Extensions of states on MV-quantales

We consider the lattice of the ordered right-continuous families of elements of a “dual quantale” with the index set [0,1] (called its “rc-extension”) and the lattice of the ordered (n+1)-tuples (with n≥1) of elements of this dual quantale (its “n-extension”) equipped with a dual quantale structure that generalizes the “canonical extension” proposed by Höhle and Weber. Of particular interest is the existence of states on n-extensions of finite MV-chains. We generalize the recent results of Weber by incorporating effectible Girard quantales, which are introduced by the author.