Repnitskij, V. B. Bases of identities of varieties of lattice ordered semigroups. (English. Russian original) Zbl 0542.06006 Algebra Logic 22, 461-472 (1983); translation from Algebra Logika 22, No. 6, 649-665 (1983). The author calls a lattice ordered semigroup S a dld-semigroup, if S is distributive as a lattice and satisfies the identities \(x(y\wedge z)=(xy)\wedge(xz),\quad(x\wedge y)z=(xz)\wedge(yz).\) The following result is proved. The variety generated by any nontrivial commutative cancellative dld-semigroup is not finitely based. Also, an equational base for the variety generated by an infinite cyclic group and for the variety generated by an infinite cyclic semigroup (both ordered naturally) is found. Reviewer: V.Novák Cited in 3 Documents MSC: 06F05 Ordered semigroups and monoids 08B05 Equational logic, Mal’tsev conditions 06B20 Varieties of lattices Keywords:lattice ordered semigroup; variety; commutative cancellative dld- semigroup; equational base PDFBibTeX XMLCite \textit{V. B. Repnitskij}, Algebra Logic 22, 461--472 (1983; Zbl 0542.06006); translation from Algebra Logika 22, No. 6, 649--665 (1983) Full Text: DOI EuDML References: [1] V. B. Repnitskii, ”On varieties of -semigroups,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 11, 54–58 (1982). [2] V. B. Repnitskii, ”On cross and almost-cross varieties of commutative -semigroups,” in: Algebraic Systems and Their Varieties [in Russian], Sverdlovsk (1982), pp. 102–116. [3] V. B. Repnitskii, ”Chain varieties of commutative -semigroups,” in: Algebraic Systems. Varieties. Lattices of Subsystems [in Russian], Sverdlovsk (1983), pp. 132–145. [4] V. B. Repnitskii, ”On lattice ordered semigroups, approximable by linearly ordered semigroups,” in: Ordered Sets and Lattices. Some Algebraic Applications of Lattice Theory [in Russian], No. 8, Saratov (1982), pp. 91–96. [5] V. B. Repnitskii, ”On subdirectly indecomposable lattice ordered semigroups,” Manuscripts deposited at VINITI, No. 2852–82, Dep. (1982). [6] T. Merlier, ”Sur les demi-groupes réticulés et les -demi-groupes,” Semigroup Forum,2, No. 1, 64–70 (1971). · Zbl 0224.06009 · doi:10.1007/BF02572273 [7] K. A. Baker, ”Primitive satisfaction and equational problems for lattices and other algebras,” Trans. Am. Math. Soc.,190, 125–150 (1974). · Zbl 0291.08001 · doi:10.1090/S0002-9947-1974-0349532-4 [8] O. A. Ivanova, ”On verbal sums and the orderability of linear algebras, rings, and universal algebras,” Author’s Abstract of Candidate’s Dissertation, Moscow (1970). 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.