  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.
