{2, 3}-perfect m-cycle systems are equationally defined for m = 5, 7, 8, 9, and 11 only

✍ Scribed by E. M. Li Marzi; C. C. Lindner; F. Rania; R. M. Wilson

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 133 KB
• Volume: 12
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20016

✦ Synopsis

Abstract

An m‐cycle system (S__,C__) of order n is said to be {2,3}‐perfect provided each pair of vertices is connected by a path of length 2 in an m‐cycle of C and a path of length 3 in an m‐cycle of C. The class of {2,3}‐perfect m‐cycle systems is said to be equationally defined provided, there exists a variety of quasigroups V with the property that a finite quasigroup (Q, $\circ$, , /) belongs to V if and only if its multiplicative (Q, $\circ$) part can be constructed from a {2,3}‐perfect m‐cycle system using the 2‐construction (a $\circ$ a = a for all a ∈ Q and if a ≠ b, a $\circ$ b = c and b $\circ$ a = d if and only if the m‐cycle (…, d, x, a, b, y, c, …) ∈ C). The object of this paper is to show that the class of {2,3}‐perfect m‐cycle systems cannot be equationally defined for all m ≥ 10, m ≠ 11. This combined with previous results shows that {2, 3}‐perfect m‐cycle systems are equationally defined for m = 5, 7, 8, 9, and 11 only. © 2004 Wiley Periodicals, Inc.