 Given two algebras \((X, f_1 , \ldots f_k)\) and \((Y, g_1 , \ldots g_k)\), a homomorphism \(h: (X, f_1 , \ldots f_k) \rightarrow (Y, g_1 , \ldots g_k)\) is a function \(h: X \rightarrow Y\) satisfying \(h(f_i (x_0 , \ldots ,x_{n_{ i}-1 })) = g_i (h(x_0), \ldots ,h(x_{n_{ i}-1 })))\) for each \(i\) from 1 to \(k\) where \(n_i\) is the arity of both \(f_i\) and \(g_i\).