Rectangularity

TL;DR

This paper introduces a new array condition called rectangularity, unifies various algebraic and combinatorial structures, and explores their generality, specific cases, and overlaps with partial Latin squares.

Contribution

It defines the rectangularity condition, shows its equivalence across different structures, and investigates their properties, generality, and relationships to partial Latin squares.

Findings

Rectangularity unifies diverse algebraic and combinatorial structures.

Structures satisfying rectangularity are extremely general, with no nontrivial equations.

Partial arrays satisfying the condition have significant overlap with partial Latin squares.

Abstract

We introduce a condition on arrays in some way maximally distinct from Latin square condition, as well as some other conditions on algebras, graphs and $0,1$-matrices. We show that these are essentially the same structures, generalising a similar collection of models presented by Knuth in 1970. We find ways in which these structures can be made more specific, relating to existing investigations, then show that they are also extremely general; the groupoids satisfy no nontrivial equations. Some construction methods are presented and some conjectures made as to how certain structures are preserved by these constructions. Finally we investigate to what degree partial arrays satisfying our conditions and partial Latin squares overlap.