A construction of orthogonal arrays and applications to embedding theorems

An n^t by k orthogonal array is a collection of k-tuples of elements from an n-set, such that if a matrix is formed with the k-tuples as rows then each ordered t-tuple of elements appears exactly once as a row of each t columned and n^t rowed submatrix. If such an array has its set of k-tuples invariant under the elements of a subgroup G of S_t then the array is referred to as a G-array. A method is described for constructing a G-array of order nr from an array of order n and G-arrays of order r.The above described construction is used to produce finite embedding theorems for partial 3-quasigroups of various types. For a class of 3-quasigroups, such a theorem shows that a finite partial member of the class can be embedded in a finite complete member of the class. Theorems included produce finite embedding theorems for 3-quasigroups satisfying the identities 〈x,y,〈y,x,z〉〉=z and 〈〈z,x,y〉,y,x〉=z, for cyclic 3-quasigroup s, and conditional embedding theorems are presented for semi-symmetric 3-quasigroups.