GALAI Seminar

The Finite Embeddability Property states that each algebra in a given class locally embeds, in a precise sense, into a finite algebra in that class. The FEP is perhaps the main algebraic tool for establishing the decidability of the universal theory of some class of algebras. A pair of fundamental results of Galatos and Jipsen from 2013 and 2017, whose proofs rely on the machinery of residuated frames, states that any variety of integral residuated lattices axiomatized by equations in the signature { v, *, 1 } has the FEP, and the same holds for varieties of distributive integral residuated lattices. We provide a clean universal algebraic perspective which unifies and extends these results.

Betweenness relations – well known from geometry – are one of the most thoroughly studied ternary relations in logic and mathematics. In [1] we initiated a theory of betweenness algebras as an approach to study the general algebraic and logical properties of the ternary relation of betweenness and its associated complex algebras. Our overall framework is a logic K^# with two binary modalities which in some sense complement each other. This logic and its corresponding algebraic models – Boolean algebras augmented with two binary modal operators with a connecting condition – can be seen as a binary extension of the unary mixed logic of [3], significantly extending the expressive power of classical modal logic.

In the presentation I will introduce betweenness relations and the logic K^#, and will report some of their properties as developed in [1, 2].

References

[1] Düntsch, I., Gruszczyński, R., and Menchón, P. (2026). Betweenness algebras. J. Symb. Log., 91(2):574–598. MR5081272.

[2] Düntsch, I., Gruszczyński, R., and Menchón, P. (2026). A mixed logic with binary operators. J. Symb. Log. Published online 2026:1-24.

[3] Düntsch, I., Orłowska, E., and Tinchev, T. (2017). Mixed algebras and their logics. J. Appl. Non-Class. Log., 27(3-4):304–320. MR3779213.

Relevant logics are those for which, roughly, it is not possible to go from some collection of premises to some unrelated conclusions. Robert Meyer in the 1970's and in later work by Weber, Slaney, and others studied various ways in which we might study mathematics or fragments thereof using a relevant logic, but the arithmetic and set theory studied there are classically very complicated theories. The theory of vector spaces over the rational numbers is classically much simpler. Specifically, it enjoys such properties as quantifier elimination, strong minimality, ω-stability, and they have countably many countable models. Adapting a proof of Badia and Tedder on the failure of quantifier elimination for algebraically closed fields over any logic L between B and RM, we show an analogous result for rational vector spaces. The proof also shows the existence of a non-minimal model, by giving us a definable subset which is neither finite nor cofinite in any sense of the word. Furthermore, up to a notion of isomorphism of these structures, there are continuum many countable models. A similar trick lets us embed a non-trivial linear ordering, showing that these are unstable, even over RM.

Our aim is to introduce a framework sufficiently general to describe the kinematics of a wide variety of open systems in classical mechanical physics while uniquely characterizing systems with specified simplest components. The data describing a physical system are local, so the construction of a global configuration space requires compatibility among local interactions. We model open systems as morphisms in a category Kin(F), where composition encodes how subsystems attach to one another and embed as open subsystems of larger systems. The framework supports a precise treatment of geometric constraints and clarifies when locally specified subsystems are compatible. It also yields a structural approach to the study of linkages: we prove the non-constructibility of sliding hinges and universal joints from two rigid bodies attached by a surface constraint compatible with rigid-motion symmetries.