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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)
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