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Free Steiner loops. (English) Zbl 0978.08007
Summary: A Steiner loop, or a sloop, is a groupoid $$(L;\cdot,1)$$, where $$\cdot$$ is a binary operation and 1 is a constant, satisfying the identities $$1\cdot x = x$$, $$x\cdot y = y\cdot x$$, $$x\cdot(x\cdot y) =y$$. 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 a recursively defined set. The main results, besides the constructions, are: Each subsloop of a free sloop is free, too. A free sloop $$\mathbf S$$ with a free finite base $$X$$, $$|X|\geq 3$$, has a free subsloop with a free base of any finite cardinality and a free subsloop with a free base of cardinality $$\omega$$ as well: also $$\mathbf S$$ has a (non free) base of any finite cardinality $$k\geq |X|$$. We also show that the word problem for the variety of sloops is solvable, due to embedding property.
##### MSC:
 08B20 Free algebras 20N05 Loops, quasigroups 05B07 Triple systems 08A50 Word problems (aspects of algebraic structures)