Glasnik matematički, Vol. 36 No. 1, 2001.
Izvorni znanstveni članak
Free Steiner loops
Smile Markovski
Ana Sokolova
Sažetak
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.
Ključne riječi
Free algebra; sloop; Steiner loop; variety; word problem
Hrčak ID:
4852
URI
Datum izdavanja:
1.6.2001.
Posjeta: 814 *