Quantale

Abstractly, a quantale is a Semigroup Object in the category of Suplattices.

Less abstractly, a quantale is

• A sublattice $(X,\bigvee)$ that is
• equipped with an associative operation $\otimes:X\to X\to X$, such that
• For all $x:X$ and $y_{i}:I\to X$, we have

  $x\otimes\left(\bigvee_{i:I}y_{i}\right)=\bigvee_{i:I}(x\otimes y_{i})$

  and

  $\left(\bigvee_{i:I}y_{i}\right)\otimes x=\bigvee_{i:I}(y_{i}\otimes x)$

As Frames

A quantale is a sort of nonabelian version of a Frame; if the tensor is a meet, and there is a Top Element that is a unit for the tensor, then we get a frame.

Automata Theory

Quantales are closely connected to automata theory; see Quantales, Observational Logic and Process Semantics.

Metric Spaces

A Quantale serves as a good "algebra of distance".