On the axiomatic sources of fundamental algebraic structures: the achievements of Hermann Grassmann and Robert Grassmann

Abstract

The axiomatic source of fundamental algebraic structures, such as groups and rings, is traced to the achievements of the brothers Hermann and Robert Grassmann. Algebra is the source of model structures for the theory of algorithms. In this respect, the work of the Grassmann brothers is the basis, for example, of Markov's constructivist theory of algorithms.

The concept of the semigroup is to be traced to Hermann and Robert Grassmann's general doctrine of forms, or Formenlehre, as developed in Robert Grassmann's Die Begriffslehre oder Logik: Zweites Buch der Formenlehre oder Mathematik; and in such of Hermann Grassmann's works as the Ausdehnungslehre is to be found the definition of an abstract group (ten years before Cayley's work on groups), and the concept of ring is developed, yielding both left and right rings. In addition to semigroups, quasigroups, groups, rings, and fields, a more general development in the Formenlehre provides the axiomatic basis for lattices and Boolean algebra.