit Dube and Naidoo at UNISA during which time some of the work here was done. Naidoo also acknowledges the National Research Foundation (NRF) of South Africa for the grant 93410 towards this project. References [1] B. Banaschewski, Wallman compactifications, (unpublished notes). [2] B. Banaschewski, Completion in pointfree topology, Lecture Notes in Math. and Appl. Math., Univ. of Cape. Town, SoCat94, No2/(1996).

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[3] B. Banaschewski, Uniform completion in pointfree topology, Topological and Algebraic Structures in Fuzzy Sets, Trends in Logic Vol. 20, 2003, pp. 19-26. [4] B. Banaschewski and S.S. Hong, Filters and Strict Extensions of Frames, Kyungpook Math. J. 39(1999), 215-230. [5] B. Banaschewski and S.S. Hong, Extension by continuity in pointfree topology, Appl. Categ. Struct. 8 (2000), 475-486, 2000. [6] B. Banaschewski and S.S. Hong, General filters and strict extensions in pointfree topology, Kyungpook Math. J. 42 (2002) 273-281. [7] B. Banaschewski and S.S. Hong, Variants of compactness in pointfree topology, Kyungpook Math. J. 45 (2005) 455-470. [8] B. Banaschewski, S.S. Hong and A. Pultr, On the completion of nearness frames, Quaest. Math. 21 (1998), 19-37. [9] B. Banaschewski and A. Pultr, Samuel compactification and completion of uniform frames, Math. Proc. Camb. Phil. Soc. 108 (1990), 63-78. [10] B. Banaschewski and A. Pultr, Cauchy points of uniform and nearness frames, Quaest. Math. 19 (1996), 107-127. [11] P. Bhattacharjee, Two spaces of minimal primes, J. of Alg. and its Appl., Vol. 11, No. 1 (2012), 1250014 (18 pages). [12] T. Dube, Structures in Frames, PhD Thesis, University of Durban-Westville (1992). [13] T. Dube, Balanced and closed-generated filters in frames, Quaest. Math. 26 (2003) 73-81. [14] T. Dube and I. Naidoo, More on uniform paracompactness in pointfree topology, Math. Slovaca 65 (2015), 273-288. [15] H. Herrlich, Topological structures, Math. Centre Tracts, 52 (1974) 59 - 122. [16] S.S. Hong, Convergence in frames, Kyungpook Math. J. 35 (1995), 85-91. [17] I. Naidoo, A note on precompact uniform frames, Topology and its Applications, 153 (2005), 941-947. [18] I. Naidoo, Strong Cauchy completeness in uniform frames, Acta Math. Hung. 116(3)(2007), 273 - 281. ˇ [19] J. Paseka and B. Smarda, T2 -frames and almost compact frames, Czech. Math. J. 42 (1992), 385-402. [20] J. Picado and A. Pultr, Frames and Locales: Topology without points, Frontiers in Mathematics, Springer, Basel, 2012. 1 Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science, 777 Glades Road, Boca Raton, FL 33431, UNITED STATES OF AMERICA Email address: [email protected] 2 University of South Africa, College of Science, Engineering and Technology, Department of Mathematical Sciences, Florida, Johannesburg, P. O. Box 392, 0003 Pretoria, SOUTH AFRICA. Email address: [email protected]