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Quasigroups with distributive lattice of subquasigroups. (Russian) Zbl 0746.20054

The quasigroups (loops) mentioned in the title are called \(D\)-quasigroups (\(D\)-loops). The author proves the following theorems. Every diassociative \(D\)-loop is a group. If \(n\in N\), \(n\geq 3\) or \(n=\aleph_ 0\) there exists a cyclic \(D\)-quasigroup of order \(n\) which is not a one- sided loop; there also exists a medial one-sided \(D\)-loop of order \(n\) which is not a loop. There exists a commutative and non-associative \(D\)- loop of order \(2n\). There also exists a commutative and non-associative loop of order \(2n\) for which the equality \(x^ 2=1\) holds.

MSC:

20N05 Loops, quasigroups
06D05 Structure and representation theory of distributive lattices
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