Quantales and Hyperstructures: Monads, Mo' Problems

Abstract: We present a theory of lattice-enriched semirings, called quantic semirings, which generalize both quantales and powersets of hyperrings. Using these structures, we show how to recover the spectrum of a Krasner hyperring (and in particular, a commutative ring with unity) via universal constructions, and generalize the spectrum to a new class of hyperstructures, hypersemirings. (These include hyperstructures currently studied under the name "semihyperrings", but we have weakened the distributivity axioms.) Much of the work consists of background material on closure systems, suplattices, quantales, and hyperoperations, some of which is new. In particular, we define the category of covered semigroups, show their close relationship with quantales, and construct their spectra by exploiting the construction of a universal quotient frame by Rosenthal. We extend these results to hypersemigroups, demonstrating various folkloric correspondences between hyperstructures and lattice-enriched structures on the powerset. Building on this, we proceed to define quantic semirings, and show that they are the lattice-enriched counterparts of hypersemirings. To a quantic semiring, we show how to define a universal quotient quantale, which we call the quantic spectrum, and using this, we show how to obtain the spectrum of a hypersemiring as a topological space in a canonical fashion. Finally, we we conclude with some applications of the theory to the ordered blueprints of Lorscheid.