Editing Quantale

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In [[mathematics]], '''quantales''' are certain [[partially ordered set|partially ordered]] [[algebraic structure]]s that generalize [[Complete_Heyting_algebra#Frames_and_locales|locale]]s ([[pointless topology|point free topologies]]) as well as various multiplicative [[lattice (order)|lattices]] of [[Ideal (ring theory)|ideal]]s from [[ring theory]] and [[functional analysis]] ([[C-star algebra|C*-algebras]], [[von Neumann algebra]]s).<ref>{{Cite book |last1=Paeska |first1=Jan |last2=Slesinger |first2=Radek |chapter=A Representation Theorem for Quantale Valued sup-algebras |date=2018 |title=2018 IEEE 48th International Symposium on Multiple-Valued Logic (ISMVL) |chapter-url=https://ieeexplore.ieee.org/document/8416927 |pages=91–96 |doi=10.1109/ISMVL.2018.00024 |arxiv=1810.09561 |isbn=978-1-5386-4464-5 |via=IEEE Xplore}}</ref> Quantales are sometimes referred to as ''complete [[residuated lattice#residuated_semilattice|residuated semigroup]]s''.

==Overview==
A '''quantale''' is a [[complete lattice]] <math>Q</math> with an [[associative]] [[binary operation]] <math>\ast\colon Q \times Q \to Q</math>, called its '''multiplication''', satisfying a distributive property such that

:<math>x*\left(\bigvee_{i\in I}{y_i}\right) = \bigvee_{i\in I}(x*y_i)</math>

and

:<math>\left(\bigvee_{i\in I}{y_i}\right)*{x}=\bigvee_{i\in I}(y_i*x)</math>

for all <math>x, y_i \in Q</math> and <math>i \in I</math> (here <math>I</math> is any [[index set]]). The quantale is '''unital''' if it has an [[identity element]] <math>e</math> for its multiplication:

:<math>x*e = x = e*x</math>

for all <math>x \in Q</math>. In this case, the quantale is naturally a [[monoid]] with respect to its multiplication <math>\ast</math>.

A unital quantale may be defined equivalently as a [[Monoid (category theory)|monoid]] in the category '''[[Complete lattice#Morphisms of complete lattices|Sup]]''' of complete [[join-semilattice]]s.

A unital quantale is an idempotent [[semiring]] under join and multiplication.

A unital quantale in which the identity is the [[Greatest element|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 [[complete Heyting algebra|frame]], with its multiplication given by the [[Meet (mathematics)|meet]] operation, is a typical example of a strictly two-sided commutative quantale. Another simple example is provided by the [[unit interval]] together with its usual [[multiplication]].

An '''idempotent quantale''' is a quantale whose multiplication is [[idempotent]]. A [[complete Heyting algebra|frame]] is the same as an idempotent strictly two-sided quantale.

An '''involutive quantale''' is a quantale with an involution

:<math>(xy)^\circ = y^\circ x^\circ</math>
that preserves joins:

:<math>\biggl(\bigvee_{i\in I}{x_i}\biggr)^\circ =\bigvee_{i\in I}(x_i^\circ).</math>

A '''quantale [[homomorphism]]''' is a [[map (mathematics)|map]] <math>f\colon Q_1 \to Q_2</math> that preserves joins and multiplication for all <math>x, y, x_i \in Q_1</math> and <math>i \in I</math>:

:<math>f(xy) = f(x) f(y),</math>

:<math>f\left(\bigvee_{i \in I}{x_i}\right) = \bigvee_{i \in I} f(x_i).</math>

== See also ==
* [[Relation algebra]]

==References==
{{Reflist}}
*{{springer|id=Q/q130010|title=Quantale|author=C.J. Mulvey}} [http://encyclopediaofmath.org/index.php?title=Quantale&oldid=42430]
* J. Paseka, J. Rosicky, Quantales, in: [[Bob Coecke|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.&nbsp;245–262. 
* M. Piazza, M. Castellan, ''Quantales and structural rules''. [[Journal of Logic and Computation]], 6 (1996), 709–724.
* K. Rosenthal, ''Quantales and Their Applications'', Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990.

[[Category:Order theory]]


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