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Exploring New Topologies for the Theory of Clones

Antonio Bucciarelli, Antonino Salibra

TL;DR

This paper develops a unified topological-algebraic framework for omega-clones and omega-polymorphisms by introducing X-topologies on omega-operations and parametric invariant relations. It establishes a Galois-theoretic correspondence between omega-polymorphisms and invariant omega-relations under these topologies, and provides a characterization of X-closed infinitary omega-clones via Pol^ω_I and Inv^ω_I. A categorical perspective introduces canonical ideal maps and matrical polymorphisms, linking finitary and infinitary settings and connecting Inv^ω-Pol^ω to Inv-Pol in a convergent manner. The work further develops omega-relational clones, cut-closedness, and limits of decreasing sequences to relate relation clones with omega-relations, including results for trace and uniform closures. Overall, it offers a cohesive topological treatment that generalizes finite clone theory to arity ω, with implications for CSP complexity and model theory, and proposes a pathway toward a broader cl-monoid framework for future theory-building.

Abstract

Clones of operations of arity omega (referred to as omega-operations) have been employed by Neumann to represent varieties of infinitary algebras defined by operations of at most arity omega. More recently, clone algebras have been introduced to study clones of functions, including omega-operations, within the framework of one-sorted universal algebra. Additionally, polymorphisms of arity omega, which are omega-operations preserving the relations of a given first-order structure, have recently been used to establish model theory results with applications in the field of complexity of CSP problems. In this paper, we undertake a topological and algebraic study of polymorphisms of arity omega and their corresponding invariant relations. Given a set A and a Boolean ideal X on the set of omega-sequences of elements of A, we propose a method to endow the set of omega-operations on A with a topology, which we refer to as X-topology. Notably, the topology of pointwise convergence can be retrieved as a special case of this approach. Polymorphisms and invariant relations are then defined parametrically, with respect to the X-topology. We characterise the X-closed clones of omega-operations in terms of polymorphisms and invariant relations of arity omega, and present a method to relate those infinitary invariant relation and polymorphisms to the classical (finitary) Inv-Pol.

Exploring New Topologies for the Theory of Clones

TL;DR

This paper develops a unified topological-algebraic framework for omega-clones and omega-polymorphisms by introducing X-topologies on omega-operations and parametric invariant relations. It establishes a Galois-theoretic correspondence between omega-polymorphisms and invariant omega-relations under these topologies, and provides a characterization of X-closed infinitary omega-clones via Pol^ω_I and Inv^ω_I. A categorical perspective introduces canonical ideal maps and matrical polymorphisms, linking finitary and infinitary settings and connecting Inv^ω-Pol^ω to Inv-Pol in a convergent manner. The work further develops omega-relational clones, cut-closedness, and limits of decreasing sequences to relate relation clones with omega-relations, including results for trace and uniform closures. Overall, it offers a cohesive topological treatment that generalizes finite clone theory to arity ω, with implications for CSP complexity and model theory, and proposes a pathway toward a broader cl-monoid framework for future theory-building.

Abstract

Clones of operations of arity omega (referred to as omega-operations) have been employed by Neumann to represent varieties of infinitary algebras defined by operations of at most arity omega. More recently, clone algebras have been introduced to study clones of functions, including omega-operations, within the framework of one-sorted universal algebra. Additionally, polymorphisms of arity omega, which are omega-operations preserving the relations of a given first-order structure, have recently been used to establish model theory results with applications in the field of complexity of CSP problems. In this paper, we undertake a topological and algebraic study of polymorphisms of arity omega and their corresponding invariant relations. Given a set A and a Boolean ideal X on the set of omega-sequences of elements of A, we propose a method to endow the set of omega-operations on A with a topology, which we refer to as X-topology. Notably, the topology of pointwise convergence can be retrieved as a special case of this approach. Polymorphisms and invariant relations are then defined parametrically, with respect to the X-topology. We characterise the X-closed clones of omega-operations in terms of polymorphisms and invariant relations of arity omega, and present a method to relate those infinitary invariant relation and polymorphisms to the classical (finitary) Inv-Pol.
Paper Structure (27 sections, 46 theorems, 41 equations)

This paper contains 27 sections, 46 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\omega$-Sequences
  4. Traces
  5. Operations and $\omega$-operations
  6. Relations and $\omega$-relations
  7. Matrices
  8. The local topology on spaces of functions
  9. Clones and polymorphisms
  10. $\omega$-Clones
  11. Clones as $\omega$-clones
  12. Topologies on function spaces
  13. Topologies on $O_A^{(\omega)}$
  14. Polymorphisms and invariant relations
  15. Topologies on $A^\omega$
  16. ...and 12 more sections

Key Result

Lemma 2.1

For every relation $S\subseteq A^n$, the $\omega$-relation $S^\top$ is locally closed.

Theorems & Definitions (108)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • ...and 98 more