Exploring new topologies for the theory of clones

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 Boolean ideal X on the set \(A^\omega \), we 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 \(\textrm{Pol}^\omega \)-\(\textrm{Inv}^\omega \) and present a method to relate \(\textrm{Inv}^\omega \)-\(\textrm{Pol}^\omega \) to the classical (finitary) \(\textrm{Inv}\)-\(\textrm{Pol}\).