Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-21T23:31:46.399Z Has data issue: false hasContentIssue false
  • English
  • Français

The pseudovariety of semigroups of triangular matrices over a finite field

Published online by Cambridge University Press:  15 March 2005

Jorge Almeida ,
Stuart W. Margolis  and
Mikhail V. Volkov
Jorge Almeida
Affiliation:
Departamento de Matemática Pura, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal; jalmeida@fc.up.pt
Stuart W. Margolis
Affiliation:
Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel; margolis@math.biu.ac.il
Mikhail V. Volkov
Affiliation:
Department of Mathematics and Mechanics, Ural State University, 620083 Ekaterinburg, Russia; Mikhail.Volkov@usu.ru
Get access

Abstract

We show that semigroups representable by triangular matrices over a fixed finite field form a decidable pseudovariety and provide a finite pseudoidentity basis for it.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 1: Imre Simon, the tropical computer scientist , January 2005 , pp. 31 - 48
Copyright
© EDP Sciences, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Almeida, J., Implicit operations on finite J-trivial semigroups and a conjecture of I. Simon. J. Pure Appl. Algebra 69 (1990) 205–218. CrossRef
J. Almeida, Finite Semigroups and Universal Algebra. World Scientific (1995).
Almeida, J. and Azevedo, A., Globals of pseudovarieties of commutative semigroups: the finite basis problem, decidability, and gaps. Proc. Edinburgh Math. Soc. 44 (2001) 27–47. CrossRef
Almeida, J. and Volkov, M.V., Profinite identities for finite semigroups whose subgroups belong to a given pseudovariety. J. Algebra Appl. 2 (2003) 137–163. CrossRef
A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups. Amer. Math. Soc. Vol. I (1961); Vol. II (1967).
Cohen, R.S. and Brzozowski, J.A., Dot-depth of star-free events. J. Comp. Syst. Sci. 5 (1971) 1–15. CrossRef
S. Eilenberg, Automata, Languages and Machines. Academic Press, Vol. A (1974); Vol. B (1976).
Eilenberg, S. and Schützenberger, M.P., On pseudovarieties. Adv. Math. 19 (1976) 413–418. CrossRef
D. Gorenstein, Finite Groups. 2nd edition, Chelsea Publishing Company (1980).
Guralnick, R.M., Triangularization of sets of matrices. Linear Multilinear Algebra 9 (1980) 133–140. CrossRef
K. Henckell and J.-E. Pin, Ordered monoids and J-trivial monoids, in Algorithmic problems in groups and semigroups, edited by J.-C. Birget, S. Margolis, J. Meakin and M. Sapir. Birkhäuser (2000) 121–137.
Higgins, P., A proof of Simon's theorem on piecewise testable languages. Theor. Comp. Sci. 178 (1997) 257–264. CrossRef
Kolchin, E.R., On certain concepts in the theory of algebraic matrix groups. Ann. Math. 49 (1948) 774–789. CrossRef
G. Lallement, Semigroups and Combinatorial Applications. John Wiley & Sons (1979).
H. Neumann, Varieties of groups. Springer-Verlag (1967).
J. Okniński, Semigroup of Matrices. World Scientific (1998).
J.-E. Pin, Variétés de langages formels. Masson, 1984 [French; Engl. translation: Varieties of formal languages. North Oxford Academic (1986) and Plenum (1986)].
J.-E. Pin and H. Straubing, Monoids of upper triangular matrices, in Semigroups. Structure and Universal Algebraic Problems, edited by G. Pollák, Št. Schwarz and O. Steinfeld. Colloquia Mathematica Societatis János Bolyai 39, North-Holland (1985) 259–272.
H. Radjavi and P. Rosenthal, Simultaneous Triangularization. Springer-Verlag (2000).
Reiterman, J., The Birkhoff theorem for finite algebras. Algebra Universalis 14 (1982) 1–10. CrossRef
I. Simon, Hierarchies of Events of Dot-Depth One. Ph.D. Thesis, University of Waterloo (1972).
I. Simon, Piecewise testable events, in Proc. 2nd GI Conf. Lect. Notes Comp. Sci. 33 (1975) 214–222.
Stern, J., Characterization of some classes of regular events. Theor. Comp. Sci. 35 (1985) 17–42. CrossRef
Straubing, H., On finite J-trivial monoids. Semigroup Forum 19 (1980) 107–110. CrossRef
Straubing, H., Finite semigroup varieties of the form V ∗ D. J. Pure Appl. Algebra 36 (1985) 53–94. CrossRef
Straubing, H. and Thérien, D., Partially ordered finite monoids and a theorem of I. Simon. J. Algebra 119 (1988) 393–399. CrossRef
Thérien, D., Classification of finite monoids: the language approach. Theor. Comp. Sci. 14 (1981) 195–208. CrossRef
D. Thérien, Subword counting and nilpotent groups, in Combinatorics on Words, Progress and Perspectives, edited by L.J. Cummings. Academic Press (1983) 297–305.
Volkov, M.V., On a class of semigroup pseudovarieties without finite pseudoidentity basis. Int. J. Algebra Computation 5 (1995) 127–135. CrossRef
Volkov, M.V. and Goldberg, I.A., Identities of semigroups of triangular matrices over finite fields. Mat. Zametki 73 (2003) 502–510 [Russian; Engl. translation: Math. Notes 73 (2003) 474–481]. CrossRef