Understand
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.