Matroid representations by partitions

Abstract

A matroid on the ground set N with the rank function r is said to be partition representable of degree d⩾2 if partitions ξ i, i∈N , of a finite set Ω of the cardinality d r( N) , exist such that the meet-partition ξ I =⋀ i∈ I ξ i has d r( I) blocks of the same cardinality for every I⊂ N. Partition representable matroids are called also secret-sharing or almost affinely representable and partition representations correspond to ideal secret-sharing schemes or to almost affine codes. These notions are shown to be closely related to generalized quasigroup equations read out of the matroid structure. A special morphism of partition representations, called partition isotopy, is introduced. For a few matroids, the partition isotopy classes of partition representations are completely classified. An infinite set of excluded minors for the partition representability is constructed.