Glasnik Matematicki, Vol. 36, No.1 (2001), 85-93.
FREE STEINER LOOPS
Smile Markovski and Ana Sokolova
Faculty of Scinces and Mathematics, Institute of Informatics,
p.f. 162, Skopje, Republic of Macedonia
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,
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.
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