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) $Inv$-$Pol$.

翻译：纽曼曾利用元数$\omega$的操作克隆（称为$\omega$-操作）来表示由至多元数$\omega$的操作定义的无穷代数簇。近年来，克隆代数被引入以在单种泛代数框架下研究函数克隆（包括$\omega$-操作）。此外，元数$\omega$的多态性（即保持给定一阶结构关系的$\omega$-操作）最近被用于建立模型论结果，并在CSP问题的复杂性领域中得到应用。本文对元数$\omega$的多态性及其对应的不变关系进行了拓扑与代数研究。给定集合$A^\omega$上的布尔理想$X$，我们提出了一种方法，为$A$上的$\omega$-操作集赋予拓扑（称为$X$-拓扑）。值得注意的是，逐点收敛拓扑可作为该方法的一个特例得到。多态性与不变关系随后以参数化方式（相对于$X$-拓扑）定义。我们通过$Pol^\omega$-$Inv^\omega$刻画了$\omega$-操作的$X$-闭克隆，并提出了一种将$Inv^\omega$-$Pol^\omega$与经典（有限）$Inv$-$Pol$相关联的方法。