A special class of congruences on $\kappa$-frames

TL;DR

This paper explores special congruences on $ppa$-frames, characterizing d-reduced $ppa$-frames via Boolean embeddings and introducing the notion of clear congruences to analyze their structure.

Contribution

It introduces the concept of clear congruences and characterizes d-reduced $ppa$-frames as quotients by these congruences, extending known Boolean frame results.

Findings

D-reduced $ppa$-frames are quotients by clear congruences.

Every $ppa$-frame congruence is a meet of clear congruences.

Characterization of d-reduced $ppa$-frames via Boolean embeddings.

Abstract

Madden has shown that in contrast to the situation with frames, the smallest dense quotient of a $\kappa$-frame need not be Boolean. We characterise these so-called d-reduced $\kappa$-frames as those which may be embedded as a generating sub-$\kappa$-frame of a Boolean frame. We introduce the notion of the closure of a $\kappa$-frame congruence and call a congruence clear if it is the largest congruence with a given closure. These ideas are used to prove $\kappa$-frame analogues of known results concerning Boolean frame quotients. In particular, we show that d-reduced $\kappa$-frames are precisely the quotients of $\kappa$-frames by clear congruences and that every $\kappa$-frame congruence is the meet of clear congruences.

Equations

$${\nabla }_I\vee C=\{(x,y)\mid (x\vee i,y\vee i)\in C\text{ for some i∈I}\}.$$

$${\mathfrak{D}}_L=\{(a,b)\mid a\wedge x=0⟺b\wedge x=0\text{ for all x∈L}\}.$$

$${\partial }_I=\{(a,b)\mid a\wedge x\in I⟺b\wedge x\in I\text{ for all x∈L}\}.$$

$${\partial }_I=\{(a,b)\mid a\wedge x{\sim }_{C}0⟺b\wedge x{\sim }_{C}0\text{ for all x∈L}\}.$$

Full text

I acknowledge financial assistance from the National Research Foundation of South Africa.

1. Background

For background on frames and $\kappa$-frames see [4] and [3]. A frame is a complete lattice satisfying the distributivity law $x\wedge\bigvee_{\alpha\in I}x_{\alpha}=\bigvee_{\alpha\in I}(x\wedge x_{\alpha})$ for arbitrary families $(x_{\alpha})_{\alpha\in I}$. We denote the smallest element of a frame by [math] and the largest element by $1$. A frame homomorphism is a function which preserves finite meets and arbitrary joins.

We use $\kappa$ to denote a fixed regular cardinal. A $\kappa$-set is then a set of cardinality strictly less than $\kappa$. A $\kappa$-frame is a bounded distributive lattice which has joins of $\kappa$-sets and satisfies the frame distributivity law when $|I|<\kappa$. Morphisms of $\kappa$-frames are functions which preserve finite meets and $\kappa$-joins (i.e. joins of $\kappa$-sets). For $\kappa=\aleph_{1}$ we obtain $\sigma$-frames as a special case and for $\kappa=\aleph_{0}$ we recover bounded distributive lattices. Results we prove about $\kappa$-frames will also hold for frames after making the obvious changes.

Recall that a poset $P$ is said to be $\kappa$-directed if every $\kappa$-subset $S\subseteq P$ has an upper bound in $P$. A $\kappa$-ideal on a $\kappa$-frame is a $\kappa$-directed downset. The set of $\kappa$-ideals $\mathfrak{H}_{\kappa}L$ on a $\kappa$-frame $L$ yields a frame when ordered by inclusion. This is the free frame generated by $L$.

A congruence on a $\kappa$-frame $L$ is an equivalence relation on $L$ that is also a sub-$\kappa$-frame of $L\times L$. The following lemma is well-known.

Lemma 1.1**.**

Let $L$ be a $\kappa$-frame and let $C$ be a congruence on $L$. Then $(a,b)\in C$ if and only if there are $x,y\in L$ with $x\leq a,b\leq y$ such that $(x,y)\in C$.

The congruence generated by $(0,a)$ for some $a\in L$ is called principal closed and denoted by $\nabla_{a}$. The congruence generated by $(a,1)$ is called open and denoted by $\Delta_{a}$. We can explicitly compute $\nabla_{a}=\{(x,y)\mid x\vee a=y\vee a\}$ and $\Delta_{a}=\{(x,y)\mid x\wedge a=y\wedge a\}$. The join of one of these congruences with an arbitrary congruence $C$ can similarly be found to give $\nabla_{a}\vee C=\{(x,y)\mid(x\vee a,y\vee a)\in C\}$ and $\Delta_{a}\vee C=\{(x,y)\mid(x\wedge a,y\wedge a)\in C\}$.

The lattice $\mathbb{C}L$ of all congruences on a $\kappa$-frame $L$ is a frame and the assignment $a\mapsto\nabla_{a}$ is an injective $\kappa$-frame homomorphism. The congruences $\nabla_{a}$ and $\Delta_{a}$ are complements of each other and the principal closed and open congruences together generate $\mathbb{C}L$.

2. Closure

For frames, principal closed congruences are closed under arbitrary joins, but in the case of $\kappa$-frames, they are only closed under joins of cardinality less than $\kappa$. Nevertheless, it is useful to consider arbitrary joins of principal closed congruences.

Definition 2.1**.**

A congruence on a $\kappa$-frame is called closed if it is a join of principal closed congruences. If $I$ is a $\kappa$-ideal of a $\kappa$-frame, we define the closed congruence $\nabla_{I}=\bigvee_{a\in I}\nabla_{a}$.

We can use closure under $\kappa$-joins to reduce any join of principal closed congruences to a $\kappa$-directed join. Also note that if a principal closed congruence $\nabla_{a}$ is included in such a join, we might as well include any smaller principal closed congruences. But a $\kappa$-directed lower set is nothing but a $\kappa$-ideal. So every closed congruence is of the form $\nabla_{I}$ for some $\kappa$-ideal $I$ on $L$.

We can compute joins with general closed congruences in a similar manner to as with principal closed congruences.

Lemma 2.2**.**

If $L$ is a $\kappa$-frame, $C\in\mathbb{C}L$ and $I$ is a $\kappa$-ideal on $L$, then

[TABLE]

In particular, \nabla_{I}=\{(x,y)\mid x\vee i=y\vee i\text{ for some i\in I}\}.

Lemma 2.3**.**

The map $\nabla_{\bullet}\colon\mathfrak{H}_{\kappa}L\to\mathbb{C}L$ is an injective frame homomorphism.

The composition of $\nabla_{\bullet}$ with its right adjoint gives a map $c\ell=\nabla_{\bullet}\circ(\nabla_{\bullet})_{*}$, which is of some interest. The map $c\ell\colon\mathbb{C}L\to\mathbb{C}L$ assigns to a congruence the largest closed congruence lying below it. More explicitly we have $c\ell(C)=\nabla_{[0]_{C}}$ where the $\kappa$-ideal $[0]_{C}$ is the equivalence class of [math] in $C$. The map $c\ell$ is an analogue of the topological closure operation and, remembering that the lattice of congruences is ordered in the reverse order to the lattice of sublocales (the lattice of regular quotients [4]), we expect it to be monotone, deflationary and idempotent and to preserve finite meets. This is easily seen to be the case.

Definition 2.4**.**

If $C$ is a congruence, $c\ell(C)$ is called the closure of $C$.

We recall the correspondence between quotients of $L/C$ and quotients of $L$ by congruences lying above $C$. The canonical isomorphism $(L/C)/\nabla_{[a]}\cong L/(C\vee\nabla_{a})$ implies that this correspondence respects (principal) closed congruences. This results in the following lemma.

Lemma 2.5**.**

Suppose $A\geq C\in\mathbb{C}L$. The closure of the image of $A$ in $\mathbb{C}(L/C)$ corresponds to $C\vee c\ell(A)$ in $\mathbb{C}L$.

3. Clear congruences

Definition 3.1**.**

Suppose $C,D\in\mathbb{C}L$ and $C\leq D$. We say $D$ is dense in $C$ if $c\ell(D)\leq C$. We say $D$ is dense if it is dense in [math].

This definition is quickly seen to be compatible with the notion of dense maps.

Lemma 3.2**.**

A congruence $D\in\mathbb{C}L$ is dense in $C$ (where $C\leq D$) if and only if the canonical map $h\colon L/C\to L/D$ is dense.

Proof.

Let $q_{A}\colon L\twoheadrightarrow L/A$ denote the quotient map associated with a congruence $A$. Now $h$ is dense if and only if $q_{D}(a)=0\implies q_{C}(a)=0$. But $q_{A}(a)=0\iff\nabla_{a}\leq A$ and so $h$ is dense if and only if $\nabla_{a}\leq D\implies\nabla_{a}\leq C$, which is to say that $c\ell(D)\leq C$. ∎

Madden shows in [3] that a $\kappa$-frame $L$ has a largest dense congruence given by

[TABLE]

Now by applying lemma 2.5 we obtain the following.

Lemma 3.3**.**

For every $\kappa$-ideal $I$ on $L$, there is a largest congruence $\partial_{I}$ dense in $\nabla_{I}$ corresponding to $\mathfrak{D}_{L/\nabla_{I}}$ in $\mathbb{C}(L/\nabla_{I})$. More explicitly we find

[TABLE]

Corollary 3.4**.**

$\partial_{I}=\{(a,b)\mid\mathord{\downarrow}a\mathbin{\rightarrow}I=\mathord{\downarrow}b\mathbin{\rightarrow}I\}$* where ‘$\mathbin{\rightarrow}\!$’ denotes the Heyting arrow in $\mathfrak{H}_{\kappa}I$.*

If $C$ is a congruence and $\nabla_{I}=c\ell(C)$, then $\partial_{I}$ is the largest congruence dense in $C$ and we have $\nabla_{I}\leq C\leq\partial_{I}$. We may compute this $\partial_{I}$ explicitly as

[TABLE]

The congruences of the form $\partial_{I}$ are an important class of congruences. In the case of frames, their associated nuclei (those of the form $w_{a}(x)=(x\mathbin{\rightarrow}a)\mathbin{\rightarrow}a$) and sublocales (Boolean sublocales) have been studied in [1], [2] and elsewhere, but the congruence-oriented approach remains largely unexplored. There has consequently been little discussion in the setting of $\kappa$-frames where nuclei and sublocales are unavailable. Some related ideas and results appear in [3], but the congruences $\partial_{I}$ narrowly escape mention. Since we might imagine the smallest dense sublocale of a frame to be so rarefied as to appear translucent, we will call these clear congruences.

Definition 3.5**.**

A clear congruence is a congruence of the form $\partial_{I}$.

The assignment $I\mapsto\partial_{I}$ is injective and reflects order, but the following example shows it is not monotone.

Example 3.6**.**

The chain $\mathbbold{3}=\{0,a,1\}$ is the frame of opens of the Sierpiński space. The clear congruences $\partial_{0}$ and $\partial_{\mathord{\downarrow}a}$ correspond to the open and closed points respectively, so $\partial_{0}$ and $\partial_{\mathord{\downarrow}a}$ are incomparable.

The clear congruences generate the congruence lattice under meets.

Theorem 3.7**.**

Every congruence is a meet of clear congruences.

Proof.

Take $C\in\mathbb{C}L$. Certainly, $C\leq\bigwedge\{\partial_{I}\mid\partial_{I}\geq C,\,I\in\mathfrak{H}_{\kappa}L\}$. Now suppose $(a,b)\notin C$. We may assume $a<b$ and we need a $\kappa$-ideal $I$ such that $\partial_{I}\geq C$ and $(a,b)\notin\partial_{I}$.

Let $\nabla_{I}=c\ell(\nabla_{a}\vee C)$. We have $\nabla_{I}\leq\nabla_{a}\vee C\leq\partial_{I}$ and thus $\partial_{I}\geq C$. Also, $\nabla_{a}\leq\nabla_{I}$ and so $a\in I$. (Indeed, one may note that $I=\mathord{\downarrow}[a]_{C}$.)

Suppose that $b\in I$ as well. Then $(a,b)\in\nabla_{I}\leq\nabla_{a}\vee C$. So $(a,b)=(a\vee a,a\vee b)\in C$. This is a contradiction and so $b\notin I$. But then $(a,b)\notin\partial_{I}$ by lemma 3.3. ∎

Quotients of a $\kappa$-frame by clear congruences have a special form.

Definition 3.8**.**

A $\kappa$-frame $L$ is called d-reduced [3] if $\mathfrak{D}_{L}=0$ in $\mathbb{C}L$.

Lemma 3.9**.**

The quotient $L/C$ is d-reduced if and only if the congruence $C$ is clear.

Proof.

By lemma 2.5, $L/C$ is d-reduced if and only if $C$ is the largest congruence $D$ for which $c\ell(D)\vee C\leq C$. But $c\ell(D)\vee C\leq C\iff c\ell(D)\leq C\iff c\ell(D)\leq c\ell(C)$ and $C$ is the largest such $D$ if and only if $C$ is clear. ∎

Lemma 3.10**.**

A $\kappa$-frame $L$ is d-reduced if and only if no two distinct principal ideals on $L$ have the same pseudocomplement in $\mathfrak{H}_{\kappa}L$.

Proof.

Simply note that $\partial_{0}=\{(a,b)\mid(\mathord{\downarrow}a)^{*}=(\mathord{\downarrow}b)^{*}\}$ by corollary 3.4. ∎

As a corollary we obtain a known result about frames.

Corollary 3.11**.**

A frame is d-reduced if and only if it is Boolean.

Proof.

Suppose $L$ is a d-reduced frame. The frame $\mathfrak{H}_{\infty}L$ of principal ideals on $L$ is isomorphic to $L$ itself and so every element of $L$ has a unique pseudocomplement by lemma 3.10. But for $a\in L$, we have $(a\vee a^{*})^{*}=a^{*}\wedge a^{**}=0$ and so $a\vee a^{*}=1$. Thus, $a$ is complemented and $L$ is Boolean.

Conversely, if $L$ is Boolean, then every element of $L$ is complemented and so pseudocomplements are clearly unique. ∎

We are now in a position to prove the following characterisation of d-reduced $\kappa$-frames.

Theorem 3.12**.**

Let $L$ be a $\kappa$-frame. The following are equivalent:

1. (1)

$L$* is d-reduced* 2. (2)

The map $g\colon L\hookrightarrow\mathfrak{H}_{\kappa}L\twoheadrightarrow\mathfrak{H}_{\kappa}L/\mathfrak{D}_{\mathfrak{H}_{\kappa}L}$ sending $a$ to $[\mathord{\downarrow}a]$ is injective 3. (3)

$L$* is embeds as a sub-$\kappa$-frame of a Boolean frame $B$ and this sub-$\kappa$-frame generates $B$ under arbitrary joins.*

Proof.

($1\Rightarrow 2$) Suppose $L$ is d-reduced. Take $x,y\in L$. By corollary 3.4, $[\mathord{\downarrow}x]=[\mathord{\downarrow}y]$ in $\mathfrak{H}_{\kappa}L/\mathfrak{D}_{\mathfrak{H}_{\kappa}L}$ if and only if $(\mathord{\downarrow}x)^{*}=(\mathord{\downarrow}y)^{*}$. But by lemma 3.10, $(\mathord{\downarrow}x)^{*}$ and $(\mathord{\downarrow}y)^{*}$ are distinct if $x$ and $y$ are. Thus $g\colon a\mapsto[\mathord{\downarrow}a]$ is injective.

($2\Rightarrow 3$) The frame $\mathfrak{H}_{\kappa}L/\mathfrak{D}_{\mathfrak{H}_{\kappa}L}$ is Boolean by corollary 3.11 and is generated by the image of $L$ under the map $g$ since $\mathfrak{H}_{\kappa}L$ is generated by the image of $L$ and the quotient map is surjective.

($3\Rightarrow 1$) We now assume $L$ is a sub-$\kappa$-frame of a complete Boolean algebra $B$ and that $L$ generates $B$ under arbitrary joins. Let $a\in L$ and consider its complement $a{{}^{\mathrm{c}}}\in B$. Suppose $b\in L$ such that $b\leq a{{}^{\mathrm{c}}}$ in $B$. Then $b\wedge a=0$ and so $b\in(\mathord{\downarrow}a)^{*}\in\mathfrak{H}_{\kappa}L$. Conversely, if $b\in(\mathord{\downarrow}a)^{*}$ then $b\wedge a=0$ and so $b\leq a{{}^{\mathrm{c}}}$. Thus $(\mathord{\downarrow}a)^{*}=\;\mathord{\downarrow}a{{}^{\mathrm{c}}}\cap L$. Since $L$ generates $B$, $a{{}^{\mathrm{c}}}=\bigvee(\mathord{\downarrow}a)^{*}$ in $B$. So for any $a,b\in L$, $(\mathord{\downarrow}a)^{*}=(\mathord{\downarrow}b)^{*}$ only if $a{{}^{\mathrm{c}}}=b{{}^{\mathrm{c}}}$ only if $a=b$. Thus, $L$ is d-reduced. ∎

Example 3.13**.**

Let $M$ be the lattice of subsets of $\mathbb{N}$ that are either finite or equal to $\mathbb{N}$. Since $M$ is a generating sublattice of $2^{\mathbb{N}}$, it is a d-reduced distributive lattice by theorem 3.12. Now consider the ideal $I$ of $M$ consisting of the sets that do not contain the natural number $2$. This is the pseudocomplement of the principal ideal generated by $\{2\}$. The quotient lattice $M/\nabla_{I}$ consists of three elements, namely $[\emptyset]$, $[\{2\}]$ and $[\mathbb{N}]$, and is isomorphic to the 3-element frame $\mathbbold{3}$. Thus, quotients of d-reduced $\kappa$-frames may fail to be d-reduced. It would be interesting to know exactly which $\kappa$-frames can appear as quotients of d-reduced ones, but we do not attempt to answer this here.

Lemma 3.14**.**

Every quotient of a $\kappa$-frame $L$ is d-reduced if and only if $L$ is Boolean.

Proof.

$(\Leftarrow)$ A quotient of a Boolean $\kappa$-frame is Boolean and Boolean $\kappa$-frames are d-reduced.

$(\Rightarrow)$ Let $a\in L$. We will show that $a$ has a complement. Consider $\Delta_{a}\in\mathbb{C}L$ and let $\nabla_{I}=c\ell(\Delta_{a})$. But $\nabla_{I}$ is clear, so $\nabla_{I}=\partial_{I}$ and thus $\nabla_{I}=\Delta_{a}$.

Now $\nabla_{a}\vee\nabla_{I}=1$ and so $(0,1)\in\nabla_{a}\vee\nabla_{I}$. So $(a,1)\in\nabla_{I}$ and $a\vee i=1$ for some $i\in I$. But then $\nabla_{a}\vee\nabla_{i}=1$ and also $\nabla_{a}\wedge\nabla_{i}=0$ since $\nabla_{i}\leq\nabla_{I}$. Thus $i$ is the complement of $a$ in $L$ and $L$ is Boolean. ∎

While arbitrary closed quotients of d-reduced $\kappa$-frames typically fail to be d-reduced, quotients by principal closed congruences always are.

Lemma 3.15**.**

If $L$ is d-reduced, then $\nabla_{a}$ is clear for all $a\in L$.

Proof.

Embed $L$ as a generating sub-$\kappa$-frame of a Boolean frame $B$. Since $L/\nabla_{a}\cong\mathord{\uparrow}a\subseteq L$, this embedding induces an embedding of $L/\nabla_{a}$ as a generating sub-$\kappa$-frame of $B/\nabla_{a}$ and hence $L/\nabla_{a}$ is d-reduced. ∎

Lemma 3.16**.**

In a $\kappa$-frame $L$, $\mathfrak{D}_{L}\vee\nabla_{a}=\partial_{(\mathord{\downarrow}a)^{**}}$.

Proof.

We find that $c\ell(\mathfrak{D}_{L}\vee\nabla_{a})=\nabla_{\mathord{\downarrow}[a]_{\mathfrak{D}_{L}}}=\nabla_{(\mathord{\downarrow}a)^{**}}$. But $\mathfrak{D}_{L}\vee\nabla_{a}$ is clear by lemma 3.15 and so the result follows. ∎

We noted in example 3.6 that the map $I\mapsto\partial_{I}$ is not monotone. Nonetheless lemma 3.16 allows us to say something in a special case.

Corollary 3.17**.**

The map $a\mapsto\partial_{(\mathord{\downarrow}a)^{**}}$ is a $\kappa$-frame homomorphism from $L$ to $\mathord{\uparrow}\mathfrak{D}_{L}\subseteq\mathbb{C}L$ with kernel $\mathfrak{D}_{L}$. In particular, it is monotone and clear congruences of the form $\partial_{(\mathord{\downarrow}a)^{**}}$ are closed under finite meets and nonempty $\kappa$-joins.

Acknowledgements

I would like to thank John Frith and Christopher Gilmour for discussions about the content and presentation of this paper.