REGULATORS OF TYPE a OF LATTICE ORDERED GROUPS

The purpose of the present paper is to investigate the lattice ordered groups (/-groups) having a base by using the algebraic and topological methods. (Note that in[9,10,12],the /-groups having a base are called /-groups of kind a;see Definition 1.2 and Lemma 1.4.) The algebraic examination is carried out by means of the so-called regulators, i.e. the indexed systems of prime subgroups having the zero meet and the topological examination by means of the topology induced on a regulator (structure space). For terminology and notations, cf. [13] I and [10]. A short review is also given in sec. 0 of the present paper. Other structure spaces were dealt with by S. J. Bernau [1]. His spaces are defined on the systems of all prime z-subgroups. Similar considerations will be included in another paper. Prime subgroups need not be z-subgroups, while minimal prime subgroups do it. The regulators of type a are formed by minimal prime subgroups and are equipped with a topology inherited from the hull-kernel topology defined in [1]. In the present paper it is proved that there exists (up to equivalence) at most one regulator of type a of an /-group, namely the set of all maximal polars (1.9). Th