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The structure of commutative residuated lattices. (English) Zbl 1011.06006
A commutative residuated lattice is a structure \(\langle L;\cdot,\wedge,\vee,\to,e\rangle\) where \(\langle L;\cdot,e\rangle\) is a commutative monoid, \(\langle L;\wedge,\vee\rangle\) is a lattice, and the operations \(\cdot\), \(\to\) are connected by the equivalences: \(a\cdot b\leq c\) iff \(a\leq b\to c\) iff \(b\leq c\to a\), \(a,b,c\in L\). It is known that \(\mathcal{CRL}\), the class of all commutative residuated lattices, is a finitely based variety of algebras. Now the theory of \(\mathcal{CRL}\) is studied from the viewpoint of universal algebra. It is shown that congruences correspond to order-convex subalgebras. Further, the authors present an equational basis for the subvariety \(\mathcal{CRL}^c\) generated by all commutative residuated chains. Finally, the congruence lattice of any member of \(\mathcal{CRL}^c\) is described.

MSC:
06B05 Structure theory of lattices
06B20 Varieties of lattices
06B10 Lattice ideals, congruence relations
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[1] Blyth T. S., Bull. Soc. Math. France 93 pp 109– (1965) · Zbl 0132.26404 · doi:10.24033/bsmf.1618
[2] Blyth T. S., J. London Math. Soc. 2 (2) pp 635– (1970) · Zbl 0209.05001 · doi:10.1112/jlms/2.Part_4.635
[3] Cornish W. H., Mathematica Japonica 18 pp 203– (1973)
[4] DOI: 10.1090/S0002-9904-1938-06736-5 · Zbl 0018.34104 · doi:10.1090/S0002-9904-1938-06736-5
[5] DOI: 10.1090/S0002-9947-1939-0000230-5 · JFM 65.0084.02 · doi:10.1090/S0002-9947-1939-0000230-5
[6] DOI: 10.2307/2154571 · Zbl 0799.06019 · doi:10.2307/2154571
[7] DOI: 10.2307/2270905 · Zbl 0181.29904 · doi:10.2307/2270905
[8] P., Acta Sci. Math. (Szeged) 27 pp 63– (1966)
[9] Monteiro A., Buenos Aires) ( pp 129– (1954)
[10] DOI: 10.1007/BF01190439 · Zbl 0806.06011 · doi:10.1007/BF01190439
[11] DOI: 10.2307/1968634 · Zbl 0019.28902 · doi:10.2307/1968634
[12] DOI: 10.1215/S0012-7094-37-00351-X · Zbl 0018.19903 · doi:10.1215/S0012-7094-37-00351-X
[13] DOI: 10.1073/pnas.24.3.162 · Zbl 0018.29003 · doi:10.1073/pnas.24.3.162
[14] DOI: 10.1090/S0002-9947-1939-1501995-3 · JFM 65.0084.01 · doi:10.1090/S0002-9947-1939-1501995-3
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