Varieties of commutative semigroups
HTML articles powered by AMS MathViewer
 by Andrzej Kisielewicz PDF
 Trans. Amer. Math. Soc. 342 (1994), 275306 Request permission
Abstract:
In this paper, we describe all equational theories of commutative semigroups in terms of certain wellquasiorderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform solution to the word problems for free objects in all varieties of commutative semigroups.References

A. Ja. Aĭzenštat, On varieties of semigroups having finite number of subvarieties, Algebraic Theory of Semigroups, Colloq. Math. Soc. János Bolyai, vol. 20, Amsterdam, New York, 1979, pp. 3341.
 Jorge Almeida, Some order properties of the lattice of varieties of commutative semigroups, Canad. J. Math. 38 (1986), no. 1, 19–47. MR 835034, DOI 10.4153/CJM1986002x
 J. Almeida and N. R. Reilly, Generalized varieties of commutative and nilpotent semigroups, Semigroup Forum 30 (1984), no. 1, 77–98. MR 759698, DOI 10.1007/BF02573439
 Stanley Burris and Evelyn Nelson, Embedding the dual of $\pi _{m}$ in the lattice of equational classes of commutative semigroups, Proc. Amer. Math. Soc. 30 (1971), 37–39. MR 285639, DOI 10.1090/S00029939197102856390
 Józef Dudek and Andrzej Kisielewicz, Totally commutative semigroups, J. Austral. Math. Soc. Ser. A 51 (1991), no. 3, 381–399. MR 1125441
 Samuel Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1974. MR 0530382
 Trevor Evans, The lattice of semigroup varieties, Semigroup Forum 2 (1971), no. 1, 1–43. MR 284528, DOI 10.1007/BF02572269
 Trevor Evans, Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc. (2) 1 (1969), 399–403. MR 249344, DOI 10.1112/jlms/s21.1.399
 Trevor Evans, Some solvable word problems, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Studies in Logic and the Foundations of Mathematics, vol. 95, NorthHolland, AmsterdamNew York, 1980, pp. 87–100. MR 579941
 Michael R. Garey and David S. Johnson, Computers and intractability, A Series of Books in the Mathematical Sciences, W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NPcompleteness. MR 519066
 J. A. Gerhard and Mario Petrich, Varieties of bands revisited, Proc. London Math. Soc. (3) 58 (1989), no. 2, 323–350. MR 977480, DOI 10.1112/plms/s358.2.323 G. Grätzer, Universal algebra, 2nd ed., SpringerVerlag, Berlin and New York, 1979.
 T. J. Head, The varieties of commutative monoids, Nieuw Arch. Wisk. (3) 16 (1968), 203–206. MR 237696
 P. M. Higgins, The varieties of commutative semigroups for which epis are onto, Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), no. 12, 1–7. MR 700494, DOI 10.1017/S0308210500016073
 Graham Higman, Ordering by divisibility in abstract algebras, Proc. London Math. Soc. (3) 2 (1952), 326–336. MR 49867, DOI 10.1112/plms/s32.1.326
 John R. Isbell, Epimorphisms and dominions, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 232–246. MR 0209202
 Andrzej Kisielewicz, All pseudovarieties of commutative semigroups, Semigroups with applications (Oberwolfach, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 78–89. MR 1197847
 I. O. Korjakov, A sketch of the lattice of commutative nilpotent semigroup varieties, Semigroup Forum 24 (1982), no. 4, 285–317. MR 657908, DOI 10.1007/BF02572774
 Joseph B. Kruskal, The theory of wellquasiordering: A frequently discovered concept, J. Combinatorial Theory Ser. A 13 (1972), 297–305. MR 306057, DOI 10.1016/00973165(72)900635
 Gérard Lallement, Semigroups and combinatorial applications, Pure and Applied Mathematics, John Wiley & Sons, New YorkChichesterBrisbane, 1979. MR 530552
 S. A. Malyšev, The lattice of subvarieties of the variety $\Pi (x_{1}\cdots x_{n}=x_{1}$ $\cdots x_{n}^{k+1},\,xy=yx)$, Semigroup varieties and semigroups of endomorphisms (Russian), Leningrad. Gos. Ped. Inst., Leningrad, 1979, pp. 113–122 (Russian). MR 569923 I. I. Me’lnik, The description of some lattices of semigroup varieties, Izv. Vyssh. Uchebn. Zaved. Mat. 7 (1972), 6774. (Russian)
 Ernst W. Mayr and Albert R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. in Math. 46 (1982), no. 3, 305–329. MR 683204, DOI 10.1016/00018708(82)900482
 Evelyn Nelson, The lattice of equational classes of commutative semigroups, Canadian J. Math. 23 (1971), 875–895. MR 288197, DOI 10.4153/CJM19710980
 Evelyn Nelson, The lattice of equational classes of semigroups with zero, Canad. Math. Bull. 14 (1971), 531–534. MR 313429, DOI 10.4153/CMB19710943
 Hanna Neumann, Varieties of groups, SpringerVerlag New York, Inc., New York, 1967. MR 0215899
 Peter Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298–314. MR 233911, DOI 10.1016/00218693(69)900581
 Mario Petrich, Lectures in semigroups, John Wiley & Sons, LondonNew YorkSydney, 1977. MR 0466270
 J.E. Pin, Varieties of formal languages, Foundations of Computer Science, Plenum Publishing Corp., New York, 1986. With a preface by M.P. Schützenberger; Translated from the French by A. Howie. MR 912694, DOI 10.1007/9781461322153
 Libor Polák, On varieties of completely regular semigroups. I, Semigroup Forum 32 (1985), no. 1, 97–123. MR 803483, DOI 10.1007/BF02575527
 Pavel Pudlák and Jiří T ma, Every finite lattice can be embedded in a finite partition lattice, Algebra Universalis 10 (1980), no. 1, 74–95. MR 552159, DOI 10.1007/BF02482893
 M. V. Sapir, On Cross semigroup varieties and related questions, Semigroup Forum 42 (1991), no. 3, 345–364. MR 1092964, DOI 10.1007/BF02573430
 O. B. Sapir, The axiomatic and the basis series of varieties of commutative semigroups, Ural. Gos. Univ. Mat. Zap. 14 (1988), no. 3, Algebr. Sistemy i ikh Mnogoobr., 112–119, iv (Russian). MR 958331
 L. N. Shevrin and E. V. Sukhanov, Structural aspects of the theory of varieties of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 6 (1989), 3–39 (Russian); English transl., Soviet Math. (Iz. VUZ) 33 (1989), no. 6, 1–34. MR 1017775
 L. N. Shevrin and M. V. Volkov, Identities of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 11 (1985), 3–47, 85 (Russian). MR 829099
 Robert Schwabauer, A note on commutative semigroups, Proc. Amer. Math. Soc. 20 (1969), 503–504. MR 233912, DOI 10.1090/S00029939196902339125
 Robert Schwabauer, Commutative semigroup laws, Proc. Amer. Math. Soc. 22 (1969), 591–595. MR 244419, DOI 10.1090/S00029939196902444193 M. A. Taitslin, Algorithmic problems for commutative semigroups, Soviet. Math. Dokl. 9 (1968), 201204.
 Walter Taylor, Equational logic, Houston J. Math. Survey (1979), iii+83. MR 546853
 B. M. Vernikov and M. V. Volkov, Lattices of nilpotent varieties of semigroups, Ural. Gos. Univ. Mat. Zap. 14 (1988), no. 3, Algebr. Sistemy i ikh Mnogoobr., 53–65, ii (Russian). MR 958324
 M. V. Volkov, Commutative semigroup varieties with distributive subvariety lattices, Contributions to general algebra, 7 (Vienna, 1990) HölderPichlerTempsky, Vienna, 1991, pp. 351–359. MR 1143098 —, Semigroup varieties with modular subvariety lattices, preprint.
Additional Information
 © Copyright 1994 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 342 (1994), 275306
 MSC: Primary 20M07; Secondary 03C05, 08B05, 20M05
 DOI: https://doi.org/10.1090/S00029947199412114110
 MathSciNet review: 1211411