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Minimal varieties of residuated lattices. (English) Zbl 1082.06011
Summary: In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of P. Jipsen and C. Tsinakis’ paper “A survey of residuated lattices” [in: J. MartĂ­nez (ed.), Ordered algebraic structures. Proceedings of the conference on lattice-ordered groups and \(f\)-rings held at the University of Florida, Gainesville, FL, USA, February 28–March 3, 2001. Dordrecht: Kluwer Academic Publishers. Developments in Mathematics 7, 19–56 (2002; Zbl 1070.06005)]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Finally, we study the connections with the subvariety lattice of residuated bounded lattices. We modify the construction mentioned above to obtain a continuum of idempotent, representable minimal varieties of residuated bounded lattices and illustrate how the existing construction provides continuum many covers of the variety generated by the three-element non-integral residuated bounded lattice.

06F05 Ordered semigroups and monoids
08B15 Lattices of varieties
03G10 Logical aspects of lattices and related structures
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