## 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.Broziková

### MSC:

 20N05 Loops, quasigroups 20E15 Chains and lattices of subgroups, subnormal subgroups
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