  This paper studies the classes of semigoups and monoids with context-free and
deterministic context-free word problem. First, some examples are exhibited to
clarify the relationship between these classes and their connection with the
notions of word-hyperbolicity and automaticity. Second, a study is made of
whether these classes are closed under applying certain semigroup
constructions, including direct products and free products, or under regressing
from the results of such constructions to the original semigroup(s) or
monoid(s).
