Abstract.
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 [12]. 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.
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In Celebration of the Sixtieth Birthday of Ralph N. McKenzie
Received August 1, 2003; accepted in final form April 27, 2004.
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Galatos, N. Minimal varieties of residuated lattices. Algebra univers. 52, 215–239 (2005). https://doi.org/10.1007/s00012-004-1870-4
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DOI: https://doi.org/10.1007/s00012-004-1870-4