2018/10/09 by Wiesław A. Dudek, Dudek, Wieslaw A., R. A. R. Monzo +1
Computer Science · Engineering · Mathematics · #20M15 #20N02 #FOS: Mathematics #Mathematics and Applications #Rings and Algebras (math.RA) #graph theory and CDMA systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1810.04566
openalex publication_date 2018/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the main result that a groupoid of order n is an idempotent k-translatable quasigroup if and only if its multiplication is given by x.y = (ax+by)(mod n), where a+b = 1(mod n), a+bk = 0(mod n) and (k,n)= 1. We describe the structure of various types of idempotent, k-translatable quasigroups, some of which are connected with affine geometry and combinatorial algebra, and their parastrophes. We prove that such parastrophes are also idempotent, translatable quasigroups and determine when they are of the same type as the original quasigroup. In addition, we find several different necessary and sufficient conditions making a k-translatable quasigroup quadratical.