Identities of the natural representation of the infinitely based semigroup

Authors:
Leonid Al’shanskii and Alexander Kushkuley

Journal:
Proc. Amer. Math. Soc. **118** (1993), 931-937

MSC:
Primary 20M07; Secondary 08B05

MathSciNet review:
1132406

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Abstract: An equational theory of a very small semigroup may fail to be finitely presented. A well-known example of such a semigroup was studied in detail by Peter Perkins some twenty years ago. We prove that the *natural representation* of his semigroup has a finite basis of identical relations and discuss this fact in a general context of universal algebra.

**[1]**Peter Perkins,*Bases for equational theories of semigroups*, J. Algebra**11**(1969), 298–314. MR**0233911****[2]**B. I. Plotkin,*Varietes of representations of groups*, Uspekhi Mat. Nauk.**32**(1977), 3-68. (Russian)**[3]**Yu. P. Razmyslov,*Varieties of representations of finite-dimensional algebras in prime algebras*, Vestnik Moskov. Univ. Ser. I Mat. Mekh.**6**(1982), 31–37, 120 (Russian, with English summary). MR**685260****[4]**P. M. Cohn,*Universal algebra*, Harper & Row, Publishers, New York-London, 1965. MR**0175948****[5]**Nathan Jacobson,*𝑃𝐼-algebras*, Lecture Notes in Mathematics, Vol. 441, Springer-Verlag, Berlin-New York, 1975. An introduction. MR**0369421****[6]**A. N. Trahtman,*A base of identities of the five-element semigroup of Brandt*, Research in Modern Algebra, Sverdlovsk Univ., Sverdlovsk, 1987, pp. 147-149. (Russian)

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1132406-6

Article copyright:
© Copyright 1993
American Mathematical Society