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Abstract
In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for m≥ 2 , a set of m+ 1 partitions of a set Ω , any m of which are the minimal nontrivial elements of a Cartesian lattice, either form a Latin square (if m= 2), or generate a joinsemilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have m+ r partitions with r≥ 2 , any m of which are minimal elements of a Cartesian lattice. If m= 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For m> 2 , things are more restricted. Any m+ 1 of the partitions generate a joinsemilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixedpointfree automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality m+ r in the (m 1) dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to nonelementary abelian groups using padic techniques.
Original language  English 

Pages (fromto)  20692080 
Number of pages  12 
Journal  Designs, Codes, and Cryptography 
Volume  90 
Issue number  9 
Early online date  4 Jul 2021 
DOIs  
Publication status  Published  Sept 2022 
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Dive into the research topics of 'Diagonal groups and arcs over groups'. Together they form a unique fingerprint.Projects
 1 Finished

Permutation groups: factorisations, structure and applications
ARC Australian Research Council
1/01/16 → 2/02/19
Project: Research