A quantale is a complete lattice with an associative binary operation , called its multiplication, satisfying a distributive property such that
and
for all and (here is any index set). The quantale is unital if it has an identity element for its multiplication:
for all . In this case, the quantale is naturally a monoid with respect to its multiplication .
A unital quantale may be defined equivalently as a monoid in the category Sup of complete join semi-lattices.
A unital quantale is an idempotent semiring under join and multiplication.
A unital quantale in which the identity is the 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 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.
An involutive quantale is a quantale with an involution
that preserves joins:
A quantale homomorphism is a map that preserves joins and multiplication for all and :