| Title:
|
Small idempotent clones. I (English) |
| Author:
|
Dudek, Józef |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
105-118 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning $p_n$-sequences and free spectra of algebras to the topic “Small idempotent clones” (see Section 6 of [18]). Many authors, e.g., [8], [14, 15], [22], [25] and [29, 30] were interested in $p_n$-sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids $(G,\cdot )$ with $p_2(G,\cdot )\le 2$ (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26, 27], [25], [28, 30, 31, 32] and [34]. (English) |
| MSC:
|
08A40 |
| MSC:
|
08B05 |
| MSC:
|
20M07 |
| MSC:
|
20N02 |
| idZBL:
|
Zbl 0931.20055 |
| idMR:
|
MR1614013 |
| . |
| Date available:
|
2009-09-24T10:11:40Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127403 |
| . |
| Reference:
|
[1] J. Berman: Free spectra of 3-element algebras.Universal Algebra and Lattice Theory (Puebla, 1982). Lecture Notes in Math., 1004, Springer-Verlag, Berlin, New York, 1983, pp. 10–53. Zbl 0518.08010, MR 0716173 |
| Reference:
|
[2] S. Crvenkovi and J. Dudek: Rectangular groupoids.Czechoslovak Math. J. 35 (1985), 405–414. MR 0803035 |
| Reference:
|
[3] B. Csákány: On affine spaces over prime fields.Acta Sci. Math. (Szeged) 37 (1975), 33–36. MR 0401609 |
| Reference:
|
[4] B. Csákány: All minimal clones on the three-element set.Acta Cybernet. 6 (1983), 227–238. MR 0725722 |
| Reference:
|
[5] J. Dudek: Some remarks on distributive groupoids.Czechoslovak Math. J. 31 (1981), 58–64. Zbl 0472.20025, MR 0626918 |
| Reference:
|
[6] J. Dudek: On binary polynomials in idempotent commutative groupoids.Fund. Math. 120 (1984), 187–191. Zbl 0555.20035, MR 0755775, 10.4064/fm-120-3-187-191 |
| Reference:
|
[7] J. Dudek: Varieties of idempotent commutative groupoids.Fund. Math. 120 (1984), 193–204. Zbl 0546.20049, MR 0755776, 10.4064/fm-120-3-193-204 |
| Reference:
|
[8] J. Dudek: Polynomial characterization of some idempotent algebras.Acta Sci. Math. 50 (1986), 39–49. Zbl 0616.08011, MR 0862179 |
| Reference:
|
[9] J. Dudek: On the minimal extension of sequences.Algebra Universalis 23 (1986), 308–312. Zbl 0627.08001, MR 0903935, 10.1007/BF01230623 |
| Reference:
|
[10] J. Dudek: Polynomials in idempotent commutative groupoids.Dissertationes Math. 286 (1989), 1–55. Zbl 0687.08003, MR 1001646 |
| Reference:
|
[11] J. Dudek: $p_n$-sequences. The minimal extension of sequences. Abstract.Presented at the Conference on Logic and Algebra dedicated to Roberto Magari, on his 60$^{\mathrm th}$ Birthday, Pontignano (Siena) 26–30 April 1994. |
| Reference:
|
[12] J. Dudek and J. Tomasik: Affine spaces over GF(4).(to appear). MR 1408727 |
| Reference:
|
[13] B. Ganter and H. Werner: Equational classes of Steiner systems.Algebra Universalis 5 (1975), 125–140. MR 0404103, 10.1007/BF02485242 |
| Reference:
|
[14] J. A. Gerhard: The lattice of quational classes of idempotent semigroups.J. Algebra 15, 195–224. MR 0263953, 10.1016/0021-8693(70)90073-6 |
| Reference:
|
[15] J. A. Gerhard: The number of polynomials of idempotent semigroups.J. Algebra 18, 366–376. Zbl 0219.20044, MR 0274375 |
| Reference:
|
[16] G. Grätzer: Composition of functions.Proceedings of the conference on universal algebra (Kingston, 1969), Queen’s Univ., Kingston, Ont., 1970, pp. 1–106. MR 0276161 |
| Reference:
|
[17] G. Grätzer: Universal Algebra. Second edition.Springer-Verlag, New York-Heidelberg-Berlin, 1979. MR 0538623 |
| Reference:
|
[18] G. Grätzer and A. Kisielewicz: A survey of some open problems on $p_n$-sequences and free spectra of algebras and varieties.Universal Algebra and Quasigroup Theory, A. Romanowska and J. D. H. Smith (eds.), Helderman Verlag (Berlin), 1992, pp. 57–88. MR 1191227 |
| Reference:
|
[19] G. Grätzer and R. Padmanabhan: On commutative idempotent and nonassociative groupoids.Proc. Amer. Math. 28 (1971), 75–78. MR 0276393, 10.2307/2037760 |
| Reference:
|
[20] G. Grätzer and J. Płonka: On the number of polynomials of an idempotent algebra I.Pacific J. Math. 22 (1970), 697–709. MR 0256969, 10.2140/pjm.1970.32.697 |
| Reference:
|
[21] H. Kaiser: On a problem in the theory of primal algebras.Algebra Universalis 5 (1974), 307–311. MR 0392773, 10.1007/BF02485263 |
| Reference:
|
[22] A. Kisielewicz: On idempotent algebra with $p_n=2n$.Algebra Universalis 23 (1981), 313–323. MR 0903936, 10.1007/BF01230624 |
| Reference:
|
[23] E. Marczewski: Independence and homomorphisms in abstract algebras.Fund. Math. 50 (1961), 45–61. Zbl 0104.25501, MR 0138572, 10.4064/fm-50-1-45-61 |
| Reference:
|
[24] A. Mitschke and H. Werner: On groupoids representable by vector spaces over finite fields.Arch. Math. 24 (1973), 14–20. MR 0316601, 10.1007/BF01228164 |
| Reference:
|
[25] R. Padmanabhan: Characterization of a class of groupoids.Algebra Universalis 1 (1972), 374–382. Zbl 0236.20043, MR 0294550, 10.1007/BF02944996 |
| Reference:
|
[26] P. P. Pálfy: Minimal clones.Preprint No. 27/1984, Budapest, May 1984, Math. Inst. of the Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda u. 13–15. Hungary. |
| Reference:
|
[27] P. P. Pálfy: The arity of minimal clones.Acta Sci. Math. 50 (1986), 331–333. MR 0882043 |
| Reference:
|
[28] R. E. Park: A four-element algebra whose identities are not finitely based.Algebra Universalis 11 91980, 255–260. Zbl 0449.08005, MR 0588218 |
| Reference:
|
[29] J. Płonka: On algebras with $n$ distinct $n$-ary operations.Algebra Universalis 1 (1971), 73–79. MR 0286928, 10.1007/BF02944958 |
| Reference:
|
[30] J. Płonka: On algebras with at most $n$ distinct $n$-ary operations.Algebra Universalis 1 (1971), 80–85. MR 0286929, 10.1007/BF02944959 |
| Reference:
|
[31] J. Płonka: On equational classes of abstract algebras defined by regular equations.Fund. Math. 64 (1969), 241–247. MR 0244133, 10.4064/fm-64-2-241-247 |
| Reference:
|
[32] J. Płonka: On $k$-cyclic groupoids.Math. Japonica 30 (1985), no. 3, 371–382. MR 0803288 |
| Reference:
|
[33] J. Płonka: Subdirectly irreducible groupoids in some varieties.CMUC 24/4 (1983), 631–645. MR 0738559 |
| Reference:
|
[34] S. K. Stein: Homogeneous quasigroup.Pacific J. Math. 14 (1964), 1091–1102. MR 0170972, 10.2140/pjm.1964.14.1091 |
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