Generalized linear functional equations on almost quasigroups. I: Equations with at most two variables (Q5943226)

The author defines a new class of groupoids, so-called almost quasigroups which generalize quasigroups. For a groupoid \((S,\cdot)\) a translation is a mapping which is either left \((L_a(x)= a\cdot x)\) or right translation \((R_a(x)= x\cdot a)\) for some \(a\in S\). If \(L_a\) \((R_a)\) is a permutation, then \(a\) is a left (right) quasiunit of \((S,\cdot)\), if \(L_a\) \((R_a)\) is a constant mapping, then \(a\) is left (right) quasizero of \((S,\cdot)\). A groupoid \((S,\cdot)\) is an almost quasigroup if: -- every element of \(S\) is either a left quasizero or a left quasiunit -- every element of \(S\) is either a right quasizero or a right quasiunit -- there is at least one quasiunit in \(S\). The author proves several representation theorems for almost quasigroups and some well-known theorems on quasigroups are generalized. Then he defines the normal form of equations and shows that every generalized linear functional equation \(Eq\) on almost quasigroups is equivalent to a system consisting of several equations with at most one variable each, and one equation in the normal form, with the same number of variables as \(Eq\). The general solution of the generalized linear functional equations on almost quasigroups with at most two variables is given.