Extreme nonassociativity in order nine and beyond

Abstract

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 ( , )

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

“…Both the pseudocoboundary and the pseudococyclic frameworks over Latin rectangles described in this paper are based on the existence of non-associative triples within a Latin rectangle. As was already indicated in the introductory section and in Section 3, the study of this type of triple in the case of dealing with Latin squares has received particular attention in the recent literature [32][33][34][35][36] because of its possible application in different areas as cryptography [30,31]. A comprehensive study of non-associative triples in the case of dealing with Latin rectangles instead of Latin squares is established, therefore, as natural further work.…”

“…Of particular interest in our study, the relevant role that quasigroups with few associative triples play in cryptography [30,31] is remarkable. It is so that quasigroups with a high amount of non-associative triples are receiving particular attention [32][33][34][35][36].…”

Division Sudokus: Invariants, Enumeration, and Multiple Partitions

“…The division sudoku L 0 from the introduction (which is ds-isotopic to DS(9, 179)) is a division sudoku with respect to exactly the same sudoku partitions as L 17 . Upon considering the ds-isotopism ( (7,8,9), (7,8,9), (7, 8, 9)), we can transform L 179 into a square L 179 which also has exactly the same sudoku partitions as L 17 .…”

“…. , 9 in this order, the square L 0 is a multiplication table of a unique quasigroup of order 9 up to isomorphism with precisely nine associative triples [6,8].…”

“…). There are 104, 015, 259, 648 = 211 • 38 • 7741 standard division sudokus of rank 3. There are 186 standard division sudokus of rank 3 up to ds-isotopism.…”