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Logarithmetics and quasigroup structure. (English) Zbl 0827.20081
The logarithmetic \(L(Q)\) of a finite quasigroup \(Q\) with elements \(a_1,\dots,a_n\) is the quotient \(N/(\equiv \bmod\log Q)\), where \(r \equiv s\pmod{\log Q}\) if \(a^r_i=a^s_i\) for all \(i=1, \dots, n\); \(r, s \in N\). The author utilizes research of H. Popova [Proc. Edinb. Math. Soc., II. Ser. 9, 74-81 (1954; Zbl 0056.255); 9, 109-115 (1956; Zbl 0072.255)] and investigates the properties of \(L(Q)\) and the relations between quasigroups and their logarithmetics. Different quasigroups may have isomorphic logarithmetics. In the last part the classification of the quasigroups that have a given logarithmetic is studied.
MSC:
20N05 Loops, quasigroups
11A07 Congruences; primitive roots; residue systems
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References:
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