Totally anti-symmetric quasigroups for all orders n =/ 2, 6

A quasigroup (Q, ∗) is called totally anti-symmetric if (c ∗ x) ∗ y = (c ∗ y) ∗ x ⇒ x = y and x∗y = y∗x ⇒ x = y. A totally anti-symmetric quasigroup can be used for the definition of a check digit system. Ecker and Poch [9] conjectured that there are no totally anti-symmetric quasigroups of order 4k + 2. This article will completely disprove their conjecture (except for n = 2, 6) as we will give constructions for totally anti-symmetric quasigroups for all orders n = 2, 6. Additionally we prove that the class of totally anti-symmetric quasigroups is no variety.

• Type: Journal
• Year: 2007
• Where: DM
• Authors: H. Michael Damm
• Keywords: Conjecture, Constructions, DM 2007, Quasigroups