On Subdirectly Irreducible SQS-Skeins

Abstract

SOS-skeins correspond exactly to the Steiner quadruple systems [8,12]. Let $P_1$ be a finite simple SQS-skein of cardinality $n>4$. In this article, we will present a construction for a non-simple subdirectly irreducible (monolithic) SOS-skein $P=2{\otimes }_{\alpha }{P}_n$ of cardinality $2n$ in which each proper homomorphic image is Boolean for all $n\equiv 2$ or $4(\mathrm{mod}6)$. We can then show that if $P_1$ has a simple derived sloop, then the constructed SOS-skein $2{\otimes }_{\alpha }{P}_1$ contains a derived sloop which is subdirectly irreducible and has the same property as the SOS-skein $2{\otimes }_{\alpha }{P}_1$ that each of its proper homomorphic images is Boolean. Similar to the theory of Steiner loops and Steiner quasigroups [14], the author [1] has proven that the variety $V(P_1)$ generated by a finite simple cubic SQS-skein $P_1$ covers the smallest non-trivial subvariety (the class of all Boolean SQS-skeins). Finally, we show that the variety $V(2{\otimes }_{\alpha }{P}_1)$ generated by the constructed SQS-skein $2{\otimes }_{\alpha }{P}_1$ covers the variety $V(P_1)$ for each finite simple cubic SOS-skein $P_1$.