Multi-Sorted Logic and Logical Geometry: Some Problems

Abstract

Abstract. The paper has a form of a survey on basics of logical geometry and consists of three parts. It is focused on the relationship between many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appears. A list of problems which naturally arise in this field is another objective of the paper.

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

“…All algebraic sets constitute a category with special rational maps as morphisms [65]. The same is true with respect to definable sets [65]. So, we can formulate logical closeness of algebras geometrically.…”

“…This is our first syntactic object. Next, letΦ = (Φ(X), X ∈ Γ) be the multi-sorted Halmos algebra of first order logical formulas based on atoms w ≡ w , w, w in W(X), see References [17,65,66]. There is a special procedure to construct such an algebraic object which plays the same role with respect to First Order Logic as Boolean algebras do with respect to Propositional calculus.…”

“…Remark 1. The principal role in all considerations plays the value homomorphism Val :Φ → Hal Θ , where Hal Θ is a special Halmos algebra associated with the vector space Hom(W(X), H), see References [65,66]. Its meaning is to make the procedure of verification whether a point satisfies the formula a homomorphism.…”

“…We call the condition A rigid (or A-rigid) if two algebras H 1 and H 2 subject to A are isomorphic. AG-equivalent algebras are called also geometrically equivalent algebras, see References [3,64,65]. The closure T H is called, sometimes, a radical of T with respect to H. This is a normal subgroup and an ideal in cases of groups and associative (Lie) algebras, respectively.…”

Action of endomorphism semigroups on definable sets

“…The classical syntactical-semantical connection in Model Theory is given by correspondence between sets of formulas and the associated definable sets. In the series of papers (see, for example, [38], [42], [39]) this connection takes the form of a Galois correspondence between elements of syntactical algebra Φ(X) and elements of semantical algebra Hal X Θ (A) (see [43] for the detailed explanations and notations). In particular this means that every notion related to sets of formulas can be reformulated via definable sets and vice versa.…”