Varieties of commutative semigroups

Kisielewicz, Andrzej

doi:10.1090/S0002-9947-1994-1211411-0

Varieties of commutative semigroups
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Trans. Amer. Math. Soc. 342 (1994), 275-306 Request permission

Abstract:

In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings 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

On varieties of semigroups having finite number of subvarieties

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/CJM-1986-002-x
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/S0002-9939-1971-0285639-0
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/s2-1.1.399
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, North-Holland, Amsterdam-New 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 NP-completeness. 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/s3-58.2.323

Universal algebra

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. 1-2, 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/s3-2.1.326
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 well-quasi-ordering: A frequently discovered concept, J. Combinatorial Theory Ser. A 13 (1972), 297–305. MR 306057, DOI 10.1016/0097-3165(72)90063-5
Gérard Lallement, Semigroups and combinatorial applications, Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 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

The description of some lattices of semigroup varieties

7

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 10.1016/0001-8708(82)90048-2
Evelyn Nelson, The lattice of equational classes of commutative semigroups, Canadian J. Math. 23 (1971), 875–895. MR 288197, DOI 10.4153/CJM-1971-098-0
Evelyn Nelson, The lattice of equational classes of semigroups with zero, Canad. Math. Bull. 14 (1971), 531–534. MR 313429, DOI 10.4153/CMB-1971-094-3
Hanna Neumann, Varieties of groups, Springer-Verlag 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 10.1016/0021-8693(69)90058-1
Mario Petrich, Lectures in semigroups, John Wiley & Sons, London-New York-Sydney, 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 10.1007/978-1-4613-2215-3
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/S0002-9939-1969-0233912-5
Robert Schwabauer, Commutative semigroup laws, Proc. Amer. Math. Soc. 22 (1969), 591–595. MR 244419, DOI 10.1090/S0002-9939-1969-0244419-3

Algorithmic problems for commutative semigroups

9

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) Hölder-Pichler-Tempsky, Vienna, 1991, pp. 351–359. MR 1143098

Semigroup varieties with modular subvariety lattices