2017/09/10 by Jun, Jaiung, Rowen, Louis
#06F05 #08A05 #08A30 #08A72 #12K10 #13C60 (Secondary) #14T05 #16Y60 #20N20 (Primary) #Algebraic Geometry (math.AG) #Category Theory (math.CT) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1709.03186
We continue the theory of \tT-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Prime ground systems are introduced as a way of developing geometry. The polynomial system over a prime system is prime, and there is a weak Nullstellensatz. Also, the polynomial \mathcal A[\la1, …, \lan] and Laurent polynomial systems \mathcal A[[\la1, …, \lan]] in n commuting indeterminates over a \tT-semiring-group system have dimension n. For module systems, special attention also is paid to tensor products and \Hom. Abelian categories are replaced by "semi-abelian" categories (where \Hom(A,B) is not a group) with a negation morphism.