- Quantale
In
mathematics , quantales are certain partially orderedalgebraic structure s that generalize locales (point free topologies) as well as various lattices of multiplicativeideal s from ring theory and functional analysis (C*-algebras,von Neumann algebra s). Quantales are sometimes referred to as "complete residuated semigroups".A quantale is a
complete lattice "Q" with anassociative binary operation ∗ : "Q" × "Q" → "Q", called its multiplication, satisfying:x*(igvee_{iin I}{y_i})=igvee_{iin I}(x*y_i)
and
:igvee_{iin I}{y_i})*{x}=igvee_{iin I}(y_i*x)
for all "x", "yi" in "Q", "i" in "I" (here "I" is any
index set ).The quantale is unital if it is has an
identity element "e" for its multiplication:: "x" ∗ "e" = "x" = "e" ∗ "x"
for all "x" in "Q". In this case, the quantale is naturally a
monoid with respect to its multiplication ∗.A unital quantale may be defined equivalently as a monoid in the category Sup of complete join semi-lattices.
A unital quantale is an idempotent
semiring , or dioid, under join and multiplication.A unital quantale in which the identity is the
top element of the underlying lattice, is said to be strictly two-sided (or simply "integral").A commutative quantale is a quantale whose multiplication is
commutative . A frame, with its multiplication given by the meet operation, is a typical example of a strictly two-sided commutative quantale. Another simple example is provided by theunit interval together with its usualmultiplication .An idempotent quantale is a quantale whose multiplication is
idempotent . A frame is the same as an idempotent strictly two-sided quantale.References
*springer|id=Q/q130010|title=Quantales|author=C.J. Mulvey
* J. Paseka, J. Rosicky, Quantales, in: B. Coecke, D. Moore, A. Wilce, (Eds.), "Current Research in Operational Quantum Logic: Algebras, Categories and Languages", Fund. Theories Phys., vol. 111, Kluwer Academic Publishers, 2000, pp. 245–262.
* K. Rosenthal, "Quantales and Their Applications", Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990.
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