ALGEBRAIC CLOSURES IN CERTAIN ELEMENTARY CLASSES
Tsuyoshi Fujiwara · Scientiae mathematicae Japonicae · 2004
The main purpose of this paper is to give some of the necessary and suffi- cient conditions for the existence of algebraic closures in a certain kind of elementary classes. This study is essentially built on the foundation of the paper: Zur Theorie der alge- braischen Erweiterungen by the late Prof. K. Shoda (Cf. (3), (4)). However our notion of algebraic extensions will be very generalized from his notion. But in the variety with the fundamental conditions (I), (IV) and (a), (b) in his paper, our notion and his notion are equivalent (Cf. (3; Satz 6)). In this paper, we shall introduce new notions of algebraic extensions, algebraic closed- ness, and algebraic closures in a certain kind of elementary classes, which are generalizations of the usual ones in the theory of commutative fields. And we shall investigate their prop- erties. Especially, we shall state some of the necessary and sufficient conditions for the existence of algebraic closures (Theorem 4.3). Our theory (§1 ∼§ 5) can be applied not only to the class of commutative fields but also to every injectively complete universal class, for example, the variety of Abelian groups, the variety of Boolean algebras, the variety of semilattices, the class of partially ordered sets, the class of totally ordered sets, etc. The main parts (§1 ∼§ 4) of our theory can be also applied to every residually small variety with the (somewhat weak) amalgamation property.