Minimal varieties of representable commutative residuated lattices (Q1935563)

The aim of the paper is to get the cardinality and a description of minimal subvarieties of the varieties of representable and integral commutative residuated lattices, denoted, respectively, by RCRL and ICRL. A commutative residuated lattice is called \(k\)-potent if it satisfies \(x^{k +1} = x^k\). It is shown that every minimal subvariety of RCRL is 4-potent and there exist continuum many minimal subvarieties. On the other hand, there are only five 3-potent minimal subvarieties of RCRL and only two minimal subvarieties of ICRL, all of them are described by their generators. Finally, it is proved that the variety of representable ICRL is generated as a quasivariety by 1-generated finite members. Hence, several open problems possed by N. Galatos, P. Jipsen, T. Kowalski and H. Ono are solved.