. Finding particular manifestations of this general law, t. e. The establishment of connections between different phenomena is one of the main tasks of all science. Therefore, it is always pleasant when deep connections are found between seemingly completely heterogeneous mathematical objects. One such link the connection between dedekind structures with additions and regular rings was discovered at the junction of algebra, geometry and functional analysis. A more prepared reader can get acquainted with this idea in more detail by reading the following introduction. The less prepared will have to start with the main text, the reading of which formally does not require any preliminary preparation. All concepts used, except the ring ideal and the partially ordered set, are defined by. The evidence, especially at first, is very detailed. A number of interesting results not included in the main line of presentation are formulated in the last paragraph. There are also some problems with ..