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
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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.
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Received 2 May 2003
Revised 7 July 2003