Ultrafilters and maximal linked systems: basic relations

Ultrafilters and maximal linked systems with elements in the form of sets from the fixed $\pi$-system with “zero” and “unit” are investigated. Ultrafilters are maximal linked systems, but, among them, maximal linked systems that are not ultrafilters can be situated. In this paper, special attention is given to the description of the set of maximal linked systems that are not ultrafilters (they are called characteristic). In their properties these maximal linked systems differ from ultrafilters essentially. Necessary and sufficient conditions for existence of the above-mentioned systems (we mean conditions with respect to the initial $\pi$-system) and some topological properties for the set of all maximal linked systems of this type are obtained. In addition, for the construction of the corresponding equipment (just as in the ultrafilter case), the schemes ascending to procedures used under Wallman extension and Stone compactums are used; but the above-mentioned schemes are realized in the case when the anticipating measurable (by sense) structure is given by a $\pi$-system of general form. In particular, this allows one to envelop by a unique structure procedures for construction of spaces of ultrafilters and maximal linked systems for measurable and topological spaces. In the framework of this construction, bitopological spaces corresponding to Wallman and Stone variants of equipment arise naturally; in the first variant, for the case of maximal linked systems, the supercompact $T_1$-space is realized. Examples are given for which all maximal linked systems are ultrafilters; this corresponds to realization of supercompact ultrafilter space when the topology of Wallman type is used.