## Abstract

Maddux recently defined the variety *V* generated by the completions of representable relation algebras. In this note, we observe that *V* is canonical, answering Maddux’s problem 1.1(3), and show that the variety of representable relation algebras is not finitely axiomatisable over *V*.

## Introduction

In a recent paper [9, 1.10], Maddux defined the variety \(V=\mathbf{H}\mathbf{S}\mathbf{P}\,\mathsf{RRA}^c\), where \(\mathsf{RRA}\) is the variety of representable relation algebras and \(\mathsf{RRA}^c=\{\mathfrak {A}^c:\mathfrak {A}\in \mathsf{RRA}\}\). Here, \(\mathfrak {A}^c\) denotes the completion of the relation algebra \(\mathfrak {A}\). For details of these notions and further ones used below, see the very attractive introduction to [9].

In [9, problem 1.1(3)], Maddux asked whether *V* is closed under canonical extensions.

### Theorem 1.1

*V* and \(\mathbf{S}\,\mathsf{RRA}^c\) are closed under canonical extensions.

### Proof

By [4, theorem 3.8], if \(\mathsf K\) is a class of relation algebras that is closed under ultraproducts, and \(\mathsf{K}^c=\{\mathfrak {A}^c:\mathfrak {A}\in \mathsf{K}\}\), then \(\mathbf{S}\,\mathsf{K}^c\) and \(\mathbf{H}\mathbf{S}\mathbf{P}\,\mathsf{K}^c\) are both closed under canonical extensions. Theorem 1.1 follows by taking \(\mathsf{K}=\mathsf{RRA}\), which is a variety and so closed under ultraproducts. \(\square \)

RA denotes the class of all relation algebras. Using the known facts that RA is closed under completions [10] but RRA is not [7], Maddux noted that

and he showed that *V* contains a number of non-representable ‘Monk algebras’, so that the gap between \(\mathsf{RRA}\) and *V* is substantial. In [9, problem 1.1(1)], he asked whether \(V=\mathsf{RA}\). This was answered negatively by Andréka and Németi [1], where it is shown that in fact there are continuum-many varieties lying between *V* and \(\mathsf{RA}\). That might suggest that *V* is ‘nearer’ to \(\mathsf{RRA}\) than to \(\mathsf{RA}\), but as ‘evidence’ in the other direction, we show below that \(\mathsf{RRA}\) is not finitely axiomatisable over *V*.

##
\(\mathsf{RRA}\) is not finitely axiomatisable over *V*

It suffices to show that RRA contains an ultraproduct of algebras in \(V{\setminus }\mathsf{RRA}\). To this end, we use a construction from [6] of relation algebras from *graphs.*

### Graphs

Graphs here are undirected and loop-free. Let *G* be a graph whose set of nodes is *N*, say. Recall that a *cycle of length*\(l\ge 3\)*in**G* is a subset \(\{v_0,\dots {},v_{l-1}\}\subseteq N\) of size *l* such that \((v_i,v_{(i+1)\bmod l})\) is an edge of *G* for each \(i<l\). A subset \(X\subseteq N\) is said to be *independent* if no pair of nodes in *X* is an edge of *G*. The *chromatic number*\(\chi (G)\) of *G* is the least natural number *n* such that *N* is the union of *n* (possibly empty) independent sets, and \(\infty \) if there is no such *n*. It is well known (see, e.g., [2, 1.6.1]) that \(\chi (G)\le 2\) iff *G* has no cycles of odd length. We let \(+\) and \(\sum \) denote disjoint union of graphs. Then if \(G_i\)\((i\in I)\) are graphs, \(\chi (\sum _{i\in I}G_i)\) is the least upper bound (possibly \(\infty \)) of \(\{\chi (G_i):i\in I\}\).

Let *n* be a positive integer. Let \(K_n\) be a complete graph with precisely *n* nodes. Clearly, \(\chi (K_n)=n\). Let \(E_n\) be a graph with \(\chi (E_n)\ge n\) and with no cycles of length at most *n*—finite examples were constructed by Erdős [3].

We use these graphs to construct some infinite graphs (\(G^k_n\), \(G^k\), \(G^\omega \)), and compute their chromatic numbers. First define

Since \(\{\chi (E_m):m\ge n\}\) is unbounded, each \(G^k_n\) has chromatic number \(\infty \).

Next, fix a non-principal ultrafilter *D* over \(\omega {\setminus }1\). Define the ultraproduct

First consider the case when \(k=0\). Observe that \(G^0_n\) in (2.1) has no cycles of length\({}\le n\). Now for each \(l\ge 3\), the property of having no cycles of length *l* can be expressed by a first-order sentence, and is true for all but finitely many \(G^0_n\). So by Łoś’s theorem, \(G^0\) in (2.3) has no cycles of length *l*. This holds for each *l*, so in fact \(G^0\) has no cycles at all, and hence \(\chi (G^0)\le 2\).

What about \(\chi (G^k)\) for \(k>0\)? By standard ultraproduct considerations,

so we further obtain \(\chi (G^k)=\max (\chi (G^0),\chi (K_k))\le \max (2,k)<\infty \).

Finally let

For each \(m>0\), \(K_m\) embeds into \(G^k\) for every \(k\ge m\). It follows by Łoś’s theorem that \(K_m\) embeds into \(G^\omega \), so plainly, \(\chi (G^\omega )\ge m\). This holds for every *m*, so \(\chi (G^\omega )=\infty \).

### Relation algebras from graphs

Let *G* be an *infinite* graph with set of nodes *N*. We write \(N\times 3\) for the set \(N\times \{0,1,2\}\), and \(G\times 3\) for the graph whose set of nodes is \(N\times 3\) and where ((*v*, *i*), (*w*, *j*)) is an edge of \(G\times 3\) iff \(i\ne j\) or (*v*, *w*) is an edge of *G*. In simple words, \(G\times 3\) consists of three disjoint copies of *G*, with all possible edges added between the copies.

We now define a relation algebra atom structure isomorphic to one in [6, section 4] and [5, chapter 14]. We stipulate that \(A=(N\times 3)\cup \{1^{\mathrm{'}}\}\), \(I=\{1^{\mathrm{'}}\}\), \({\breve{x}}=x\) for every \(x\in A\), and for each \(x,y,z\in A\), *C*(*x*, *y*, *z*) holds iff

- (1)
one of

*x*,*y*,*z*is \(1^{\mathrm{'}}\) and the other two are equal, or - (2)
\(\{x,y,z\}\subseteq N\times 3\) and \(\{x,y,z\}\) is not independent (in the graph \(G\times 3\)).

One can check that for any graphs \(G_n\)\((0<n<\omega )\),

We write \(\mathfrak {A}(G)\) for the complex algebra of \(\alpha (G)\) (see [9, 1.13]). By [8, lemma 6.2], \(\mathfrak {A}(G)\) is a relation algebra; of course it is atomic and its atom structure is \(\alpha (G)\). We will need the following fact about it, from [6, theorems 10–11] or [5, theorems 14.12–13] or (for \(\Leftarrow \)) [5, exercise 14.2(7)]. It uses our assumption that *G* is infinite.

### Fact 2.1

\(\mathfrak {A}(G)\in \mathsf{RRA}\) if and only if \(\chi (G)=\infty \).

### Why \(\mathsf{RRA}\) is not finitely axiomatisable over *V*

Let \(k<\omega \). For \(0<n<\omega \), we saw that \(\chi (G^k_n)=\infty \), so \(\mathfrak {A}(G^k_n)\in \mathsf{RRA}\) by fact 2.1. Define the ultraproduct

Then \(\mathfrak {A}^k\in \mathsf{RRA}\), since \(\mathsf{RRA}\) is a variety and closed under ultraproducts. So by definition of *V*,

By Łoś’s theorem, \(\mathfrak {A}^k\) is atomic. So (e.g., [5, remark 2.67]) its completion \(\mathfrak {C}^k\) is isomorphic to the complex algebra of the atom structure of \(\mathfrak {A}^k\). This atom structure is \(\prod _D\alpha (G^k_n)\cong \alpha (\prod _DG^k_n)=\alpha (G^k)\) by (2.6, 2.5, 2.3). Hence, \(\mathfrak {C}^k\cong \mathfrak {A}(G^k)\). We saw that \(\chi (G^k)<\infty \), so by fact 2.1, \(\mathfrak {C}^k\notin \mathsf{RRA}\).

Finally let \(\mathfrak {C}\) be the ultraproduct \(\prod _D\{\mathfrak {C}^k:0<k<\omega \}\). As before, this is an atomic relation algebra with atom structure isomorphic to \(\prod _D\alpha (G^k)\cong \alpha (\prod _DG^k)=\alpha (G^\omega )\), so its completion \(\mathfrak {C}^c\) is isomorphic to \(\mathfrak {A}(G^\omega )\). But \(\chi (G^\omega )=\infty \), so \(\mathfrak {A}(G^\omega )\in \mathsf{RRA}\) by fact 2.1. Then \(\mathfrak {C}\subseteq \mathfrak {C}^c\in \mathsf{RRA}\), and as RRA is closed under subalgebras, we obtain \(\mathfrak {C}\in \mathsf{RRA}\).

We can now prove our main theorem.

### Theorem 2.2

\(\mathsf{RRA}\) is not finitely axiomatisable over *V*.

### Proof

We have shown that \(\mathfrak {C}^k\in V{\setminus }\mathsf{RRA}\) for \(k>0\), and \(\mathfrak {C}=\prod _D\mathfrak {C}^k\in \mathsf{RRA}\). It follows by Łoś’s theorem that \(\mathsf{RRA}\) is not finitely axiomatisable over *V*. \(\square \)

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## Acknowledgements

I thank Robin Hirsch and the anonymous referee for helpful comments.

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### Cite this article

Hodkinson, I. On the variety generated by completions of representable relation algebras.
*Algebra Univers.* **81**, 10 (2020). https://doi.org/10.1007/s00012-020-0643-z

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DOI: https://doi.org/10.1007/s00012-020-0643-z

### Keywords

- Non-finitely axiomatisable variety
- Canonical variety
- Graph
- Chromatic number
- Cycle

### Mathematics Subject Classification

- 03G15
- 03C05
- 05C15
- 06B23