S-sets and semigroups of quotients

During the last two decades, a good deal of work has been conducted on S-sets and semigroups of quotients, establishing several concepts and results which are more or less analogous to the classicalas well as ring-theoretical ones. Clearly there are different (and sometimes, we feel, too restrictive) assumptions in these papers, in particular on the semigroups S and on the S-sets under consideration, and various notations need to be unified. Generalizing diverging dispositions, we try to give a self-contained survey of the most fundamental parts (and other considerable parts) of these investigations, and add some new results. A forthcoming paper will deal with S-semimodules over semirings S, including some relationships with rings and modules.