Local Theory of t-bonded sets

The local theory for regular and multi-regular systems was developed in the assumption that these systems are Delone sets, or (r;R)-systems. The requirement for a set to be a (r;R)-system particularly implies that any two points in a Delone set can be connected by a sequence of points from the set with sequel interpoint distances bounded by 2R. In the terminology we adopted in this paper, it means that a Delone set is a 2R-bonded set. Meanwhile, there are crystals, e.g. zeolites, whose atomic structure is multi-regular microporous point set. In these structures there are cavities that are relatively large compared to the "length"of bonds between atomes. In other words, the parameter R in this Delone set significantly exceeds a natural link parameter. For a better description of such "microporous"structures it is worthwhile to take into consideration a parameter that represents atomic bonds within the matter. In the paper we generalize some results of the local theory to the sets that we called t-bonded sets even without making an assumption that a set is a Delone set.