I quote from Petsche's Hermann Graßmann: Biography (emphasis mine):
The mathematical part of the book begins with the conception of the “General Theory of Forms”. Starting with a perspective on mathematics as a theory of forms, Graßmann analyses in the most abstract way possible the general structures of concrete “conjunctions of forms”. Here, he places special emphasis on “elementary conjunctions”, demanding they have module properties, that is, associativity, commutativity and an inverse and neutral element. The so-defined conjunction of the first order, or “formal addition”, is then followed up by an investigation of a conjunction of the second order (“formal multiplication”), for which he only requires distributivity with respect to formal addition. Graßmann directly posits the validity of the module properties for formal addition and distributivity for formal multiplication as the principles for constructing these conjunctions: “This generally is the way”, he wrote, “that initially, that is when no species of conjunction is yet given, such a conjunction of next higher order is defined.”
Since Graßmann does not require the forms generated by conjunctions of the second order to be embedded in the fundamental domain, he can use this form of conjunction for the formal generation of new mathematical objects in the further course of the text.
After Graßmann has laid down the foundation for all mathematical disciplines by presenting these uniquely generalized group-theoretical and structural abstractions he starts with the actual presentation of his new mathematical discipline.
What is the modern terminology for Grassmann's "General Theory of Forms"? What research work has been done in order to continue this line of thinking? Which resources could I acquaint myself with in order to answer these questions?
I think the answer is simply (and very generally, thus unhelpfully) "universal algebra": http://en.wikipedia.org/wiki/Algebraic_structure
Am I on the right track? I am not sure. See also this question of mine, which is looking for something similar in spirit.
See also (again, unsure of the relevancy): http://arxiv.org/pdf/0904.3349v1.pdf
Can someone with familiarity weave together a proper answer from these three resources and others as appropriate?