Totally anti-symmetric quasigroups for all orders \(n\neq 2,6\). (English) Zbl 1128.20053
In 1984, A. Ecker and G. Poch [Computing 37, 277-301 (1986; Zbl 0595.94012)] searched for TA-quasigroups, and they proved that there are no TA-quasigroups of order \(4k+2\). In fact they were not able to find TA-quasigroups for orders \(n=2,6\).
In this paper the author supports to prove this conjecture to be wrong, except for \(n=2,6\), via the main Theorem: There are TA-quasigroups of order \(n\) for all \(n\neq 2,6\). On the other hand, in 2003 the author gave counterexamples to the conjecture of Ecker and Poch [M. Damm, Computing 70, No. 4, 349-357 (2003; Zbl 1021.05016)].
In this paper the author supports to prove this conjecture to be wrong, except for \(n=2,6\), via the main Theorem: There are TA-quasigroups of order \(n\) for all \(n\neq 2,6\). On the other hand, in 2003 the author gave counterexamples to the conjecture of Ecker and Poch [M. Damm, Computing 70, No. 4, 349-357 (2003; Zbl 1021.05016)].
Reviewer: C. Pereira da Silva (Curitiba)
Keywords:
totally anti-symmetric quasigroups; TA-quasigroups; loops; Latin squares; check digit systems; varieties; groupsReferences:
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