Finite basis problem for 2-testable monoids

Edmond Lee

Open Mathematics (2011)

  • Volume: 9, Issue: 1, page 1-22
  • ISSN: 2391-5455

Abstract

top
A monoid S 1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S 1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.

How to cite

top

Edmond Lee. "Finite basis problem for 2-testable monoids." Open Mathematics 9.1 (2011): 1-22. <http://eudml.org/doc/269230>.

@article{EdmondLee2011,
abstract = {A monoid S 1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S 1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.},
author = {Edmond Lee},
journal = {Open Mathematics},
keywords = {Semigroups; Monoids; Varieties; Finitely based; Hereditarily finitely based; varieties of monoids; varieties of semigroups; finite basis problem; hereditarily finitely based monoids; inherently non-finitely based semigroups},
language = {eng},
number = {1},
pages = {1-22},
title = {Finite basis problem for 2-testable monoids},
url = {http://eudml.org/doc/269230},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Edmond Lee
TI - Finite basis problem for 2-testable monoids
JO - Open Mathematics
PY - 2011
VL - 9
IS - 1
SP - 1
EP - 22
AB - A monoid S 1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S 1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.
LA - eng
KW - Semigroups; Monoids; Varieties; Finitely based; Hereditarily finitely based; varieties of monoids; varieties of semigroups; finite basis problem; hereditarily finitely based monoids; inherently non-finitely based semigroups
UR - http://eudml.org/doc/269230
ER -

References

top
  1. [1] Almeida J., Finite Semigroups and Universal Algebra, Series in Algebra, 3, World Scientific, Singapore, 1994 Zbl0844.20039
  2. [2] Burris S., Sankappanavar H.P., A Course in Universal Algebra, Grad. Texts in Math., 78, Springer, New York-Heidelberg-Berlin, 1981 Zbl0478.08001
  3. [3] Edmunds C.C., On certain finitely based varieties of semigroups, Semigroup Forum, 1977, 15(1), 21–39 http://dx.doi.org/10.1007/BF02195732 Zbl0375.20057
  4. [4] Edmunds C.C., Varieties generated by semigroups of order four, Semigroup Forum, 1980, 21(1), 67–81 http://dx.doi.org/10.1007/BF02572537 Zbl0449.20060
  5. [5] Hall T.E., Kublanovskii S.I., Margolis S., Sapir M.V., Trotter P.G., Algorithmic problems for finite groups and finite 0-simple semigroups, J. Pure Appl. Algebra, 1997, 119(1), 75–96 http://dx.doi.org/10.1016/S0022-4049(96)00050-3 Zbl0880.20040
  6. [6] Lee E.W.H., Identity bases for some non-exact varieties, Semigroup Forum, 2004, 68(3), 445–457 http://dx.doi.org/10.1007/s00233-003-0029-5 Zbl1059.20054
  7. [7] Lee E.W.H., Subvarieties of the variety generated by the five-element Brandt semigroup, Internat. J. Algebra Comput., 2006, 16(2), 417–441 http://dx.doi.org/10.1142/S0218196706002998 Zbl1103.20051
  8. [8] Lee E.W.H., On identity bases of exclusion varieties for monoids, Comm. Algebra, 2007, 35(7), 2275–2280 http://dx.doi.org/10.1080/00927870701328722 Zbl1125.20044
  9. [9] Lee E.W.H., Combinatorial Rees-Sushkevich varieties are finitely based, Internat. J. Algebra Comput., 2008, 18(5), 957–978 http://dx.doi.org/10.1142/S0218196708004755 Zbl1160.20057
  10. [10] Lee E.W.H., On the variety generated by some monoid of order five, Acta Sci. Math. (Szeged), 2008, 74(3–4), 509–537 
  11. [11] Lee E.W.H., Hereditarily finitely based monoids of extensive transformations, Algebra Universalis, 2009, 61(1), 31–58 http://dx.doi.org/10.1007/s00012-009-0001-7 Zbl1184.20048
  12. [12] Lee E.W.H., Combinatorial Rees-Sushkevich varieties that are Cross, finitely generated, or small, Bull. Aust. Math. Soc., 2010, 81(1), 64–84 http://dx.doi.org/10.1017/S0004972709000616 Zbl1192.20044
  13. [13] Lee E.W.H., Varieties generated by 2-testable monoids (submitted) Zbl1274.20057
  14. [14] Lee E.W.H., Li J.R., Minimal non-finitely based monoids (submitted) 
  15. [15] Lee E.W.H., Volkov M.V., On the structure of the lattice of combinatorial Rees-Sushkevich varieties, In: Semigroups and Formal Languages, Lisboa, 2005, World Scientific, Singapore, 2007, 164–187 
  16. [16] Lee E.W.H., Volkov M.V., Limit varieties generated by completely 0-simple semigroups, Internat. J. Algebra Comput. (in press) 
  17. [17] Luo Y.F., Zhang W.T., On the variety generated by all semigroups of order three (submitted) Zbl1243.20071
  18. [18] Perkins P., Bases for equational theories of semigroups, J. Algebra, 1969, 11(2), 298–314 http://dx.doi.org/10.1016/0021-8693(69)90058-1 
  19. [19] Sapir M.V., Problems of Burnside type and the finite basis property in varieties of semigroups, Math. USSR, Izv., 1988, 30(2), 295–314 http://dx.doi.org/10.1070/IM1988v030n02ABEH001012 
  20. [20] Shevrin L.N., Volkov M.V., Identities of semigroups, Soviet Math. (Iz. VUZ), 1985, 29(11), 1–64 Zbl0629.20029
  21. [21] Torlopova N.G., Varieties of quasi-orthodox semigroups, Acta Sci. Math. (Szeged), 1984, 47(3–4), 297–301 (in Russian) Zbl0577.20052
  22. [22] Trahtman A.N., The finite basis question for semigroups of order less than six, Semigroup Forum, 1983, 27(1), 387–389 http://dx.doi.org/10.1007/BF02572749 
  23. [23] Trahtman A.N., Some finite infinitely basable semigroups, Ural. Gos. Univ. Mat. Zap., 1987, 14(2), 128–131 (in Russian) 
  24. [24] Trahtman A.N., Identities of a five-element 0-simple semigroup, Semigroup Forum, 1994, 48(3), 385–387 http://dx.doi.org/10.1007/BF02573687 
  25. [25] Trahtman A.N., Identities of locally testable semigroups, Comm. Algebra, 1999, 27(11), 5405–5412 http://dx.doi.org/10.1080/00927879908826762 Zbl0939.20058
  26. [26] Volkov M.V., The finite basis question for varieties of semigroups, Math. Notes, 1989, 45(3), 187–194 
  27. [27] Volkov M.V., The finite basis problem for finite semigroups, Sci. Math. Jpn., 2001, 53(1), 171–199 Zbl0990.20039
  28. [28] Zhang W.T., Some Studies on Varieties Generated by Finite Semigroups and Their Subvarieties Lattices, PhD thesis, Lanzhou University, Gansu, China, 2009 (in Chinese) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.