Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 2, pp. 325-331 (2002)

On Subdirectly Irreducible Steiner Loops of Cardinality $2n$

Magdi H. Armanious

El-Metawakel Ala Alah St. 3, El-Zeiton, P.C. 11321, Cairo, Egypt, e-mail:

Abstract: Let $ L_1$ be a finite simple sloop of cardinality $n$ or the 8-element sloop. In this paper, we construct a subdirectly irreducible (monolithic) sloop $ L=2\otimes_\alpha L_1$ of cardinality $2n$, for each $n\geq 8$, with $n\equiv 2$ or 4 (mod 6), in which each proper homomorphic image is a Boolean sloop. Quackenbush [Q] has proved that the variety $V( L_1)$ generated by a finite simple planar sloop $ L_1$ covers the smallest non-trivial subvariety (the class of all Boolean sloops). For any finite planar sloop $ L_1$, the variety $V( L)$ generated by the constructed sloop $ L= 2\otimes_\alpha L_1$ covers the variety $V( L_1)$.

[Q] Quackenbush, R. W.: {\it Varieties of Steiner Loops and Steiner Quasigroups. Canad. J. Math. 18 (1978), 1187-1198.

Classification (MSC2000): 05B07; 20N05

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