  We show that any finite monoid or semigroup presentation satisfying the small
overlap condition C(4) has word problem which is a deterministic rational
relation. It follows that the set of lexicographically minimal words forms a
regular language of normal forms, and that these normal forms can be computed
in linear time. We also deduce that C(4) monoids and semigroups are rational
(in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in
the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy
analogues of Kleene's theorem, and admit decision algorithms for the rational
subset and finitely generated submonoid membership problems. We also prove some
automata-theoretic results which may be of independent interest.
