The main result of this paper is a description of totally commutative idempotent groupoids. In particular, we show that if an idempotent groupoid (G,·) has precisely m ≥ 2 distinct essentially binary polynomials and they are all commutative, then G contains a subgroupoid isomorphic to the groupoid described below. In [2], this fact was proved for m = 2.
@article{bwmeta1.element.bwnjournal-article-cmv71i2p335bwm,
author = {J. Dudek},
title = {The minimal extension of sequences III. On problem 16 of Gr\"atzer and Kisielewicz},
journal = {Colloquium Mathematicae},
volume = {70},
year = {1996},
pages = {335-338},
zbl = {0865.08001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv71i2p335bwm}
}
Dudek, J. The minimal extension of sequences III. On problem 16 of Grätzer and Kisielewicz. Colloquium Mathematicae, Tome 70 (1996) pp. 335-338. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv71i2p335bwm/
[000] [1] J. Dudek, Variety of idempotent commutative groupoids, Fund. Math. 120 (1984), 193-204. | Zbl 0546.20049
[001] [2] J. Dudek, On the minimal extension of sequences, Algebra Universalis 23 (1986), 308-312. | Zbl 0627.08001
[002] [3] J. Dudek, -sequences. The minimal extension of sequences (Abstract), presented at the Conference on Logic and Algebra dedicated to Roberto Magari on his 60th birthday, Pontignano (Siena), 26-30 April 1994, 1-6.
[003] [4] G. Grätzer, Composition of functions, in: Proc. Conference on Universal Algebra, Kingston, 1969, Queen's Univ., Kingston, Ont., 1970, 1-106.
[004] [5] G. Grätzer, Universal Algebra, 2nd ed., Springer, New York, 1979.
[005] [6] G. Grätzer and A. Kisielewicz, A survey of some open problems on -sequences and free spectra of algebras and varieties, in: Universal Algebra and Quasigroup Theory, A. Romanowska and J. D. H. Smith (eds.), Heldermann, Berlin, 1992, 57-88. | Zbl 0772.08001