Help defining a shift cipher mathematically
My understanding is given that $E_K(x)=(x+K)~mod~26$ is the character by character encryption of a shift cipher, you want to extend it to words as the function $E'$ given by
$$E'_K(x_1,\ldots,x_m)=(E_K(x_1),E_K(x_m)).$$
If you intended the key $K$ to change per character you can modify.