Acta Mathematica Sinica, English Series

, Volume 29, Issue 3, pp 571–590

# On the finite basis problem for certain 2-limited words

Article

## Abstract

Let X* be a free monoid over an alphabet X and W be a finite language over X. Let S(W) be the Rees quotient X*/I(W), where I(W) is the ideal of X* consisting of all elements of X* that are not subwords of W. Then S(W) is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S(W) is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.

### Keywords

Finite basis problem 2-limited words discrete syntactic monoid

20M07 08B05

## Supplementary material

10114_2012_193_MOESM1_ESM.tex (80 kb)
Supplementary material, approximately 80.0 KB.

### References

1. [1]
McKenzie, R.: Tarski’s finite basis problem is undecidable. Internat. J. Algebra Comput., 6, 49–104 (1996)
2. [2]
Edmunds, C. C.: On certain finitely based varieties of semigoups. Semigroup Forum, 15, 21–39 (1977)
3. [3]
Edmunds, C. C.: Varieties generated by semigroups of order four. Semigroup Forum, 21, 67–81 (1980)
4. [4]
Jackson, M.: Finite semigroups whose varieties have uncountably many subvarieties. J. Algebra, 228, 512–535 (2000)
5. [5]
Lee, E. W. H.: Subvarieties of the variety generated by the five-element Brandt semigroup. Internat. J. Algebra Comput., 16(2), 417–441 (2006)
6. [6]
Lee, E. W. H.: Identity bases for some non-exact varieties. Semigroup Forum., 68, 445–457 (2004)
7. [7]
Perkins, P.: Bases for equational theories of semigroups. J. Algebra, 11, 298–314 (1969)
8. [8]
Sapir, O.: Finitely based words. Internat. J. Algebra Comput., 10, 457–480 (2000)
9. [9]
Volkov, M. V.: The finite basis problem for finite semigroups. Sci. Math. Jpn., 53, 171–199 (2001)
10. [10]
Zhang, W. T., Luo, Y. F.: The subvariety lattice of the join of two semigroup varieties. Acta Mathematica Sinica, English Series, 25(6), 971–982 (2009)
11. [11]
Jackson, M., Sapir, O.: Finitely based, finite sets of words. Internat. J. Algebra Comput., 10, 683–708 (2000)
12. [12]
Jackson, M.: On the finite basis problem for finite Rees quotients of free monoids. Acta Sci. Math. (Szeged) 67, 121–159 (2001)
13. [13]
Shevrin, L. N., Volkov, M. V.: Identities of semigroups (in Russian). Izv. Vyssh. Uchebn. Zaved. Mat., 11, 3–47 (1985)
14. [14]
Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra, Springer-Verlag, New York, 1981
15. [15]
Howie, J. M.: Fundamentals of Semigroup Theory, Charendon Press, Oxford, 1995
16. [16]
Jackson, M.: Finiteness properties of varieties and the restriction to finite algebras. Semigroup Forum, 70, 159–187 (2005)
17. [17]
Jackson, M.: Finite semigroups with infinite irredundant identity bases. Internat. J. Algebra Comput., 15, 405–422 (2005)