No finite axiomatizations for posets embeddable into distributive lattices

Abstract

Let m and n be cardinals with 3 ≤ m, n ≤ ω. We show that the class of posets that can be embedded into a distributive lattice via a map preserving all existing meets and joins with cardinalities strictly less than m and n respectively cannot be finitely axiomatized.

Non-elementary classes of representable posets

“…Thus the stated classes are not closed under elementary equivalence. Figure 2 below summarizes the known results collated from here, [6] and [5].…”

Closure Operators, Frames and Neatest Representations

“…Moreover, existence of a split triple can be defined in first-order logic, so if having a split triple were also a necessary condition for a poset to fail to be 3-representable, it would follow that the class of 3-representable posets is finitely axiomatisable. This is not the case [6], so there must be other, less obvious obstructions to 3-representability. This is not surprising, as even in the simpler semilattice case the existence of a split triple is not necessary for failure of 3-representability.…”

“…However, the semilattices that fail to be (ω, 3)-representable can be characterised as those that contain either a split triple or an indeterminate triple (by [2,Theorem 2.2], phrased in the terminology of triples), which is also a first-order property. Since the class of (m, n)-representable posets is elementary for all m and n with 2 < m, n ≤ ω, it follows that the class of posets that fail to be 3representable cannot be axiomatised in first-order logic at all (otherwise it would be finitely axiomatisable, in contradiction with [6]). This contrasts starkly with the intuitive finite axiomatisation of the semilattice case.…”

Recursive axiomatizations for representable posets

“…Of course, the meat of any such proof is to be found in the constructions themselves, but this can be a useful approach, where it applies. For example, this method is essentially the engine of the proofs of the titular result of [14], and the results of [23, section 5], though the work in these examples is phrased in terms of ultraproducts. Note that the argument as described here has an advantage over the originals as reasoning about properties of the ultraproduct is not required.…”

“…Proof. We have shown that the class of (α, β)-representable posets is a basic separation subclass of the class of posets whenever 2 ≤ α, β < ω, and this class is also known to not be finitely axiomatizable for α, β ≥ 3 [14].…”