Closed and Open Maps for Partial Frames

This paper concerns the notions of closed and open maps in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices, σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}- and κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-frames and full frames. We define closed and open maps using geometrically intuitively appealing conditions involving preservation of closed, respectively open, congruences under certain maps. We then characterize them in terms of algebraic identities involving adjoints. We note that partial frame maps need have neither right nor left adjoints whereas frame maps of course always have right adjoints. The embedding of a partial frame in either its free frame or its congruence frame has proved illuminating and useful. We consider the conditions under which these embeddings are closed, open or skeletal. We then look at preservation and reflection of closed or open maps under the functors providing the free frame or the congruence frame. Points arise naturally in the construction of the spectrum functor for partial frames to partial spaces. They may be viewed as maps from the given partial frame to the 2-chain or as certain kinds of filters; using the former description we consider closed and open points. Any point of a partial frame extends naturally to a point on its free frame and a point on its congruence frame; we consider the closedness or openness of these.


Introduction
Topological spaces and frames or locales have been contexts in which closed and open maps have proved to be important tools. For topological spaces, for example, the second projection from X × Y to Y is closed for any space Y iff X is compact. Another example states that if there exists an open map f : X → Y of a locally compact space X onto a Hausdorff space Y , then Y is locally compact. The uses to which closed and open mappings, respectively, have been put are somewhat different, but in this paper we begin by emphasizing their formally similar properties.
Our context will be that of partial frames. Partial frames are meet-semilattices where, in contrast with frames, not all subsets need have joins. A selection function, S, specifies, for all meet-semilattices, certain subsets under consideration, which we call the "designated" ones; an S-frame then must have joins of (at least) all such subsets and binary meet must distribute over these. A small collection of axioms suffices to specify our selection functions; these axioms are sufficiently general to include as examples of partial frames, bounded distributive lattices, σ -frames, κ-frames and frames.
In this paper we provide definitions of closed and open maps between partial frames. We use a geometrically intuitively appealing condition involving preservation of closed, respectively open, congruences under certain maps. We then characterize them in terms of algebraic identities involving adjoints. We follow the ideas of Chen (see [6]) whose work we gratefully acknowledge. We note however that partial frame maps need have neither right nor left adjoints whereas frame maps of course always have right adjoints.
A weakening of the notion of an open map leads to the idea of a skeletal map; these played an important rôle in out study of the Madden quotient in [12] and arise here also.
The embedding of a partial frame in either its free frame or its congruence frame has proved illuminating and useful. (See [9,12].) We consider the conditions under which these embeddings are closed, open or skeletal. We then look at preservation and reflection of closed or open maps under the functors providing the free frame or the congruence frame.
Points arise naturally in the construction of the spectrum functor for partial frames to partial spaces. (See [13].) They may be viewed as maps from the given partial frame to the 2-chain or as certain kinds of filters; using the former description we consider closed and open points. Any point of a partial frame extends naturally to a point on its free frame and a point on its congruence frame; we consider the closedness or openness of these.

Background
See [16,23] as references for frame theory; see [2,3] for σ -frames; see [20,21] for κ-frames; see [1,19] for general category theory. The basics of our approach to partial frames can be found in [7][8][9]. An example of a paper of ours with a more topological flavour is [11]. Our papers with a more algebraic flavour, especially relevant to the current topic, are [10,12,14,15]. For earlier work by other authors in this field see [22,[24][25][26]. For those interested in a comparison of the various approaches, see [8].
A meet-semilattice is a partially ordered set in which all finite subsets have a meet. In particular, we regard the empty set as finite, so a meet-semilattice comes equipped with a top element, which we denote by 1. We do not insist that a meet-semilattice should have a bottom element, which, if it exists, we denote by 0. A function between meet-semilattices f : L → M is a meet-semilattice map if it preserves finite meets, including the top element. A sub meet-semilattice is a subset for which the inclusion map is a meet-semilattice map.
The essential idea for a partial frame is that it should be "frame-like" but that not all joins need exist; only certain joins have guaranteed existence and binary meets should distribute over these joins. The guaranteed joins are specified in a global way on the category of meetsemilattices by specifying what is called a selection function; the details are given below.

Definition 2.1
A selection function is a rule, which we usually denote by S, which assigns to each meet-semilattice A a collection S A of subsets of A, such that the following conditions hold (for all meet-semilattices A and B): (By X ≤ Y we mean, as usual, that for each x ∈ X there exists y ∈ Y such that x ≤ y.) Of course (SFin) implies (S1) but there are situations where we do not impose (SFin) but insist on (S1). Note that we always have ∅ ∈ S A. Once a selection function, S, has been fixed, we speak informally of the members of S A as the designated subsets of A.
Definition 2.2 An S-frame L is a meet-semilattice in which every designated subset has a join and for any such designated subset B of L and any a ∈ L Of course such an S-frame has both a top and a bottom element which we denote by 1 and 0 respectively.
In particular such an f preserves the top and bottom element. If, in addition, f (x) = 0 implies that x = 0, we say that f is dense.
A sub S-frame T of an S-frame L is a subset of L such that the inclusion map i : T → L is an S-frame map.
The category SFrm has S-frames as objects and S-frame maps as arrows.
We use the terms "partial frame" and "S-frame" interchangeably, especially if no confusion about the selection function is likely. We also use the term full frame in situations where we wish to emphasize that all joins exist.

Note 2.3 Here are some examples of different selection functions and their corresponding
S-frames.
1. In the case that all joins are specified, we are of course considering the notion of a frame. 2. In the case that (at most) countable joins are specified, we have the notion of a σ -frame. 3. In the case that joins of subsets with cardinality less than some (regular) cardinal κ are specified, we have the notion of a κ-frame. 4. In the case that only finite joins are specified, we have the notion of a bounded distributive lattice.
The remainder of this section gives a lot of information about H S L, the free frame over the S-frame L, as well as C S L, the frame of S-congruences of L, and the relationship between the two. These results come from [9] on H S L, [10,12,14] on C S L.
In the definition below, L is an S-frame. (d) The collection of all S-congruences on L is denoted by C S L; we refer to it as the congruence frame of L. It is in fact a full frame with meet given by intersection. (e) (i) Let A ⊆ L × L. We use the notation A to denote the smallest S-congruence containing A. This exists by completeness of C S L. (ii) For a ∈ L we define ∇ a = {(x, y) : x ∨ a = y ∨ a} and a = {(x, y) : x ∧ a = y ∧ a}; these are S-congruences on L. (iii) It is easily seen that ∇ a = (0, a) and that a = (a, 1) . (h) We also note that for frame maps f and g with domain C S L, if f • ∇ = g • ∇ then f = g. (i) A useful congruence for our purposes is the Madden congruence, π, described below.
(i) For x ∈ L, set P x = {t ∈ L : t ∧ x = 0}. (ii) For x ∈ L, P x is an S-ideal, and in H S L, P x = (↓ x) * , the pseudocomplement of ↓ x. (iii) We define π = {(x, y) : P x = P y }; π is an S-congruence of L. (iv) The quotient map induced by the Madden congruence, p : L → L/π is dense, onto and the universal such. We refer to this as the Madden quotient. (See [12].)

Definition 2.5
For any S-frame L, define e L : H S L → C S L to be the unique frame map such that e L (↓ a) = ∇ a for all a ∈ L; that is, making the following diagram commute: That this map e L exists follows from the freeness of H S L as a frame over L. See [9]. Where no confusion can arise, we omit the subscript L.

Remark 2.6
The range of the map e L : H S L → C S L mentioned above consists of all the S-congruences of L that can be written as arbitrary joins of ∇ a 's, for a ∈ L. For a proof, see [14]. In the same place we show the following:

Note 2.7
For any S-frame L, H S L is isomorphic to a subframe of C S L; that is, the free frame over L is isomorphic to a subframe of the frame of S-congruences of L.

Right and Left Adjoints
Much of what is mentioned in this section is well known but it is as well to remind the reader that, since S-frames are in general not complete, some of what is used in frame theory concerning adjoints needs to be reconsidered.
We make no claim that all S-frame maps have right or left adjoints; this is false (see Example 3.4). However, clearly if an S-frame map has a right or left adjoint, such is unique. The following are well-known.

(a) Suppose that h has a right adjoint r . Then: (i) hr ≤ id M and r h ≥ id L , (ii) h is one-one ⇐⇒ rh = id L and (iii) h is onto ⇐⇒ hr = id M . (b) Suppose that h has a left adjoint . Then: (i) id M ≤ h and id L ≥ h. (ii) h is one-one ⇐⇒ lh = id L and (iii) h is onto ⇐⇒
Since arbitrary meets or joins in S-frames need not exist, the following prove useful.

(a) Suppose that h has a right adjoint, r . Then h preserves all existing joins and r preserves all existing meets. (b) Suppose that h has a left adjoint, . Then h preserves all existing meets and preserves all existing joins.
Proof This is a categorical fact about adjoints but can of course be checked directly.
We mention some cases in which existence of adjoints ensures completeness. This result will prove useful when embedding an S-frame in its free frame or congruence frame.

Corollary 3.6 Suppose that h : L → N is a one-one S-frame map, where L is an S-frame and N is a (full) frame. If h has a right or left adjoint, then L is a complete lattice.
Proof Assume h has a right adjoint r and {x j : j ∈ J } an arbitrary subset of L. Since N is a frame, A possible converse to Corollary 3.6 fails as the following example shows.

Example 3.7
Let L consist of all countable subsets of R, with R itself added as top element. Let N be the power set of R and h : L → N the identical embedding. Then L is a σ -frame and a complete lattice while N is a frame, and h is a one-one σ -frame map. However h does not preserve all existing joins. This is demonstrated by considering the set {{i} : i is irrational}: Considering the σ -frame consisting of all co-countable subsets of R with the empty set added as bottom element, provides a similar example, this time of a function with no left adjoint.

Closed and Open Maps
There are various equivalent characterizations of closed and open maps for frames, see for example [23] or [6]. For partial frames we choose a definition using closed and open congruences and then provide an equivalent approach using right and left adjoints.
The above corresponds with one's intuition for topological spaces and closed and open continuous functions: for example, such a function f : We note that (see [12]) C S is a functor from S-frames to frames such that, for any S-frame map h : L → M we have a frame map C S h : C S L → C S M making the following diagram commute: In particular, notice that We now provide characterizations of closed and open maps in which right and left adjoints arise naturally.

(b) The map h is open iff h has a left adjoint, , and for all x ∈ L, m ∈ M,
and

Note 4.4
Let L be an S-frame and a ∈ L. The following is well-known in the case of frames; for future reference we describe quotients using closed congruences explicitly.
(a) Consider the diagram We note that j is an isomorphism making this diagram commute. The right adjoints of h and g are given explicitly by (b) A similar situation applies for open quotients with L/ a ↓ a. Here are the details.
Consider the diagram Again j is an isomorphism making this diagram commute. The left adjoints of h and g are given explicitly by The following natural and useful results show that closed maps and closed congruences are related appropriately as are the open ones.  The following result for frames appears in [6]. ii) By assumption, for all n ∈ N , where r is the right adjoint of g f . Now for m ∈ M and x, y ∈ L, Thus f is closed. We note that (as can of course be checked directly) the right adjoint of f is rg. iii) By assumption, for all n ∈ N , (g f × g f ) −1 (∇ n ) = ∇ r (n) , where r is the right adjoint of g f . Using f onto we need only consider pairs of the form ( f (x), f (y)) for x, y ∈ L: Thus g is closed. We note that the right adjoint of g is f r.

Note 4.8 Theorem 3.2.1 of [6] states that any frame map with a Boolean domain is closed.
The corresponding statement for S-frames is false, as Example 3.4 shows. In Theorem 3.5 of [15] we noted that if H S L is Boolean then L is a Boolean frame but not conversely. The counterexample used there also shows that an S-frame map with Boolean frame as domain need not be closed. However, an application of Lemma 4.7 shows that, if H S L is Boolean, any S-frame map with domain L is closed.
The following result, due to [6] in the frame case, characterizes closed and open maps with domain 3.

Lemma 4.9 Let 3 denote the 3-element frame with middle element μ, and let M denote any
The final example of this section gives a simple example showing that maps that are both closed and open need not be isomorphisms. For j ∈ I , the right adjoint, r j , of p j is given by r j (x) = (a α ) α∈I where a j = x and a k = 1 for k = j. Then Thus p j is closed. A similar idea shows that each p j is open, using as left adjoint j (x) = (a α ) α∈I where a j = x and a k = 0 for k = j.

A Skeletal Interlude
Weakly open maps in the context of Boolean frames were considered by Banaschewski and Pultr in [4,5]. Such maps are precisely the skeletal maps used in [17,18]; the latter terminology is perhaps more widely known and we use it here. In [12] we considered the category of partial frames with skeletal maps and showed that the d-reduced partial frames form a reflective subcategory of this. In this paper we are interested in the fact that skeletal maps are a generalization of open ones.

Definition 5.1 An
. For the definition of the Madden congruences π L , π M see Definition 2.4.

Lemma 5.2 An S-frame map h : L → M is dense and skeletal if and only if
Proof The proof is straightforward and omitted.

Proof Suppose that h : L → M is open. By Theorem 4.2, h has a left adjoint and
To show that h is skeletal, we assume (x, y) ∈ π L and show (h(x), h(y)) ∈ π M . So suppose that P x = P y . Take m ∈ M with m ∧h(x) = 0. We show that m ∧h(y) = 0. Applying gives On the other hand, by [12], L is d-reduced, so every S-frame map with domain L is skeletal. (a) If f , g are skeletal, so is g f .

(b) If g f is skeletal and g is one-one then f is skeletal. (c) If g f is skeletal and g is dense then f is skeletal.
Proof (a) See [12].

The Embedding of a Partial Frame into Its Free Frame and Its Congruence Frame
We investigate two important embeddings of a partial frame: first into its free frame and then into its congruence frame. In each case we characterize when the embeddings are closed, open and skeletal. x ∧ (J ) for all x ∈ L, J ∈ H S L. By (c) above, this amounts to { j ∈ J : j ≤ x} = x ∧ J . The proof of the required frame distribution law is then straightforward using the idea of replacing the join of an arbitrary subset by the join of the S-ideal generated by it. (e) Suppose that (x, y) ∈ π L , so that P x = P y . By Lemma 4.19 of [12], To put the above result into some context, we note that in [15] we established several conditions equivalent to the embedding ↓: L → H S L being an isomorphism. These are For details, see [15], Theorem 5.2.
We turn now to the analogous embedding of a partial frame into its frame of S-congruences. See Definition 2.5 for some details. Proposition 6.2 Let L be an S-frame and ∇ : L → C S L the embedding into its congruence frame.
Proof (a) Suppose that ∇ has right adjoint R and consider the commuting diagram By Lemma 4.7, since e is one-one, ∇ closed implies ↓closed. By Proposition 6.1, ↓: L → H S L is an isomorphism. For a ∈ L, a ∧ R( a ) = R(∇ a ∧ a ) = 0. This uses the fact that R preserves meets and ∇ is one-one. Since ∇ is also closed, a ∨ R( a ) = R(∇ a ∨ a ) = 1. As always, R(1) = 1 and R(0) = 0 since ∇ is dense. So R( a ) is the complement of a in L. This shows that L is Boolean, so by Proposition 3.3 of [15] e : H S L → C S L is an isomorphism. So ∇ : L → C S L is an isomorphism.

The Free Functor on Closed and Open Maps
We note that the map ↓is a natural transformation from the identity functor on S-frames to the functor H S . (See [9].) So for any S-frame map h : L → M there exists a unique frame map H S h : H S L → H S M making the following diagram commute.
We examine the relative strengths of h being closed versus H S h being closed. In fact the former condition is stronger; to ensure that h is closed, the right adjoint of H S h must send principal S-ideals to principal ones as we show below. Proof (⇒) Suppose that h is closed. Since H S h is a frame map, it automatically has a right adjoint R; we now give an explicit description of R. For J ∈ H S M, define R(J ) to be the S-ideal of L generated by {r ( j) : j ∈ J } where r is the right adjoint that must exist for h. We check that for all (⇐) By assumption, if m ∈ M, then there is a (necessarily unique) a ∈ L such that R(↓ m) =↓ a. We define r (m) = a. We check then that r is the right adjoint of h; for x ∈ L, m ∈ M:  Of course a partial frame need not have all Heyting arrows. However openness of S-frame maps is related to the preservation of Heyting arrows in the free frame as considered in the next proposition.

Note 7.4
Let h : L → M be an S-frame map. If H S h is skeletal then h is skeletal: By Proposition 6.1 (5), ↓: L → H S L is always skeletal. Since a composite of skeletal maps is skeletal, ↓ h : L → H S M is skeletal. By Lemma 5.6, h is skeletal. We do not know if the converse holds.

Points of Partial Frames
The classical adjunction between frames and topological spaces is given by an open set functor and a spectrum functor. The latter can be described using completely prime filters or, equivalently, frame maps to the 2-chain. These are usually referred to as the "points" of the frame; we call them frame points below.
The analogous situation for S-frames was presented in [13], where the definition below appeared. Definition 8.1 An S-point of an S-frame L is an S-frame map h : L → 2 where 2 is the two-element S-frame.
We turn our attention to closed and open S-points. Since points are obviously onto maps, Theorem 4.5 applies. We note that, given an S-point h : L → 2, this extends quite naturally to a frame point h : H S L → 2 such thath ↓= h. This in turn extends naturally to a frame pointh : C S L → 2 withhe =h. The existence ofh andh follows from the universal properties of H S L and C S L. This is made clearer in the following diagram in which all triangles commute:

Theorem 8.3
Let h : L → 2 be an S-point of an S-frame L. Using the notation above, we have: (a) If h is closed, thenh is closed, but not conversely.
(b) Ifh is closed thenh is closed, but not conversely.
Proof (a) We note that, in the following diagram,h and H S h amount to the same map. So h closed impliesh closed by Proposition 7.1. • ∇ J = ∇ because J =↓ 1.