Polynomials in idempotent commutative groupoids
Józef Dudek
GDML_Books, (1989), p.

In this paper we investigate the variety of idempotent commutative groupoids. In particular, we improve the results of Grätzer and Padmanabhan on the number pn of essentially n-ary polynomials in idempotent commutative groupoids. They have shown that if an idempotent commutative groupoid (G,•) is different from a semilattice, thenpn(G,•)≥1/3(2n-(-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), thenpn(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.

CONTENTS1. Introduction......................................................................52. Terminology......................................................................73. Applied results.................................................................84. Nonmedial groupoids.....................................................105. Sterner quasigroups......................................................136. Near-semilattices...........................................................177. Totally commutative groupoids.......................................218. Some lemmas on idempotent algebras..........................279. Ternary polynomials.......................................................2910. Nonmedial groupoids (continued)................................3311. Medial groupoids..........................................................3912. Proof of Theorem 1......................................................4813. Proof of Theorem 2......................................................48References........................................................................52

EUDML-ID : urn:eudml:doc:268496 
@book{bwmeta1.element.zamlynska-e19ebdc7-ed54-4052-8ce7-5365e2b1e70b,
     author = {J\'ozef Dudek},
     title = {Polynomials in idempotent commutative groupoids},
     series = {GDML\_Books},
     publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
     address = {Warszawa},
     year = {1989},
     zbl = {0687.08003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.zamlynska-e19ebdc7-ed54-4052-8ce7-5365e2b1e70b}
}

Józef Dudek. Polynomials in idempotent commutative groupoids. GDML_Books (1989),  http://gdmltest.u-ga.fr/item/bwmeta1.element.zamlynska-e19ebdc7-ed54-4052-8ce7-5365e2b1e70b/