#### 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

*e-mail:* `smile@pmf.ukim.edu.mk`

*e-mail:* `anas@pmf.ukim.edu.mk`

**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.

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