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On the theory of F-quasigroups. (Russian) Zbl 0681.20044
Webs and quasigroups, Interuniv. thematic Collect. sci. Works, Kalinin 1988, 127-130 (1988).
[For the entire collection see Zbl 0632.00006.]
A groupoid \((Q,A)\) is a semigroup iff the following law holds: \(A(A(x,y),z)=A(x,A(y,z))\). \((Q,A)\) is a quasigroup iff the equations \(A(a,x)=b\) and \(A(y,a)=b\) are uniquely solvable for all \(a,b\in Q\). A group is an associative quasigroup. Investigation of quasigroups naturally yields the following question: how near to groups are special quasigroups. D. C. Murdoch [Am. J. Math. 61, 509-522 (1939; Zbl 0020.34702)] treated this problem in the following way. In every quasigroup \((Q,A)\) the equation \(A(A(a,b),c)=A(a,A(b,x))\) is uniquely solvable for all \(a,b,c\in Q\). If the solution is denoted by \(f_{(a,b)}c\), then \(f_{(a,b)}\) is a permutation of the set \(Q\) which in general depends on \(a\) and \(b\). Then especially the following cases are of interest: a) \(f_{(a,b)}\) depends on \(b\) only; b) \(f_{(a,b)}\) depends on \(a\) only; and c) \(f_{(a,b)}\) does not depend neither on \(a\) nor on \(b\). Quasigroups for which b) holds are said to be F-quasigroups. In this article a relationship between F-quasigroups and loops (quasigroups with a neutral element) satisfying some special laws is stated.
Reviewer: J.Ušan

20N05 Loops, quasigroups