Exploring New Topologies for the Theory of Clones

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 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 $Pol^\omega$-$Inv^\omega$ and present a method to relate $Inv^\omega$-$Pol^\omega$ to the classical (finitary) $Inv$-$Pol$.