Locally finite M-solid varieties of semigroups

Klaus Denecke ; Bundit Pibaljommee

Discussiones Mathematicae - General Algebra and Applications, Tome 23 (2003), p. 139-148 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $B¹₂ = { \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix}}$ 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 $V = H_{M}Mod{F(x₁,F(x₂,x₃)) ≈ F(F(x₁,x₂),x₃)}$, 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.

Détail

Publié le : 2003-01-01

Zbl 1069.08007

EUDML-ID : urn:eudml:doc:287680

Bibliographie

[000] [1] Sr. Arworn, Groupoids of Hypersubstitutions and G-Solid Varieties, Shaker-Verlag, Aachen 2000.

[001] [2] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, Berlin-Heidelberg-New York 1981. | Zbl 0478.08001

[002] [3] Th. Changphas and K. Denecke, Complexity of hypersubstitutions and lattices of varieties, Discuss. Math. - Gen. Algebra Appl. 23 (2003), 31-43. | Zbl 1069.08005

[003] [4] K. Denecke and J. Koppitz, M-solid varieties of semigroups, Discuss. Math. - Algebra & Stochastics Methods 15 (1995), 23-41. | Zbl 0842.20050

[004] [5] K. Denecke, J. Koppitz and N. Pabhapote, The greatest regular-solid variety of semigroups, preprint 2002. | Zbl 1158.20030

[005] [6] O.G. Kharlampovich and M.V. Sapir, Algorithmic problems in varieties, Internat. J. Algebra Comput. 5 (1995), 379-602. | Zbl 0837.08002

[006] [7] A.Yu. Olshanski, Geometry of Defining Relations in Groups, (Russian), Izdat. 'Nauka', Moscow 1989.

[007] [8] G. Paseman, A small basis for hyperassociativity, preprint, University of California, Berkeley, CA, 1993.

[008] [9] P. Perkins, Decision Problems for Equational Theories of Semigroups and General Algebras, Ph.D. Thesis, University of California, Berkeley, CA, 1966.

[009] [10] P. Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298-314. | Zbl 0186.03401

[010] [11] J. P onka, Proper and inner hypersubstitutions of varieties, Proceedings of the International Conference: 'Summer School on General Algebra and Ordered Sets', Palacký University of Olomouc 1994, 106-116.

[011] [12] L. Polák, On hyperassociativity, Algebra Universalis, 36 (1996), 363-378.

[012] [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.