Glasnik Matematicki, Vol. 36, No.1 (2001), 85-93.

FREE STEINER LOOPS

Smile Markovski and Ana Sokolova

Abstract.   A Steiner loop, or a sloop, is a grupoid (L; · ,1), where · is a binary operation and 1 is a constant, satisfying the identities 1 · x = x, x · y = y · x, x · (x · y) = y. There is a one-to-one correspondence between Steiner triple systems and finite sloops. Two constructions of free objects in the variety of sloops are presented in this paper. They both allow recursive construction of a free sloop with a free base X, provided that X is recursively defined set. The main results besides the constructions are: Each subsloop of a free sloop is free two. A free sloop S with a free finite bases X, |X| ≥ 3, has a free subsloop with a free base of any finite cardinality and a free subsloop with a free base of cardinality ω as well; also S has a (non free) base of any finite cardinality k ≥ |X|. We also show that the word problem for the variety of sloops is solvable, due to embedding property.

1991 Mathematics Subject Classification.   08B20, 05B07, 08A50.

Key words and phrases.   Free algebra, sloop, Steiner loop, variety, word problem.