On (i,j)-associative n-groupoids with non-empty center
Let (G,f) is an n-groupoid. Elements \(a_ 2,a_ 3,...,a_ n\in G\) are cancellable if the condition \(f(a^ p_ 2,x,a^ n_{p+1})=f(a^ p_ 2,y,a^ n_{p+1})\Rightarrow x=y\) holds for all x,y\(\in G\) and every \(p=1,2,...,n\). (G,f) is (i,j)-associative if \[ f(x_ 1^{i-1},f(x_ i^{n+i-1}),x_{n+i}^{2n-1})=f(x_ 1^{j-1},f(x_ j^{n+j- 1}),x_{n+j}^{2n-1}) \] for all \(x_ 1,...,x_{2n-1}\in G\) and fixed \(1\leq i<j\leq n\). The main result of the paper is the theorem: An n- groupoid (G,f) with cancellable elements in the center is an n-semigroup iff it is (i,j)-associative for some \(1\leq i<j\leq n\). If (G,f) is an n- quasigroup, then we get the following corollary: An n-quasigroup with non-empty center is an n-group iff it is (i,j)-associative for some pair (i,j) with \(1\leq i,j\leq n\).
scientific article; zbMATH DE number 3998102