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Polynomials in idempotent commutative groupoids

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Rozprawy Matematyczne tom/nr w serii: 286 wydano: 1989
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EN

In this paper we investigate the variety of idempotent commutative groupoids. In particular, we improve the results of Grätzer and Padmanabhan on the number $p_n$ of essentially n-ary polynomials in idempotent commutative groupoids. They have shown that if an idempotent commutative groupoid (G,•) is different from a semilattice, then
$p_n(G,•) ≥ 1/3(2^n - (-1)^n)$
for all n. Moreover, they have proved that the equality is achieved if and only if (G,•) is polynomially equivalent to an affine space over GF(3).
We prove that if (G,•) is different from a semilattice and not polynomially equivalent to an affine space over GF(3), then
$p_n(G,•) ≥ 3^(n-1)$
for all n ≥ 4. Also, we give a complete characterization of those groupoids for which the lower bound is attained. These results we obtain by detailed analysis of the variety of idempotent commutative groupodis, proving a series of theorems and lemmas which give an insight into the complexity of this variety.
EN

CONTENTS
1. Introduction......................................................................5
2. Terminology......................................................................7
3. Applied results.................................................................8
4. Nonmedial groupoids.....................................................10
5. Sterner quasigroups......................................................13
6. Near-semilattices...........................................................17
7. Totally commutative groupoids.......................................21
8. Some lemmas on idempotent algebras..........................27
9. Ternary polynomials.......................................................29
10. Nonmedial groupoids (continued)................................33
11. Medial groupoids..........................................................39
12. Proof of Theorem 1......................................................48
13. Proof of Theorem 2......................................................48
References........................................................................52
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 286
Liczba stron
52
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCLXXXVI
Daty
wydano
1989
Twórcy
autor
  • Instytut Matematyczny Uniwersytetu Wrocławskiego, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
  • Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
  • [1] J. Berman, Free spectra of 3-element algebras, Universal Algebra and Lattice Theory, Proceedings, Puebla 1982, R. S. Freese and O. C. Garcia, eds., Springer-Verlag, Lecture Notes in Math. 1004, 1983, 10-51.
  • [2] J. Berman, Algebraic properties of k-valued logics, Proceedings of the Tenth International Symposium on Multiple-valued Logic, Evanston Illinois, 1980, 195-204.
  • [3] A. H. Clifford, Semigroups admitting relative inverses, Ann. of Math. 42 (4) (1941).
  • [4] B. Csákány, On affine spaces over prime fields, Acta Sci. Math., 37 (1975), 33-36.
  • [5] J. Dudek, A characterization of some idempotent abelian groupoids, Colloq. Math., 30 (1974), 219-223.
  • [6] J. Dudek, Medial groupoids and Mersenne numbers, Fund. Math., 114 (1981), 109-112.
  • [7] J. Dudek, On binary polynomials in idempotent commutative groupoids, Fund. Math., 120 (1984), 187-191.
  • [8] J. Dudek, Varieties of idempotent commutative groupoids. Fund. Math., 120 (1984), 193-204.
  • [9] J. Dudek, A characterization of distributive lattices, Colloquia Mathematica Societatis János Bolyai, 33, Contributions to Lattice Theory, Szeged (Hungary), 1980 (North-Holland 1983), 325-335.
  • [10] J. Dudek and E. Graczyńska, The lattice of varieties of algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Phys. Astronom., 29 (1981), 337-340.
  • [11] G. Grätzer, Universal Algebra, Van Nostrand (1979).
  • [12] G. Grätzer, Composition of functions, Proceedings of the Conference on Universal Algebra, Queen's University (Kingston, Ont., 1970), 1-106.
  • [13] G. Grätzer, Universal Algebra and Lattice Theory: A story and three research problems, Universal Algebra and its Links with Logic, Algebra, Combinatorics and Computer Science, Proceedings "25 Arbeitstagung über Allgemeine Algebra", Darmstadt 1983, P. Burmeister et. al. (eds.) Copyright Helderman Verlag 1984, 1-13.
  • [14] G, Grätzer and R. Padmanabhan, On commutative idempotent and nonassociative groupoids, Proc. Amer. Math. Soc., 28 (1971), 75-78.
  • [15] G. Grätzer and J. Płonka, On the number of polynomials of an idempotent algebra I, Pacific J. Math., 22 (1970), 697-709.
  • [16] E. Marczewski, Independence and homomorphisms in abstract algebras. Fund. Math., 50 (1961), 45-61.
  • [17] E. Marczewski, Independence in abstract algebras. Results and Problems, Colloq. Math., 14 (1966), 169-188.
  • [18] R. McKenzie, Abstracts, Notices Amer. Math. Soc., June, 1983, pp. 315.
  • [19] J. Pionka, On equational classes of abstract algebras defined by regular identities, Fund. Math., 64 (1969), 241-247.
  • [20] J. Pionka, On the arity of idempotent reduct of groups, Colloq. Math., 21 (1970), 35-37.
  • [21] J. Pionka, On free algebras and algebraic decomposition of algebras from some equational classes defined by regular equations, Algebra Universalis 1 (1971), 261-264.
  • [22] W. Taylor, Equational logic, Houston J. Math., Survey, 1979, 1-83.
Języki publikacji
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Uwagi
Identyfikator YADDA
bwmeta1.element.zamlynska-e19ebdc7-ed54-4052-8ce7-5365e2b1e70b
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ISBN
83-01-09221-1
ISSN
0012-3862
Kolekcja
DML-PL
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