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{2, 3}‐perfect m‐cycle systems are equationally defined for m = 5, 7, 8, 9, and 11 only
Author(s) -
Marzi E. M. Li,
Lindner C. C.,
Rania F.,
Wilson R. M.
Publication year - 2004
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20016
Subject(s) - mathematics , combinatorics , class (philosophy) , multiplicative function , quasigroup , order (exchange) , path (computing) , variety (cybernetics) , discrete mathematics , mathematical analysis , computer science , artificial intelligence , statistics , finance , economics , programming language
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.