We introduce a derivative of a relation over the ring of integers modulo an odd number which is based on the very fundamental concepts which helped in the evolution of derivative of a function over the real number field, namely slope. Then, for a prime field GF(p), we use the derivatives to construct an algo-rithm that find all the directions, in the sense of Redei [1], of graphs of certain exponential relations over R.
Hasse, H. (1936) Theorie der Hoheren Differentiale in Einem Algebraischen Funk-tionenkorper mit Vollkommenen Konstantenkorper bei Beliebiger Charakteristik. Journal für die reine und angewandte Mathematik, 175, 50-54.
Massey, J.L., von Seeman, N. and Schoeller, P. (1986) Hasse Derivatives and Repeated-Root Cyclic Codes. IEEE International Symposium on Information Theory, USA.
de Souza, M.M.C., de Oliveira, H.M., de Souza, R.M.C. and Vasconcelos, M.M. (2004) The Discrete Cosine Transform over Prime Finite Fields. LNCS, 3124, 37-59.
https://doi.org/10.1007/978-3-540-27824-5_65
Blockhuis, A., Ball, S., Brouwer, A.E., Storme, L. and Szonyi, T. (1999) On the Number of Slopes of the Fraph of a Function Defined on a Finite Field. Journal of Combinatorial Theory, Series A, 86, 187-196.
https://doi.org/10.1006/jcta.1998.2915
Ball, S. (2003) The Number of Directions Determined by a Function over Finite Field. Journal of Combinatorial Theory, Series A, 104, 341-435.
https://doi.org/10.1016/j.jcta.2003.09.006
Ball, S. (2007) Functions over Finite Fields That Fetermine Few Directions. Electronic Notes in Discrete Mathematics, 29, 185-188.
https://doi.org/10.1016/j.endm.2007.07.032
Garcia-Colin, N., Montejano, A., Mon-tejano, L. and Oliveros, D. (2017) Transitive Oriented 3-Hypergraphs of Cyclic Orders.
https://arxiv.org/pdf/1210.6828.pdf
Campello de Souza, R.M., de Oliveira, H.M., Kauffman, A.N. and Paschoal, A.J.A. (1998) Trigonometry in Finite Fields and a New Hartley Transform. ISIT, Cambridge. https://doi.org/10.1109/ISIT.1998.708898
