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Lattice definability of diassociative commutative loops without torsion. (Russian) Zbl 0648.20068
A loop \({\mathcal G}\) is said to be diassociative if any two elements of \({\mathcal G}\) generate a subgroup and \({\mathcal G}\) is called without torsion if \({\mathcal G}\) has no elements of finite order. Let \({\mathcal G}\) and \({\mathcal G}_ 1\) be quasigroups, \({\mathcal L}({\mathcal G})\) and \({\mathcal L}({\mathcal G}_ 1)\) their lattices of subgroupoids, then the lattice isomorphism \(\phi\) : \({\mathcal L}({\mathcal G})\to {\mathcal L}({\mathcal G}_ 1)\) is said to be induced by the mapping \(\psi\) : \(G\to G_ 1\) if \(\phi\) (\({\mathcal Q})=\psi ({\mathcal Q})\) for every subgroupoid \({\mathcal Q}\subseteq {\mathcal G}\). The author proves: (1) The number of quasigroup isomorphisms \(\psi\) : \({\mathcal G}\to {\mathcal G}_ 1\) inducing a given lattice isomorphism \(\phi\) : \({\mathcal L}({\mathcal G})\to {\mathcal L}({\mathcal G}_ 1)\) is equal to the number of quasigroup automorphisms of \({\mathcal G}\) inducing an identical lattice automorphism of \({\mathcal G}\). (2) In the class of diassociative loops the commutative diassociative loops without torsion are determined by the lattice of their subgroupoids. (3) A lattice isomorphism of a diassociative loop containing an element of infinite order can be induced by only one isomorphism.
Reviewer: E.Brozikov√°
MSC:
20N05 Loops, quasigroups
20E15 Chains and lattices of subgroups, subnormal subgroups
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