On Automorphisms of Polyadic Algebras

Introduction. This paper belongs to the theory of polyadic algebras as developed by Haimos [11-15], but it has a bearing on the theory of models. Two central concepts are those of homogeneous(2) and of normal extensions of a polyadic algebra (see the beginning of §2 and of §6). These concepts bear some resemblance to concepts of the theory of algebraic extensions of fields; however, we have been influenced more immediately by M. Krasner's general Galois theory [33-35] which can be given a polyadic interpretation. Aside from the short preliminary Chapter 0, the paper is divided into sections grouped into three chapters. Each section begins by an outline of its contents. We proceed to an analysis of the main results of the paper. All polyadic algebras considered are locally finite of infinite degree. The highlights of Chapter 1 are: (i) the possibility of extending any simple extension of a (simple polyadic) algebra to a simple homogeneous and normal extension (Corollary 4.6); (ii) a first step on a Galois theory (Theorem 4.5) which, in the case of a full simple functional algebra with finite domain, takes a definite form (Theorem 4.7) closely related to Krasner's theory in that case; (iii) the existence and unicity of a+-universaI-homogeneous algebras in the sense of B. Jönsson [21; 22] in certain classes of polyadic algebras (Theorem 6.3, Theorem 6.6). The results in (iii) make use of Jönsson's work which is dependent on the continuum hypothesis (see also Theorem 5.5 which is independent of this hypothesis). Chapter II departs from the theme of automorphisms and uses little aside from Chapter 0 and §1. The main result here is Theorem 8.1 which gives a description of the functional representations of a full simple functional algebra in terms of a new generalization of the concept of reduced power. In Chapter III, we propose to show that the algebraic results of the first two chapters, when put together, embody several known model-theoretic results or new forms of such results. After a section devoted to the connections between Model theory and polyadic algebras, we give new proofs of Beth's theorem