Cubic and Semi-Cubic SQS-Skeins

There is a one to one correspondence between quadruple systems and SQS-skeins (Quackenbush 1973). A quadruple system is said to be subsystem free, if it contains no nontrivial subsystems, or equivalently if it is generated by each independent 4-element subset. Mendelsohn and Phelps (1982) have given several recursive constructions of subsystem free quadruple systems. It is more convenient to call the SQS-skein associated with a subsystem free quadruple system cubic. The author (1981) has proved that a cubic SQS-skein is simple, if it is of cardinality n ≥ 8. Let (P1;q1) be a simple SQS-skein of cardinality n. In this paper, we construct a simple SQS-skein denoted by (2 ⊗ P1; qα,A) of cardinality 2n for any n ≡ 2 or 4 (mod 6), n ≥ 8. Consequently, we may deduce that there is a simple SQS-skein of order n for each n ≡ 2 or 4 (mod 6), n ≠ 4, 8. The quadruple systems associated with these SQS-skeins are not subsystem free. If there is a cubic SQS-skein (P1; q1) of cardinality n, then we will prove that there exists a semi-cubic SQS-skein of cardinality 2n. Accordingly, one may say that there is a semi-cubic quadruple system which is not subsystem free of cardinality m for all m ≡ 4 or 8 (mod 12), m ≥ 16. The author (1981) has proved that any finite cubic SQS-skein of cardinality n ≥ 8 generates a subvariety covers the smallest subvariety (the Boolean SQS-skeins). Similarly, one can directly show that each semi-cubic SQS-skein generates another variety that also covers the variety of all Boolean SQS-skeins.