Has anyone every considered the following class of quantales? If * is the tensor and I its unit, the additional axiom u*v = I implies u=v=I holds. If anybody has used them how are they called? Actually I am interested in the ones among them that are also commutative and where I is a dualizing element. In other words they are a specific class of Girard quantales. There is an interesting subclass of them: look at the following list of multiplication tables. *| I 1 -1 *| I 1 0 -1 *| I 2 1 -1 -2 ----------- -------------- ---------------- I| I 1 -1 I| I 1 0 -1 I| I 2 1 -1 -2 1| 1 1 -1 1| 1 1 0 -1 2| 2 2 1 -1 -2 -1|-1 -1 -1 0| 0 0 -1 -1 1| 1 1 1 -2 -2 -1|-1 -1 -1 -1 -1|-1 -1 -2 -2 -2 -2|-2 -2 -2 -2 -2 It is easy to see this generalizes for any set of the form { I, n,n-1,...,1,0,-1,...,-(n-1),-n } or { I, n,n-1,...,1, -1,...,-(n-1),-n }. The slogan is "take the table for inf in the total ordering given by the writing order and overwrite the lower right triangle with -n". The dualizing operator is multiplication by -1 (keeping I fixed, natch). It suffices to say what is the set of elements >= 1 to define the order on the quantale, and it is always {n,I}. Thus the Hasse diagrams look like 1 1 2 | / \ / \ I 0 I 1 \ | \ / | I ... -1 -1 -1 / \ / -2 This can be generalized even further: given a complete lattice with an involution (-)~ : A^{op} -> A , on the set A+{I} define * as follows: / a if b = I, ( b if a = I ) a*b = < bottom if a =< b~ \ a inf b if a not =< b~ Here as before the set of elements >= I is {top, I} Have these ever been considered? Do they have a name? Francois Lamarche