Quantale

Remarks

Example: the ideals of a commutative ring $R$, ordered by inclusion, with the product of ideals, form a quantale. The join of a family of ideals is their sum, and $J \sum_i K_i = \sum_i J K_i$ because the product of ideals distributes over sums. Example: a complete lattice in which $\wedge$ distributes over arbitrary joins is a quantale with $a \cdot b = a \wedge b$; these are the frames. Non-example: on the two-element lattice $\set{0, 1}$, let $a \cdot b = 1$ for all $a, b$. This is associative, but for the empty family $a \cdot \bigvee \emptyset = a \cdot 0 = 1 \ne 0 = \bigvee \emptyset$, so distributivity fails. The case $I = \emptyset$ says that $a \cdot 0 = 0 \cdot a = 0$, where $0$ is the least element. A quantale is unital if the semigroup has an identity element, and commutative if the multiplication is commutative.