The finitely axiomatizable complete theories of non-associative arrow frames

Arrow logic is a modal logic that is designed to talk about objects that can be illustrated as arrows. In this article, we consider the non-associative arrow logic NAL. We list all the finitely axiomatizable, complete and consistent theories of NAL. This gives an answer to the open problem, posed by I. Nemeti, addressing the atomicity of the free algebras of the class NA of non-associative relation algebras. We use the method of games as introduced to the fields of logic and algebra by R. Hirsch and I. Hodkinson. We also give a simple proof for the known fact that NAL is decidable through the finite model property. (C) 2019 Elsevier Inc. All rights reserved.