On the extension of involutorial Bol loops. (English) Zbl 1016.20051

Kiechle, H.; Nagy, G. P.

Abh. Math. Semin. Univ. Hamb. 72, 235-250 (2002).

A loop \((L,\cdot)\) with neutral element \(1\) is called involutorial if for all \(x\in L\), \(x\cdot x=1\), and a Bol-loop if for all \(a,b,c\in L\), \(a(b\cdot ac)=(a\cdot bc)c\).
In this note the authors present a method (called “loop extension”) to obtain an involutorial Bol-loop \((E,\bullet)\) starting from an involutorial Bol-loop \((L,\cdot)\) and from a Bol-loop \((L,*)\) with the same identity 1, enjoying suitable properties. The loop extension \((E,\bullet)\) has cardinality \(|E|=2|L|\) and contains \((L,\cdot)\) as a proper subloop of index 2. Conversely, they also prove that any involutorial Bol-loop \((E,\bullet)\) containing the involutorial Bol-loop \((L,\cdot)\) as a subloop of index \(2\) arises in this way. They also address the problem of formulating conditions ensuring that two loop extensions of the same loop \((L,\cdot)\) are isomorphic, and they apply this theorem to determine all 34 non-isomorphic involutorial Bol-loops of order 16. The latter result is obtained by means of the computer algebra program GAP4, since loops can be represented by their set of left translations and this allows the use of group theoretical tools. Finally, they characterize all involutorial Bol-loops \((L,\cdot)\) with right nucleus of index 2 in \(L\).

Reviewer: Elena Zizioli (Brescia)

MSC:

20N05

Loops, quasigroups

Keywords:

involutorial Bol-loops; loop extensions; left translations

Software:

GAP

References:

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