DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

SJR: 0.203

SNIP: 0.562

MCQ: 0.12

Index Copernicus: 121.02

Discussiones Mathematicae - General Algebra and Applications

PDF

Discussiones Mathematicae General Algebra and Applications 23(2) (2003) 139-148
DOI: https://doi.org/10.7151/dmgaa.1069

LOCALLY FINITE M-SOLID VARIETIES OF SEMIGROUPS

Klaus Denecke and Bundit Pibaljommee

Universität Potsdam, Institut für Mathematik
D-14415 Potsdam, PF 601553, Germany
e-mail:
kdenecke@rz.uni-potsdam.de
     bunpib@rz.uni-potsdam.de

Abstract

An algebra of type τ is said to be locally finite if all its finitely generated subalgebras are finite. A class K of algebras of type τ is called locally finite if all its elements are locally finite. It is well-known (see [2]) that a variety of algebras of the same type τ is locally finite iff all its finitely generated free algebras are finite. A variety V is finitely based if it admits a finite basis of identities, i.e. if there is a finite set σ of identities such that V = ModΣ , the class of all algebras of type τ which satisfy all identities from Σ . Every variety which is generated by a finite algebra is locally finite. But there are finite algebras which are not finitely based. For semigroup varieties, Perkins proved that the variety generated by the five-element Brandt-semigroup
B21 =



0
0
0
0


,

1
0
0
0


,

0
1
0
0


,

0
0
1
0


,

0
0
0
1




is not finitely based ([9], [10]). An identity s ≈ t is called a hyperidentity of a variety V if whenever the operation symbols occurring in s and in t, respectively, are replaced by any terms of V of the appropriate arity, the identity which results, holds in V. A variety V is called solid if every identity of V also holds as a hyperidentity in V. If we apply only substitutions from a set M we speak of M-hyperidentities and M-solid varieties. In this paper we use the theory of M-solid varieties to prove that a type (2) M-solid variety of the form V = HMMod{F(x1,F(x2,x3)) ≈ F(F(x1,x2),x3)} , which consists precisely of all algebras which satisfy the associative law as an M-hyperidentity is locally finite iff the hypersubstitution which maps F to the word x1x2x1 or to the word x2x1x2 belongs to M and that V is finitely based if it is locally finite.

Keywords: locally finite variety, finitely based variety, M-solidvariety.

2000 Mathematics Subject Classification: 08A15, 08B15, 20M01.

 References

[1]Sr. Arworn, Groupoids of Hypersubstitutions and G-Solid Varieties, Shaker-Verlag, Aachen 2000.
[2]S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, Berlin-Heidelberg-New York 1981.
[3]Th. Changphas and K. Denecke, Complexity of hypersubstitutions and lattices of varieties, Discuss. Math. - Gen. Algebra Appl. 23 (2003), 31-43.
[4]K. Denecke and J. Koppitz, M-solid varieties of semigroups, Discuss. Math. - Algebra & Stochastics Methods 15 (1995), 23-41.
[5]K. Denecke, J. Koppitz and N. Pabhapote, The greatest regular-solid variety of semigroups, preprint 2002.
[6]O.G. Kharlampovich and M.V. Sapir, Algorithmic problems in varieties, Internat. J. Algebra Comput. 5 (1995), 379-602.
[7]A.Yu. Olshanski, Geometry of Defining Relations in Groups, (Russian), Izdat. ``Nauka'', Moscow 1989.
[8]G. Paseman, A small basis for hyperassociativity, preprint, University of California, Berkeley, CA, 1993.
[9]P. Perkins, Decision Problems for Equational Theories of Semigroups and General Algebras, Ph.D. Thesis, University of California, Berkeley, CA, 1966.
[10]P. Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298-314.
[11]J. P onka, Proper and inner hypersubstitutions of varieties, p. 106-116 in: Proceedings of the International Conference: ``Summer School on General Algebra and Ordered Sets", Palacký University of Olomouc 1994.
[12]L. Polák, On hyperassociativity, Algebra Universalis, 36 (1996), 363-378.
[13]M. Sapir, Problems of Burnside type and the finite basis property in varieties of semigroups, (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 319-340, English transl. in Math. USSR-Izv. 30 (1988), 295-314.

Received 2 May 2003
Revised 7 July 2003


Close