“…If r ≥ 3 then R(r, n, k, h) and T (r, n, k, h) are infinite by Theorem A, so we may assume r = 2. If α ≡ 0 and β ≡ 0 mod n then R(2, n, k, h), and hence T (2, n, k, h), is infinite by [17], [18]. If α ≡ 0 or β ≡ 0 mod n then T (r, n, k, h) is the semigroup free product of (n, h) copies of T (2, n, k, 1) = S(2, n, k) which, by [16,Theorem 4], is the union of n trivial ideals, and hence T (r, n, k, h) is infinite.…”