nLab semiquantale

A right semiquantale is a complete lattice QQ with an associative binary operation \bullet satisfying

a αb α= α(ab α),a,b αQ. a\bullet \Vee_\alpha b_\alpha = \Vee_\alpha (a \bullet b_\alpha),\,\,\,\,\,a, b_\alpha\in Q.

Symmetrically, a left semiquantale is a complete lattice PP with an associative binary operation \bullet satisfying

( αb α)c= α(b αc),c,b αP. \left(\Vee_\alpha b_\alpha\right)\bullet c= \Vee_\alpha (b_\alpha \bullet c),\,\,\,\,\,c, b_\alpha\in P.

The main example of a right semiquantale is the lattice of the topologizing filters of right (or left) ideals. These are the filters I rR\mathcal{F}\subset I_r R such that for all II\in\mathcal{F}

(I:r):={sR|rsI} (I:r) := \{s\in R \,|\, r s\in I\}

is also in \mathcal{F} (i.e. \mathcal{F} is a uniform filter) and if JJ\in \mathcal{F} and (I:r)I(I:r)\in I for all rJr\in J then II\in \mathcal{F}.

The ordering is the reverse inclusion, thus the intersection is the supremum. The intersection of topologizing filters is topologizing, the lattice is complete and the product is the Gabriel multiplication

𝒢={KI rR|L𝒢,KL,rK,(K:r)}\mathcal{F}\bullet \mathcal{G} = \{ K\in I_r R\,|\,\exists L\in\mathcal{G}, K\subset L, \forall r\in K, (K:r)\in\mathcal{F}\}

The operation \bullet preserves arbitrary intersections in the right variable.

Last revised on November 6, 2015 at 14:23:32. See the history of this page for a list of all contributions to it.