States on semi-divisible residuated lattices

摘要

Given a residuated lattice L, we prove that the subset MV(L) of complement elements x * of L generates an MV-algebra if, and only if L is semi-divisible. Rie?an states on a semi-divisible residuated lattice L, and Rie?an states on MV(L) are essentially the very same thing. The same holds for Bosbach states as far as L is divisible. There are semi-divisible residuated lattices that do not have Bosbach states.