×

Structure theorems for idempotent residuated lattices. (English) Zbl 1481.06033

Summary: In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has the amalgamation property. Finally, we address an open problem in the literature by giving an example of a noncommutative variety of idempotent residuated lattices that has the amalgamation property.

MSC:

06F05 Ordered semigroups and monoids
03G10 Logical aspects of lattices and related structures
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)

Software:

Prover9; Mace4
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] van Alten, CJ, Congruence properties in congruence permutable and in ideal determined varieties, with applications, Algebra Universalis, 53, 4, 433-449 (2005) · Zbl 1077.08003 · doi:10.1007/s00012-005-1911-7
[2] van Alten, CJ, The finite model property for knotted extensions of propositional linear logic, J. Symb. Logic, 70, 1, 84-98 (2005) · Zbl 1089.03015 · doi:10.2178/jsl/1107298511
[3] Blount, K.; Tsinakis, C., The structure of residuated lattices, Int. J. Algebr. Comput., 13, 4, 437-461 (2003) · Zbl 1048.06010 · doi:10.1142/S0218196703001511
[4] Chen, W.; Chen, Y., Variety generated by conical residuated lattice-ordered idempotent monoids, Semigroup Forum, 98, 3, 431-455 (2019) · Zbl 1467.06014 · doi:10.1007/s00233-019-10014-3
[5] Chen, W.; Zhao, X., The structure of idempotent residuated chains, Czech. Math. J., 59, 134, 453-479 (2009) · Zbl 1224.06025 · doi:10.1007/s10587-009-0031-5
[6] Chen, W.; Zhao, X.; Guo, X., Conical residuated lattice-ordered idempotent monoids, Semigroup Forum, 79, 2, 244-278 (2009) · Zbl 1185.06010 · doi:10.1007/s00233-009-9158-9
[7] Couceiro, M.; Devillet, J.; Marichal, JL, Quasitrivial semigroups: characterizations and enumerations, Semigroup Forum, 98, 3, 472-498 (2019) · Zbl 1468.20099 · doi:10.1007/s00233-018-9928-3
[8] Dunn, JM, Algebraic completeness for \({R}\)-mingle and its extensions, J. Symb. Logic, 35, 1-13 (1970) · Zbl 0231.02024 · doi:10.2307/2271149
[9] Galatos, N.; Jipsen, P.; Kowalski, T.; Ono, H., Residuated Lattices: An Algebraic Glimpse at Substructural Logics (2007), Amsterdam: Elsevier, Amsterdam · Zbl 1171.03001
[10] Galatos, N.; Raftery, JG, A category equivalence for odd Sugihara monoids and its applications, J. Pure Appl. Algebra, 216, 10, 2177-2192 (2012) · Zbl 1279.03043 · doi:10.1016/j.jpaa.2012.02.006
[11] Galatos, N.; Raftery, JG, Idempotent residuated structures: some category equivalences and their applications, Trans. Am. Math. Soc., 367, 5, 3189-3223 (2015) · Zbl 1402.06005 · doi:10.1090/S0002-9947-2014-06072-8
[12] Jipsen, P.; Tsinakis, C.; Martinez, J., A survey of residuated lattices, Ordered Algebraic Structures, 19-56 (2002), Dordrecht: Kluwer, Dordrecht · Zbl 1070.06005
[13] Kamide, N., Substructural logics with mingle, J. Logic Lang. Inf., 11, 227-249 (2002) · Zbl 1003.03019 · doi:10.1023/A:1017586008091
[14] Maksimova, LL, Craig’s theorem in superintuitionistic logics and amalgamable varieties, Algebra i Logika, 16, 6, 643-681 (1977) · Zbl 0403.03047
[15] Marchioni, E.; Metcalfe, G., Craig interpolation for semilinear substructural logics, Math. Log. Q., 58, 6, 468-481 (2012) · Zbl 1273.03075 · doi:10.1002/malq.201200004
[16] McCune, W.: Prover9 and Mace4 (2005-2010). http://www.cs.unm.edu/ mccune/Prover9
[17] McKinsey, JCC; Tarski, A., On closed elements in closure algebras, Ann. Math., 47, 1, 122-162 (1946) · Zbl 0060.06207 · doi:10.2307/1969038
[18] Metcalfe, G.; Montagna, F.; Tsinakis, C., Amalgamation and interpolation in ordered algebras, J. Algebra, 402, 21-82 (2014) · Zbl 1318.06012 · doi:10.1016/j.jalgebra.2013.11.019
[19] Metcalfe, G.; Paoli, F.; Tsinakis, C.; Hosni, H.; Montagna, F., Ordered algebras and logic, Uncertainty and Rationality, 1-85 (2010), Pisa: Publications of the Scuola Normale Superiore di Pisa, Pisa
[20] Olson, JS, The subvariety lattice for representable idempotent commutative residuated lattices, Algebra Universalis, 67, 1, 43-58 (2012) · Zbl 1253.06017 · doi:10.1007/s00012-012-0167-2
[21] Raftery, JG, Representable idempotent commutative residuated lattices, Trans. Am. Math. Soc., 359, 9, 4405-4427 (2007) · Zbl 1117.03070 · doi:10.1090/S0002-9947-07-04235-3
[22] Stanovský, D., Commutative idempotent residuated lattices, Czech. Math. J., 57, 132, 191-200 (2007) · Zbl 1174.06332 · doi:10.1007/s10587-007-0055-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.