A complete description is given for semigroup varieties, which are distributive elements of the lattice of all semigroup varieties.
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References
L. N. Shevrin, B. M. Vernikov, and M. V. Volkov, “Lattices semigroup varieties,” Izv. Vyssh. Uch. Zav., Mat., No. 3, 3–36 (2009).
G. Grätzer, General Lattice Theory, Academie-Verlag, Berlin (1978).
M. V. Volkov, “Modular elements of the lattice of semigroup varieties,” Contr. Gen. Alg., 16, 275–288 (2005).
B. M. Vernikov, “Special elements in the lattice of overcommutative semigroup varieties,” Mat. Zametki, 70, No. 5, 670–678 (2001).
V. Yu. Shaprynskǐ and B. M. Vernikov, “Special elements in the lattice of overcommutative semigroup varieties revisited,” Order, submitted.
B. M. Vernikov, “Upper-modular elements of the lattice of semigroup varieties. II,” Fundam. Prikl. Mat., 14, No. 7, 43–51 (2008).
J. Ježek and R. N. McKenzie, “Definability in the lattice of equational theories of semigroups,” Semigroup Forum, 46, No. 2, 199–245 (1993).
B. M. Vernikov, “On modular elements of the lattice of semigroup varieties,” Comment. Math. Univ. Carol., 48, No. 4, 595–606 (2007).
B. M. Vernikov, “Lower-modular elements of the lattice of semigroup varieties,” Semigroup Forum, 75, No. 3, 554–566 (2007).
B. M. Vernikov, “Lower-modular elements of the lattice of semigroup varieties. II,” Acta Sci. Math. (Szeged), 74, Nos. 3/4, 539–556 (2008).
B. M. Vernikov, “Upper-modular elements of the lattice of semigroup varieties,” Alg. Univ., 59, Nos. 3/4, 405–428 (2008).
B. M. Vernikov and M. V. Volkov, “Modular elements of the lattice of semigroup varieties. II,” Contr. Gen. Alg., 17, 173–190 (2006), 173—190.
B. M. Vernikov and M. V. Volkov, “Lattices of nilpotent semigroup varieties,” in Algebraic Systems and Their Varieties, Ural State Univ., Sverdlovsk (1988), pp. 53–65.
´E. A. Golubov and M. V. Sapir, “Finitely approximable semigroup varieties,” Izv. Vyssh. Uch. Zav., Mat., No. 11, 21–29 (1982).
M. V. Volkov, “Semigroup varieties with a modular lattice of subvarieties,” Izv. Vyssh. Uch. Zav., Mat., No. 6, 51–60 (1989).
T. J. Head, “The varieties of commutative monoids,” Nieuw Arch. Wiskd., III. Ser., 16, No. 3, 203–206 (1968).
M. V. Sapir, “On cross semigroup varieties and related questions,” Semigroup Forum, 42, No. 3, 345–364 (1991).
M. V. Sapir and E. V. Sukhanov, “Varieties of periodic semigroups,” Izv. Vyssh. Uch. Zav., Mat., No. 4, 40–55 (1981).
M. S. Putcha and A. Yaqub, “Semigroups satisfying permutation identities,” Semigroup Forum, 3, No. 1, 68–73 (1971).
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Supported by RFBR (project No. 09-01-12142) and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project No. 2.1.1/3537). (B. M. Vernikov and V. Yu. Shaprynskii)
Translated from Algebra i Logika, Vol. 49, No. 3, pp. 303–330, May-June, 2010.
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Vernikov, B.M., Shaprynskii, V.Y. Distributive elements of the lattice of semigroup varieties. Algebra Logic 49, 201–220 (2010). https://doi.org/10.1007/s10469-010-9090-9
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DOI: https://doi.org/10.1007/s10469-010-9090-9