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 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.