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Interpolation and Amalgamation

George Metcalfe

TL;DR

This chapter develops a unified universal-algebra framework to connect interpolation properties of non-classical propositional logics with amalgamation properties of their algebraic semantics. Central to the approach is a bridge between equational consequence and congruences of free algebras, enabling transfer theorems among CEP, EP, AP, RP, DIP, and CIP across a wide range of logics (Heyting, Gödel, MV, FL, and modal). The work establishes key equivalences (e.g., AP ↔ RP, DIP with EP plus AP) and surveys which algebraic varieties possess these properties, including detailed treatment of FL_e-algebras, FL-algebras, and semilinear residuated lattices. It also clarifies when uniform and Craig interpolation can be derived syntactically or algebraically, and highlights instances where interpolation properties fail. Overall, the chapter provides a comprehensive toolkit for analyzing non-classical logics via their algebraic counterparts, with broad implications for both logic and universal algebra.

Abstract

This chapter presents a state-of-the-art survey of relationships, traditionally referred to as `bridges', between interpolation properties for propositional logics -- including superintuitionistic, modal, and substructural logics -- and amalgamation properties for corresponding classes of algebraic structures. These bridges are developed in the framework of universal algebra and illustrated with a broad range of examples from logic and algebra, demonstrating their use in establishing properties for both fields.

Interpolation and Amalgamation

TL;DR

This chapter develops a unified universal-algebra framework to connect interpolation properties of non-classical propositional logics with amalgamation properties of their algebraic semantics. Central to the approach is a bridge between equational consequence and congruences of free algebras, enabling transfer theorems among CEP, EP, AP, RP, DIP, and CIP across a wide range of logics (Heyting, Gödel, MV, FL, and modal). The work establishes key equivalences (e.g., AP ↔ RP, DIP with EP plus AP) and surveys which algebraic varieties possess these properties, including detailed treatment of FL_e-algebras, FL-algebras, and semilinear residuated lattices. It also clarifies when uniform and Craig interpolation can be derived syntactically or algebraically, and highlights instances where interpolation properties fail. Overall, the chapter provides a comprehensive toolkit for analyzing non-classical logics via their algebraic counterparts, with broad implications for both logic and universal algebra.

Abstract

This chapter presents a state-of-the-art survey of relationships, traditionally referred to as `bridges', between interpolation properties for propositional logics -- including superintuitionistic, modal, and substructural logics -- and amalgamation properties for corresponding classes of algebraic structures. These bridges are developed in the framework of universal algebra and illustrated with a broad range of examples from logic and algebra, demonstrating their use in establishing properties for both fields.

Paper Structure

This paper contains 7 sections, 14 theorems, 19 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Logic and Algebra
  3. Consequence and Congruence
  4. Local Deduction Theorems and the Congruence Extension Property
  5. Amalgamation and the Robinson Property
  6. Deductive Interpolation Properties
  7. The Craig Interpolation Property

Key Result

Theorem 8

For any surjective homomorphism $\pi\colon{\bf {A}}\to{\bf {B}}$ and homomorphism $\varphi\colon{\bf {A}}\to{\bf {C}}$ with $\ker(\pi)\subseteq\ker(\varphi)$, there exists a unique homomorphism $\psi\colon{\bf {B}}\to{\bf {C}}$ satisfying $\psi\pi=\varphi$; moreover, $\psi$ is injective if, and only

Figures (3)

  • Figure 1: Commutative diagrams for algebraic properties
  • Figure 2: Commutative diagrams for the proof of Theorem \ref{['t:AP<=>RP']}
  • Figure 3: Bridges between algebraic and syntactic properties

Theorems & Definitions (38)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Remark 7
  • Theorem 8: cf. MPT23
  • Lemma 9
  • Example 10
  • ...and 28 more