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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.
08B20 Free algebras
20N05 Loops, quasigroups
05B07 Triple systems
08A50 Word problems (aspects of algebraic structures)