States on semi-divisible generalized residuated lattices reduce to states on MV-algebras

摘要

A semi-divisible residuated lattice is a residuated lattice L satisfying an additional condition weaker than that of divisibility. Such structures are related to mathematical fuzzy logic as well as to extended probability theory by the fact that the subset of complemented elements induces an MV-algebra. We define generalized residuated lattices by omitting commutativity of the corresponding monoidal operation and study semi-divisibility in such structures. We show that, given a good generalized residuated lattice L, the set of complemented elements of L, denoted by M V(L), forms a pseudo-MV-algebra if and only if L is semi-divisible. Maximal filters on a semi-divisible generalized residuated lattice L are in one-to-one correspondence with maximal filters on MV(L). We study states on semi-divisible generalized residuated lattices. Riecan states on a semi-divisible generalized residuated lattice L are determined by Riecan states on MV(L). The same holds true for Bosbach states whenever L is a good divisible generalized residuated lattice. Extremal Riecan states on a semi-divisible generalized residuated lattice L are in one-to-one correspondence with maximal and semi-normal filters on L.