Some logical invariants of algebras and logical relations between algebras

Abstract

Abstract. Let Θ be an arbitrary variety of algebras and H an algebra in Θ. Along with algebraic geometry in Θ over the distinguished algebra H, a logical geometry in Θ over H is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras H ∈ Θ are introduced and logical relations between different H 1 and H 2 in Θ are analyzed. The paper contains a brief review of ideas of logical geometry ( §1), the necessary material from algebraic logic ( §2), and a deeper introduction to the subject ( §3). Also, a list of problems is given. 0.1. Introduction. The paper consists of three sections. A reader wishing to get a feeling of the subject and to understand the logic of the main ideas can confine himself to §1. A more advanced look at the topic of the paper is presented in § §2 and 3.In §1 we give a list of the main notions, formulate some results, and specify problems. Not all the notions used in §1 are well formalized and commonly known. In particular, we operate with algebraic logic, referring to §2 for precise definitions. However, §1 is self-contained from the viewpoint of ideas of universal algebraic geometry and logical geometry.Old and new notions from algebraic logic are collected in §2. Here we define the Halmos categories and multisorted Halmos algebras related to a variety Θ of algebras.§3 is a continuation of §1. Here we give necessary proofs and discuss problems. The main problem we are interested in is what are the algebras with the same geometrical logic.The theory described in the paper has deep ties with model theory, and some problems are of a model-theoretic nature.We emphasize once again that §1 gives a complete insight on the subject, while §2 and §3 describe and decode the material of §1. §1. Preliminaries. General view 1.1. Main idea. We fix an arbitrary variety Θ of algebras. Throughout the paper we consider algebras H in Θ. To each algebra H ∈ Θ one can attach an algebraic geometry (AG) in Θ over H and a logical geometry (LG) in Θ over H.In algebraic geometry we consider algebraic sets over H, while in logical geometry we consider logical (elementary) sets over H. These latter sets are related to the elementary logic, i.e., to the first order logic (FOL).Consideration of these sets gives grounds to geometries in an arbitrary variety of algebras. We distinguish algebraic and logical geometries in Θ. However, there is very 2000 Mathematics Subject Classification. Primary 03G25.

Multi-Sorted Logic and Logical Geometry: Some Problems

“…where the automorphism of categories ϕ : Φ Θ → Φ Θ is coordinated with the lattice structures of closed filters by additional conditions (see [33]).…”

Typical equivalence of linear groups and other algebraic systems

“…The full first-order theory of the algebraic structure H is the set Th(H) of closed first-order formulas over Ω that hold in H. Algebraic structures H 1 and H 2 are called elementarily equivalent when Th(H 1 ) and Th(H 2 ) coincide. Now we introduce the notion of typical equivalence (see [4], where it is called "logical equivalence"). Here and further we suppose that the only predicate symbol in Ω is the equality sign, which is always interpreted as coincidence of two elements.…”

“…The precise definition of the algebra ϕ = ϕ(X) is given in [4]. In fact, ϕ(X) is a set of first-order formulas over X which is converted in a special way into an algebra of formulas.…”

“…Now we just mention that ϕ(X) is a Boolean algebra with equalities of the form w ≡ w and, moreover, with acting quantifiers ∃ x for all x ∈ X. We call such algebras extended Boolean algebras (see [4]). …”

New algebraic invariants for definable subsets in universal algebra

“…In [9,10], a similar idea was implemented for subsets definable by first-order formulas in universal algebras (logical geometry of universal algebras). There, instead of homomorphisms of finitely generated free algebras into the algebras in question, use was made of homomorphisms of multibase Boolean algebras (Lindenbaum algebras) into Boolean algebras of subsets of degrees in the universe of a certain algebra.…”