This talk deals with the problem whether a function $f \colon A^n \to B$ is uniquely determined, up to equivalence, by its identification minors, i.e., by the functions derived from $f$ by identifying a pair of its arguments. Some recent results, both positive and negative, about this reconstruction problem are discussed (see [1, 2, 3, 4]). For example, totally symmetric functions and affine functions over finite fields are reconstructible. On the other hand, the class of order-preserving functions is not reconstructible, not even weakly. Some open problems are also presented.

This talk is partly based on work done in collaboration with Miguel Couceiro (Université Paris-Dauphine) and Karsten Schölzel (University of Luxembourg).

References:

[1] E. Lehtonen, Totally symmetric functions are reconstructible from identification minors, Electron. J. Combin. 21(2) (2014) #P2.6.

[2] E. Lehtonen, Reconstructing multisets over commutative groupoids and affine functions over nonassociative semirings, Internat. J. Algebra Comput. 24 (2014) 11-31.

[3] M. Couceiro, E. Lehtonen, K. Schölzel, Hypomorphic Sperner systems and nonreconstructible functions, arXiv:1306.5578.

[4] M. Couceiro, E. Lehtonen, K. Schölzel, Set-reconstructibility of Post classes, arXiv:1310.7797.