  We present a new sufficient criterion to prove that a non-sofic
half-synchronized subshift is direct prime. The criterion is based on conjugacy
invariant properties of Fischer graphs of half-synchronized shifts. We use this
criterion to show as a new result that all n-Dyck shifts are direct prime, and
we also give new proofs of direct primeness of non-sofic beta-shifts and
non-sofic S-gap shifts. We also construct a class of non-sofic synchronized
direct prime subshifts which additionally admit reversible cellular automata
with all directions sensitive.
