Integer domains have at least two operations. One of them is typically
considered to be (commutatively) multiplicative in resemblance and one is
typically taken to be additive in resemblance (where additivity is taken
to be commutative pretty much in every context). Usually the former
(multiplicative) operation is the one being considered for primes. But I
suppose that (especially with relatively exotic operations, and therefore
integer domains, in question) one could consider numbers to be composite
or prime under the additive (or, more generally, other) operation. For
example, 1 and -1 the only "additive primes" (if we include units) in the
typical ring of integers under addition and multiplication. I think that
it could be helpful and interesting to include a third terbri for "under
operation x3"; at the very least, what could it hurt?