论文标题
克隆的代数理论，并将其应用于伯克霍夫和玛尔特塞夫的问题
An algebraic theory of clones with an application to a question of Birkhoff and Maltsev
论文作者
论文摘要
我们介绍了克隆代数的概念，该概念旨在找到一种单排，纯粹的代数克隆理论。克隆代数由真实身份定义，因此在普遍代数的意义上形成了多种多样。最自然的克隆代数，即公理的特征是函数的代数，称为功能性克隆代数。功能性克隆代数的宇宙称为欧米茄 - 克隆，是一组包含投影的无限行动，并在限制组成下封闭。我们表明，存在克隆（限制操作）与合适的功能克隆代数的合适子类（称为块代数）之间存在着一种徒。给定克隆，通过可计数的许多虚拟参数扩展克隆的操作来获得相应的块代数。 本文的主要结果之一是一般表示定理，其中表明每个克隆代数与功能性克隆代数同构。在本文的另一个结果中，我们证明了块代数类别产生的克隆代数。这意味着每个欧米茄克隆都是通过使用直接产品，亚代词和同型图像来由合适的克隆家族产生的。 我们以两种应用结束了论文。在第一个中，我们使用克隆代数来回答有关方程理论晶格的经典问题。第二个应用是对所有品种类别VAR的研究。我们介绍了所有克隆代数（任意相似性类型）的类别Ca，并将纯同质性作为箭头。我们表明，类别VAR是对CA的完整子类别的同构。我们使用此结果来提供对独立品种的经典定理的概括。
We introduce the notion of clone algebra, intended to found a one-sorted, purely algebraic theory of clones. Clone algebras are defined by true identities and thus form a variety in the sense of universal algebra. The most natural clone algebras, the ones the axioms are intended to characterise, are algebras of functions, called functional clone algebras. The universe of a functional clone algebra, called omega-clone, is a set of infinitary operations containing the projections and closed under finitary compositions. We show that there exists a bijective correspondence between clones (of finitary operations) and a suitable subclass of functional clone algebras, called block algebras. Given a clone, the corresponding block algebra is obtained by extending the operations of the clone by countably many dummy arguments. One of the main results of this paper is the general representation theorem, where it is shown that every clone algebra is isomorphic to a functional clone algebra. In another result of the paper we prove that the variety of clone algebras is generated by the class of block algebras. This implies that every omega-clone is algebraically generated by a suitable family of clones by using direct products, subalgebras and homomorphic images. We conclude the paper with two applications. In the first one, we use clone algebras to answer a classical question about the lattices of equational theories. The second application is to the study of the category VAR of all varieties. We introduce the category CA of all clone algebras (of arbitrary similarity type) with pure homomorphisms as arrows. We show that the category VAR is categorically isomorphic to a full subcategory of CA. We use this result to provide a generalisation of a classical theorem on independent varieties.
