Minimal varieties of residuated lattices |
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Authors: | Nikolaos Galatos |
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Affiliation: | (1) School of Information Science, Japan Advanced Institute of Science and Technology, 1–1 Asahidai, Tatsunokuchi, Ishikawa 923-1292, Japan |
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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.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 1, 2003; accepted in final form April 27, 2004. |
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Keywords: | 06F05 08B15 03B47 03G10 |
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