Structure Theory of Semirings

• Applicant: Professor Dr. Jens Zumbrägel
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552130058

Basic arithmetic operations are described in classical algebra by means of notions such as ring and field. These comprise the usual domains of numbers, but also structures like polynomials or modulo computations. It is this generality which makes the abstract concepts so successful and allows for a wide variety of applications in mathematics and computer science. The notion of a semiring generalizes that of a ring, insofar that only addition and multiplication are available, but no subtraction. In the literature there is a growing interest of semirings, since they allow the description of structures which are not captured by more traditional concepts. Examples for this are multi-valued logic or max-plus-algebras which occur in optimization problems. The theory of semirings is located at an exciting crossroads of ring theory and semigroup theory. The area is relatively young and still features a range of open questions. A fundamental characterization of so-called congruence-simple semirings was achieved. The planned research project is now focusing on the next step. It is aimed to develop a structure theory of semirings, akin to the very fruitful structure theory of rings. As in ring theory, the notion of a module plays a pivotal role in the investigation of congruence-simple semirings. In order to investigate further classes of semirings, it is aimed to generalize the concept of semi-simple modules in a suitable way. This paves the way for new versions of the celebrated structure theorems of semi-simple rings. This task is challenging, since modules over semirings do not allow a subtraction, whence different novel notions emerge that coincide in the classical theory. Research on the general structure theory of semirings has a great potential, since it allows to describe very different objects by the same concept. In particular, it is planned to investigate convolutional algebras in a wide sense. These structures appear frequently in the current literature and offer an interesting test bed to evaluate the applicability of the theory being developed.

• DFG Programme: Research Grants