Reconstructing multisets over commutative groupoids and affine functions over nonassociative semirings

TL;DR

This paper investigates the conditions under which multisets over commutative groupoids and affine functions over nonassociative semirings can be reconstructed from partial information, providing new insights into algebraic reconstruction problems.

Contribution

It establishes necessary and sufficient conditions for reconstructibility of multisets and affine functions over nonassociative semirings, extending previous work to new algebraic structures.

Findings

Multisets over commutative groupoids are reconstructible under certain conditions.

Affine functions over nonassociative semirings are weakly reconstructible.

Affine functions of large arity over finite fields are fully reconstructible.

Abstract

A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. Necessary and sufficient conditions for the reconstructibility of multisets are determined. These results find an application in a different kind of reconstruction problem for functions of several arguments and identification minors: classes of linear or affine functions over nonassociative semirings are shown to be weakly reconstructible. Moreover, affine functions of sufficiently large arity over finite fields are reconstructible.