Abstract :
It is well known that there is a planar sloop of cardinality n for each n≡2 or 4 (mod 6) (Math. Z. 111 (1969) 289–300). A semi-planar sloop is a simple sloop in which each triangle either generates the whole sloop or the 8-element sloop. In fact, Quackenbush (Canad. J. Math. 28 (1976) 1187–1198) has stated that there should be such semi-planar sloops. In this paper, we construct a semi-planar sloop of cardinality 2n for each n≡2 or 4 (mod 6). Consequently, we may say that there is a semi-planar sloop that is not planar of cardinality m for each m>16 and m≡4 or 8 (mod 12). Moreover, Quackenbush (Canad. J. Math. 28 (1976) 1187–1198) has proved that each finite simple planar sloop generates a variety, which covers the smallest non-trivial subvariety (the variety of all Boolean sloops) of the lattice of the subvarieties of all sloops. Similarly, it is easy to show that each finite semi-planar sloop generates another variety, which also covers the variety of all Boolean sloops. Furthermore, for any finite simple sloop L of cardinality n, the author (Beiträge Algebra Geom. 43 (2) (2002) 325–331) has constructed a subdirectly irreducible sloop S=2⊗pL of cardinality 2n and containing L as the only proper normal subsloop. Accordingly, if L is a semi-planar sloop, then the variety V(S) generated by S=2⊗pL properly contains the subvariety V(L).
