Aleksandar Krapež – Generalized quadratic functional equations on ternary quasigroups

Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia

Quadratic quasigroup functional equations were introduced in [1] as a
natural generalization of permutational (usually called balanced) functional equations
on quasigroups. A fairly wide class of quadratic equations were solved in the same paper.
General case for binary quasigroups was solved by S. Krstić in his PhD thesis
[3], using the correspondence between such equations and certain connected
multigraphs. A slightly different version of such graphs is named Krstić graphs in
[2].
We generalize Krstić's results proving the existence of analogous correspondence between generalized quadratic functional equations on ternary quasigroups and connected multigraphs of degree 4. This also generalizes some recent results by F.M. Sokhatsky et all. [4], [5]. We also pose some related problems.

**References **

- [1]
- A. Krapež,
*Strictly quadratic functional equations on quasigroups I,*Publ. Math, Inst. (Beograd) 29 (43) (1981), 125–138,

http://elib.mi.sanu.ac.rs/files/journals/publ/49/n043p125.pdf - [2]
- A. Krapež, M.A. Taylor,
*Gemini functional equations on quasigroups,*Publ. Math. Debrecen 47 (1995), 281–292. - [3]
- S. Krstić,
*Quadratic quasigroup identities (Serbian),*PhD thesis, University of Belgrade (1985),

http://elibrary.matf.bg.ac.rs/bitstream/handle/123456789/83/phdSavaKrstic.pdf?sequence=1 (acessed: Nov. 23, 2020). - [4]
- F.M. Sokhatsky, H. Krainichuk, A. Tarasevich,
*A classification of generalized functional equations on ternary quasigroups,*Bull. Donetsk Nat. Univ. Series A: Natural Sci. 1–2 (2017), 35–64. - [5]
- F.M. Sokhatsky, A. Tarasevich,
*Classification of generalized ternary quadratic quasigroup functional equations of the length three,*Carpathian Math. Publ. 11(1) (2019), 179–192, doi:10.15330/cmp.11.1.179-192.