  We define a pair of simple combinatorial operations on subshifts, called
existential and universal extensions, and study their basic properties. We
prove that the existential extension of a sofic shift by another sofic shift is
always sofic, and the same holds for the universal extension in one dimension.
However, we also show by a construction that universal extensions of
two-dimensional sofic shifts may not be sofic, even if the subshift we extend
by is very simple.
