Varieties of \(\ell\)-groups with the identity \([x^ p,y^ p]=e\) have finite bases. (English. Russian original) Zbl 0544.06013
Algebra Logic 23, 20-35 (1984); translation from Algebra Logika 23, No. 1, 27-47 (1984).
Let p be a prime. Let us denote by \({\mathcal L}_ p\) the variety of lattice ordered groups defined by the identity \([x^ p,y^ p]=e\). This paper contains the following two main results: 1) All proper subvarieties of the variety \({\mathcal L}_ p\) are constructively described. 2) It is proved that each subvariety of \({\mathcal L}_ p\) possesses a finite basis of identities.
Reviewer: J.Jakubík
MSC:
06F15 | Ordered groups |
20F60 | Ordered groups (group-theoretic aspects) |
08B15 | Lattices of varieties |
20E10 | Quasivarieties and varieties of groups |
06B20 | Varieties of lattices |
References:
[1] | V. M. Kopytov and N. Ya. Medvedev, ”On varieties of lattice-ordered groups,” Algebra Logika,16, No. 4, 417–423 (1977). · Zbl 0395.17015 · doi:10.1007/BF01670004 |
[2] | T. Feil, ”An uncountable tower ofl-group varieties,” Algebra Universalis,14, No. 1, 129–131 (1982). · Zbl 0438.06003 · doi:10.1007/BF02483914 |
[3] | N. Ya. Medvedev, ”l-Varieties without an independent basis of identities,” Math. Slovaka,32, No. 4, 417–425 (1982). · Zbl 0503.06018 |
[4] | S. A. Gurchenkov, ”On varieties of nilpotent lattice-ordered groups,” Algebra Logika,21, No. 5, 499–510 (1982). |
[5] | M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of the Theory of Groups [in Russian], 3rd edn, Nauka, Moscow (1982). · Zbl 0508.20001 |
[6] | A. I. Kokorin and V. M. Kopytov, Fully Ordered Groups, Wiley, New York (1974). · Zbl 0192.36401 |
[7] | L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford (1963). · Zbl 0137.02001 |
[8] | E. B. Scrimger, ”A large class of small varieties of lattice-ordered groups,” Proc. Am. Math. Soc.,51, No. 2, 301–306 (1975). · Zbl 0312.06010 · doi:10.1090/S0002-9939-1975-0384644-7 |
[9] | R. D. Bleier, ”The SP-hull of a lattice-ordered group,” Can. J. Math.,26, No. 4, 866–878 (1974). · Zbl 0298.06021 · doi:10.4153/CJM-1974-081-x |
[10] | A. M. W. Glass, W. C. Holland, and S. H. McCleary, ”The structure ofl-group varieties,” Algebra Universalis,10, 1–20 (1980). · Zbl 0439.06013 · doi:10.1007/BF02482885 |
[11] | S. J. Bernau, ”Varieties of lattice groups are closed under -completion,” in: Symposia Mathematica, Vol. 21 (Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975), Academic Press, London (1977), pp. 349–355. |
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.