On the variety generated by completions of representable relation algebras

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 = HSP RRA c , where RRA is the variety of representable relation algebras and RRA c = {A c : A ∈ RRA}. Here, A c denotes the completion of the relation algebra 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. and he showed that V contains a number of non-representable 'Monk algebras', so that the gap between RRA and V is substantial. In [9, problem 1.1(1)], he asked whether V = 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 RA. That might suggest that V is 'nearer' to RRA than to RA, but as 'evidence' in the other direction, we show below that RRA is not finitely axiomatisable over V .

RRA is not finitely axiomatisable over V
It suffices to show that RRA contains an ultraproduct of algebras in V \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 is an edge of G for each i < l. A subset X ⊆ N is said to be independent if no pair of nodes in X is an edge of G. The chromatic number χ(G) of G is the least natural number n such that N is the union of n (possibly empty) independent sets, and ∞ if there is no such n. It is well known (see, e.g., [2, 1.6.1]) that χ(G) ≤ 2 iff G has no cycles of odd length. We let + and denote disjoint union of graphs.
Let n be a positive integer. Let K n be a complete graph with precisely n nodes. Clearly, χ(K n ) = n. Let E n be a graph with χ(E n ) ≥ 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 ω ), and compute their chromatic numbers. First define First consider the case when k = 0. Observe that G 0 n in (2.1) has no cycles of length ≤ n. Now for each l ≥ 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 Loś'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 χ(G 0 ) ≤ 2.
Vol. 81 (2020) On the variety generated by completions Page 3 of 5 10 Finally let For each m > 0, K m embeds into G k for every k ≥ m. It follows by Loś's theorem that K m embeds into G ω , so plainly, χ(G ω ) ≥ m. This holds for every m, so χ(G ω ) = ∞.

Relation algebras from graphs
Let G be an infinite graph with set of nodes N . We write N × 3 for the set N × {0, 1, 2}, and G × 3 for the graph whose set of nodes is N × 3 and where w) is an edge of G. In simple words, G×3 consists of three disjoint copies of G, with all possible edges added between the copies. We now define a relation algebra atom structure α(G) = (A, C,˘, I) isomorphic to one in [6, section 4] and [5, chapter 14]. We stipulate that , },x = x for every x ∈ A, and for each x, y, z ∈ A, C(x, y, z) holds iff (1) one of x, y, z is 1 , and the other two are equal, or (2) {x, y, z} ⊆ N × 3 and {x, y, z} is not independent (in the graph G × 3).

Why RRA is not finitely axiomatisable over
Then A k ∈ RRA, since RRA is a variety and closed under ultraproducts. So by definition of V , By Loś's theorem, A k is atomic. So (e.g., [5, remark 2.67]) its completion C k is isomorphic to the complex algebra of the atom structure of A k . This atom structure is D α(G k n ) ∼ = α( D G k n ) = α(G k ) by (2.6, 2.5, 2.3). Hence, C k ∼ = A(G k ). We saw that χ(G k ) < ∞, so by fact 2.1, C k / ∈ RRA.
Finally let C be the ultraproduct D {C k : 0 < k < ω}. As before, this is an atomic relation algebra with atom structure isomorphic to D α(G k ) ∼ = α( D G k ) = α(G ω ), so its completion C c is isomorphic to A(G ω ). But χ(G ω ) = ∞, so A(G ω ) ∈ RRA by fact 2.1. Then C ⊆ C c ∈ RRA, and as RRA is closed under subalgebras, we obtain C ∈ RRA.
We can now prove our main theorem.
Theorem 2.2. RRA is not finitely axiomatisable over V .
Proof. We have shown that C k ∈ V \RRA for k > 0, and C = D C k ∈ RRA. It follows by Loś's theorem that RRA is not finitely axiomatisable over V .