Quantale

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In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various lattices of multiplicative ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). Quantales are sometimes referred to as complete residuated semigroups.

A quantale is a complete lattice Q with an associative binary operation ∗ : Q × Q → Q, called its multiplication, satisfying

x ∗ sup { x_i | i in I } = sup { x ∗ x_i | i in I }

sup { x_i | i in I } ∗ x = sup { x_i ∗ x | i in I }

for all x, x_i 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 ∗. If the unit is the top element of the underlying lattice, the quantale 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 the unit interval together with its usual multiplication.

An idempotent quantale is a quantale whose multiplication is idempotent. A frame is the same as an idempotent strictly two-sided quantale.

[edit] References

• C.J. Mulvey, "Quantales" SpringerLink Encyclopaedia of Mathematics (2001)
• 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.