Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T23:23:50.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Medial idempotent groupoids III

Part of: Algebraic structures Other generalizations of groups

Published online by Cambridge University Press:  09 April 2009

Jung R. Cho  and
Józef Dudek
Jung R. Cho
Affiliation:
Department of Mathematics Pusan National universityPusan 609-735Korea e-mail: [email protected]
Józef Dudek
Affiliation:
Mathematical Institute University of Wroclawpl. Grunwaldzki 2/4 50-384 WroclawPoland e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

As a sequel to the previous two papers of the second author, we investigate the structure of medial idempotent groupoids by Pn-sequences. To complete the series of research, this paper has theree purposes. First, we summarize some results in the previous papers so that this paper can cover the materials presented there. Secondly, using earlier results, we prove a few theorems which show the importance of the medial law in controlling the growth of Pn-sequences of groupoids. Finally, we state some problems and conjectures raised during the series of research.

MSC classification

Secondary: 08A40: Operations, polynomials, primal algebras 20N02: Sets with a single binary operation (groupoids)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 3 , June 2000 , pp. 312 - 320
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Cho, J. R., Varieties of medial algebras (Ph. D. Thesis, Emory University, Atlanta, 1986).Google Scholar
[2]Csákány, B., ‘On affine spaces over prime fields’, Acta Sci. Math. (Szeged) 37 (1975), 33–36.Google Scholar
[3]Crvenković, S. and Dudek, J., ‘Rectangular groupoids’, Czechoslovak Math. J. 35 (1985), 405–414.CrossRefGoogle Scholar
[4]Dudek, J., ‘Medial groupoids and Mersenne's numbers’, Fund. Math. 114 (1981), 109–112.CrossRefGoogle Scholar
[5]Dudek, J., ‘On binary polynomials in idempotent commutative groupoids’, Fund. Math. 120 (1984), 187–191.CrossRefGoogle Scholar
[6]Dudek, J., ‘Varieties of idempotent commutative groupoids’, Fund. Math. 120 (1984), 193–204.CrossRefGoogle Scholar
[7]Dudek, J., ‘Polynomial characterization of some idempotent algebras’, Acta Sci. Math. (Szeged) 50 (1986), 39–49.Google Scholar
[8]Dudek, J., ‘Polynomials in idempotent commutative groupoids’, Dissertationes Math. 286 (1989), 1–55.Google Scholar
[9]Dudek, J., ‘The unique minimal clone with three essentially binary operations’, Algebra Universalis 27 (1990), 201–209.CrossRefGoogle Scholar
[10]Dudek, J., ‘Medial idempotent groupoids I’, Czechoslovak Math. J. 41 (1991), 249–259.CrossRefGoogle Scholar
[11]Dudek, J., ‘On Csákány's problem concerning affine spaces’, Acta Sci. Math. (Szeged) 46 (1992), 3–13.Google Scholar
[12]Dudek, J., ‘Medial idempotent groupoids II’, in: Proceeding of the workshop on general algebra, Contribution to general algebras 9 (Linz, 1994), pp. 133–150.Google Scholar
[13]Dudek, J., ‘On varieties of groupoid modes’, Demonstration Math. 27 (1994), 815–818.Google Scholar
[14]Dudek, J., ‘Another unique minimal clone’, preprint.Google Scholar
[15]Dudek, J. and Kisielewicz, A., ‘Idempotent algebras with log-linear free spectra’, Algebra Universalis 28 (1991), 119–127.CrossRefGoogle Scholar
[16]Grätzer, G., Universal algebra, 2nd edition, (Springer, New York, 1979).CrossRefGoogle Scholar
[17]Grätzer, G. and Kisielewicz, A., ‘A survey of some open problems on p n-sequences and free spectra of algebras and varieties’, in: Universal algebra and quasigroup theory (eds. Romanowska, A. and Smith, J. D. H.), (Helderman, Berlin, 1992), pp. 57–88.Google Scholar
[18]Grätzer, G. and Padmanabhan, R., ‘On commutative, idempotent and nonassociative groupoids’, Proc. Amer. Math. Soc. 28 (1971), 75–78.CrossRefGoogle Scholar
[19]Ježek, J. and Kepka, T., ‘Medial groupoids’, Rozpravy ČSAV, Řada Mat. Při´rod. Věd 93 (1983).Google Scholar
[20]Kisielewicz, A., ‘The p n-sequences of idempotent algebras are strictly increasing II’, Algebra Universalis 27 (1990), 261–269.Google Scholar
[21]Lévai, L. and Pálfy, P., ‘On binary minimal clones’, Acta Cybernet. 12/3 (1996), 279–294.Google Scholar
[22]Ostermann, F. and Schmidt, J., ‘Der baryzentrische Kalkül als axiomatische Grundlage der affinen Geometrie’, J. Reine Angew. Math. 224 (1966), 44–57.CrossRefGoogle Scholar
[23]Plonka, J., ‘On equational classes of abstract algebras defined by regular equations’, Fund. Math. 64 (1969), 241–247.CrossRefGoogle Scholar
[24]Plonka, J., ‘On algebras with n distinct essentially n-ary operations’, Algebra Universalis 1 (1971), 73–79.CrossRefGoogle Scholar
[25]Plonka, J., ‘R-prime idempotent reduct of abelian groups’, Archiv. Math. (Basel) 24 (1973), 129–132.CrossRefGoogle Scholar
[26]Romanowska, A. and Smith, J. D. H., Modal theory, an algebraic approach to order, geometry and convexity (Helderman, Berlin, 1985).Google Scholar
[27]Soublin, J., ‘Étude algébrique de la notion de moyenne’, J. Math. Pures Appl. 50 (1971), 53–264.Google Scholar