Effectible residuated lattices and n-th roots

In this paper, conditions are given on residuated lattices and on Girard algebras in order for them to be generalized effect algebras and effect algebras, respectively. A new class of such “effectible” residuated lattices essentially enlarging MV-algebras is defined. Of particular interest are “pre-radicable” residuated lattices having all “n-th roots” generalizing Höhleʹs square roots to an arbitrary order n. The existence of “partial” n-th roots on effectible residuated lattices and Girard algebras is related with the concept of divisible effect algebras.