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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

##### MSC:
 20N05 Loops, quasigroups
##### Keywords:
quasigroups; F-quasigroups; loops