  This paper studies decision problems for semigroups that are word-hyperbolic
in the sense of Duncan & Gilman. A fundamental investigation reveals that the
natural definition of a `word-hyperbolic structure' has to be strengthened
slightly in order to define a unique semigroup up to isomorphism. The
isomorphism problem is proven to be undecidable for word-hyperbolic semigroups
(in contrast to the situation for word-hyperbolic groups). It is proved that it
is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously
automatic, biautomatic, or asynchronously biautomatic. (These properties do not
hold in general for word-hyperbolic semigroups.) It is proved that the uniform
word problem for word-hyperbolic semigroup is solvable in polynomial time
(improving on the previous exponential-time algorithm). Algorithms are
presented for deciding whether a word-hyperbolic semigroup is a monoid, a
group, a completely simple semigroup, a Clifford semigroup, or a free
semigroup.
