Planar Near-Rings: Thery and Applications

Planar near-rings form a special class of "algebraic systems", i.e., of "systems of objects with which one can compute as if they were numbers". These planar near-rings have a series of special properties and applications. They yield in a very easy way subcollections ("blocks") of the numbers 0,1,2,..., v-1 all of which have the same size and such that each of the numbers 0,1,2,..., v-1 appear in the same number of blocks. The set {0,1,2,3,...,v-1}, together with these blocks, then form a so-called "tactical configuration"; quite often, this is even a "balances incomplete block design (BIB-design)". This can be used for statistical purposes: Suppose you want to test the efficiency of a number k, say, of fertilizer ingredients. This requires a large number of tests: no fertilizer, just one component, exactly 2 components, and so on. With a BIB-design of the type above, a much smaller number of tests is required (quite often, the savings are 90% or more). Just take the blocks B1, B2, ..., Bk and apply on field i the fertilizer component # j if and only if i is contained in the block Bj. Wait for harvest time; then the yields on the different fields determines the efficiency of each fertilizer component via standard statistical tests. These connections with and the applications to the design of statistical experiments should be investigated much more thoroughly in this project. For instance, we want to discover more construction principles in order to have a much greater flexibility in the BIB-designs for statistical purposes. But planar near-rings are also interesting in their own rights. Their structure is still not understood completely, and there are most interesting cross-connections to other parts of algebra (in particular, to Frobenius groups), to combinatorics, and to geometry (planar near-rings are being used as coordinate domains of certain geometric planes). Presently, there are 3 centers, where planar near-rings are intensively studied: Hamburg (Germany), Linz (Austria), and Tainan (Taiwan). We plan to concentrate the whole power of these centers via a series of mutual visits and joint workshops. In addition, a powerful computer algebra package ("SONATA") was developed in Linz which allows us for the first time to study planar near-rings intensively on a computer. So this would be a unique occasion to establish a "world center of excellence" for planar near-rings. We must not spoil this chance!

Every possible logical function can be built up from the three functions AND, OR, and NOT. For example every function that maps any sequence of 0 (false) and 1 (true) of length, say 7, to 0 or 1 is expressible as some composition of the three simple functions given above. This fact of the so-called Boolean algebra is known since times immemorial. lt is the reason why Computer Chips can use the same electric circuits to do all sorts of computations. Mathematicians from Linz, Dresden, Novi Sad, and Oxford together have now proved that every function from a finite set to itself can be composed from three basic functions: a sort of addition, subtraction, and some permutation of the elements of the set. It does not even matter much which permutation you choose as your third basic function. If the set is large enough, a randomly Chosen permutation almost certainly will do the trick. This is just one result an which functions can be built up from a given set of simple ones - a topic that was studied extensively by our research group at the Department of Algebra at Linz. Computing with things in a "strange way" is the specialty of our research group. We use abstract algebra for concrete applications. For example, how do you check which ingredients cause certain effects in paint? In the straightforward approach, you mix all possible combinations and analyze the properties of every composition. But looking at all mixtures is very costly in real life. The theory of statistical designs provides methods to obtain the effect of one of, say seven, possible substances by considering only certain combinations of three seven. It is all in how to choose this combinations - as few as possible to keep testing cheap but as much as necessary so that you can also determine the interactions among the ingredients. Because of the research project at Linz, we now know better why certain ways to choose are so efficient and we have new, easily accessible plans for testing.