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| In [[mathematics]], '''quantales''' are certain [[partially ordered set|partially ordered]] [[algebraic structure]]s that generalize locales ([[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). Quantales are sometimes referred to as ''complete residuated semigroups''.
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| A '''quantale''' is a [[complete lattice]] ''Q'' with an [[associative]] [[binary operation]] ∗ : ''Q'' × ''Q'' → ''Q'', called its '''multiplication''', satisfying
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| :<math>x*(\bigvee_{i\in I}{y_i})=\bigvee_{i\in I}(x*y_i)</math>
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| :<math>(\bigvee_{i\in I}{y_i})*{x}=\bigvee_{i\in I}(y_i*x)</math>
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| for all ''x'', ''y<sub>i</sub>'' in ''Q'', ''i'' in ''I'' (here ''I'' is any [[index set]]).
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| The quantale is '''unital''' if it has an [[identity element]] ''e'' for its multiplication:
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| : ''x'' ∗ ''e'' = ''x'' = ''e'' ∗ ''x''
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| for all ''x'' in ''Q''. In this case, the quantale is naturally a [[monoid]] with respect to its multiplication ∗.
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| 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 semi-lattices.
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| A unital quantale is an idempotent [[semiring]], or dioid, under join and multiplication.
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| 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'').
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| 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]].
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| 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.
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| An '''involutive quantale''' is a quantale with an involution:
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| :<math>(xy)^\circ = y^\circ x^\circ</math>
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| that preserves joins:
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| :<math>\biggl(\bigvee_{i\in I}{x_i}\biggr)^\circ =\bigvee_{i\in I}(x_i^\circ).</math>
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| A '''quantale [[homomorphism]]''' is a [[map (mathematics)|map]] f : ''Q<sub>1</sub>'' → ''Q<sub>2</sub>'' that preserves joins and multiplication for all ''x'', ''y'', ''x<sub>i</sub>'' in ''Q'', ''i'' in ''I'':
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| :<math>f(xy) = f(x)f(y)</math>
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| :<math>f\biggl(\bigvee_{i \in I}{x_i}\biggl) = \bigvee_{i \in I} f(x_i)</math> | |
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| ==References==
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| *{{springer|id=Q/q130010|title=Quantales|author=C.J. Mulvey}} [http://www.encyclopediaofmath.org/index.php?title=Quantale&oldid=17639]
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| * 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. 245–262.
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| * K. Rosenthal, ''Quantales and Their Applications'', Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990.
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| [[Category:Order theory]]
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