Ivanova contact join-semilattices are not finitely axiomatizable

TL;DR

The paper investigates whether Ivanova contact join-semilattices admit a finite first-order axiomatization. It proves non-finite axiomatizability for the Ivanova class by constructing, for each $n\ge 2$, counterexamples that satisfy $(D1)$ and $(D2_m)$ for $m<n$ but violate $(D2_n)$, using a carefully built Boolean-algebra-based framework and compactness arguments. In contrast, it shows that the non-overlap (weak) variant admits a simple finite axiomatization via $(D1)$ (and equivalent conditions), with several embeddability characterizations into weak contact Boolean algebras or distributive lattices; some representation results hinge on the Boolean Prime Ideal Theorem. A final remark notes that the overlap-subclass is also finitely axiomatizable, while natural generalizations to hypercontact fail to be finitely axiomatizable, highlighting the nuanced role of the contact relation’s arity.

Abstract

We show that the class of Contact join-semilattices, as introduced by T. Ivanova, is not finitely axiomatizable. On the other hand, a simple finite axiomatization exists for the class of those join semilattices with a weak contact relation which can be embedded into the reduct of a Boolean algebra (equivalently, a distributive lattice) with a weak contact relation.

Theorems & Definitions (23)

• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Claim 3.2
• Corollary 3.3
• Theorem 3.4
• proof
• Lemma 3.5