On idempotent, commutative, and nonassociative groupoids
Authors:
G. Grätzer and R. Padmanabhan
Journal:
Proc. Amer. Math. Soc. 28 (1971), 7580
MSC:
Primary 20.95
MathSciNet review:
0276393
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Abstract: For an algebra and for , let denote the number of essentially ary polynomials of . J. Dudek has shown that if is an idempotent and nonassociative groupoid then for all . In this paper this result is improved for the commutative case to show that for such groupoids for all (Theorem 1) and that this is the best possible result. Those groupoids for which this lower bound is attained are completely characterized. In fact, the relevant result proved below is much stronger (Theorem 3). From these and other known results it is deduced that the sequence has the minimal extension property.
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 J. Dudek, The number of algebraic operations in an idempotent groupoid, Colloq. Math 21 (1970), 169177. MR 0263959 (41:8558)
 [2]
 G. Grätzer, Universal algebra, Van Nostrand, Princeton, N.J., 1968. MR 40 #1320. MR 0248066 (40:1320)
 [3]
 , Universal algebra, Trends in Lattice Theory, Van Nostrand, Princeton, N. J., 1969.
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 G. Grätzer and J. Płonka, On the number of polynomials of an idempotent algebra. I, Pacific J. Math. 32 (1970), 697709. MR 0256969 (41:1624)
 [5]
 G. Grätzer, J. Płonka and A. Sekanina, On the number of polynomials of a universal algebra. I, Colloq. Math. 22 (1970), 911. MR 0294216 (45:3289)
 [6]
 J. Płonka, On the number of independent elements in finite abstract algebra having a binary operation, Colloq. Math. 14 (1966), 189201. MR 33 #88. MR 0191861 (33:88)
 [7]
 , On the arity of idempotent reducts of groups, Colloq. Math. 21 (1970), 3537. MR 0253968 (40:7181)
 [8]
 , On algebras with distinct essentially ary operations, Algebra Universalis (to appear).
 [9]
 , On algebras with at most distinct essentially ary operations, Algebra Universalis (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197102763937
PII:
S 00029939(1971)02763937
Keywords:
ary polynomials and essentially ary polynomials of a universal algebra,
representable sequence,
minimal extension property,
idempotent reduct of an abelian group
Article copyright:
© Copyright 1971
American Mathematical Society
