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Extreme nonassociativity in order nine and beyond
Author(s) -
Drápal Aleš,
Valent Viliam
Publication year - 2020
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.21679
Subject(s) - quasigroup , mathematics , isomorphism (crystallography) , order (exchange) , associative property , combinatorics , division (mathematics) , arithmetic , discrete mathematics , algebra over a field , pure mathematics , crystallography , economics , finance , chemistry , crystal structure
The main concern of this paper are quasigroups of order nine that possess at most 18 associative triples. The order nine is the least order for which there exists a quasigroup ( Q , * ) such that x * ( y * z ) = ( x * y ) * z holds if and only if x = y = z . Up to isomorphism there is only one such quasigroup of this order. It has remarkable properties that bind it to a nearfield, to a PMD ( 9 , 4 ) and to a Sudoku division square.