School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China

*e-mail:* `lijr@lzu.edu.cn`

**Abstract.**
It is known that the variety *M _{n}* generated by all monoids of order

**2010 Mathematics Subject Classification.**
20M07.

**Key words and phrases.** Monoid, semigroup, variety, finitely based.

DOI: 10.3336/gm.50.2.08

**References:**

- J. Almeida, Finite semigroups and universal algebra, World Scientific, Singapore, 1994.

MathSciNet - A. P. Birjukov,
*Varieties of idempotent semigroups*, Algebra i Logika**9**(1970), 255-273 (in Russian); English transl.: Algebra and Logic**9**(1970), 153-164.

MathSciNet - A. Distler and J. D. Mitchell, Smallsemi - a GAP package, version 0.6.6, 2013, available at
`http://www.gap-system.org/Packages/smallsemi.html` - C. C. Edmunds,
*On certain finitely based varieties of semigroups*, Semigroup Forum**15**(1977/78), 21-39.

MathSciNet CrossRef - C. C. Edmunds,
*Varieties generated by semigroups of order four*, Semigroup Forum**21**(1980), 67-81.

MathSciNet CrossRef - C. F. Fennemore,
*All varieties of bands. I, II*, Math. Nachr.**48**(1971), 237-252; ibid., 253-262.

MathSciNet CrossRef - J. A. Gerhard,
*The lattice of equational classes of idempotent semigroups*, J. Algebra**15**(1970), 195-224.

MathSciNet CrossRef - M. Jackson,
*Finite semigroups whose varieties have uncountably many subvarieties*, J. Algebra**228**(2000), 512-535.

MathSciNet CrossRef - E. W. H. Lee,
*Hereditarily finitely based monoids of extensive transformations*, Algebra Universalis**61**(2009), 31-58.

MathSciNet CrossRef - E. W. H. Lee,
*Varieties generated by 2-testable monoids*, Studia Sci. Math. Hungar.**49**(2012), 366-389.

MathSciNet CrossRef - E. W. H. Lee,
*Finite basis problem for semigroups of order five or less: generalization and revisitation*, Studia Logica**101**(2013), 95-115.

MathSciNet CrossRef - E. W. H. Lee and J. R. Li,
*Minimal non-finitely based monoids*, Dissertationes Math. (Rozprawy Mat.)**475**(2011), 65 pp.

MathSciNet CrossRef - E. W. H. Lee and W. T. Zhang,
*Finite basis problem for semigroups of order six*, LMS J. Comput. Math.**18**(2015), 1-129

MathSciNet CrossRef - J. R. Li and Y. F. Luo,
*Equational property of certain transformation monoids*, Internat. J. Algebra Comput.**20**(2010), 833-845.

MathSciNet CrossRef - J. R. Li, W. T. Zhang, and Y. F. Luo,
*On the finite basis problem for the variety generated by all*, Algebra Universalis*n*-element semigroups**73**(2015), 225-248.

MathSciNet CrossRef - Y. F. Luo and W. T. Zhang,
*On the variety generated by all semigroups of order three*, J. Algebra**334**(2011), 1-30.

MathSciNet CrossRef - S. Oates and M. B. Powell,
*Identical relations in finite groups*, J. Algebra**1**(1964), 11-39.

MathSciNet CrossRef - P. Perkins,
*Bases for equational theories of semigroups*, J. Algebra**11**(1969), 298-314.

MathSciNet CrossRef - M. V. Sapir,
*Problems of Burnside type and the finite basis property in varieties of semigroups*, Izv. Akad. Nauk SSSR Ser. Mat.**51**(1987), 319-340 (in Russian); English transl.: Math. USSR-Izv.**30**(1988), 295-314.

MathSciNet - L. N. Shevrin and M. V. Volkov,
*Identities of semigroups*, Izv. Vyssh. Uchebn. Zaved. Mat.**1985(11)**, 3-47 (in Russian); English transl.: Soviet Math. (Iz. VUZ)**29(11)**(1985), 1-64.

MathSciNet - The on-line encyclopedia of integer sequences,
`http://oeis.org/A058129` - A. N. Trahtman,
*Finiteness of a basis of identities of five-element semigroups*, in: Semigroups and their homomorphisms (ed. E. S. Lyapin), Ross. Gos. Ped. Univ., Leningrad, 1991, 76-97 (in Russian).

MathSciNet - M. V. Volkov,
*The finite basis property of varieties of semigroups*, Mat. Zametki**45**(1989), 3, 12-23 (in Russian); English transl.: Math. Notes**45**(1989), 187-194.

MathSciNet - M. V. Volkov,
*The finite basis problem for finite semigroups*, Sci. Math. Jpn.**53**(2001), 171-199.

MathSciNet - M. V. Volkov,
*Reflexive relations, extensive transformations and piecewise testable languages of a given height*, Internat. J. Algebra Comput.**14**(2004), 817-827.

MathSciNet