  A language L is suffix-convex if for any words u, v,w, whenever w and uvw are
in L, vw is in L as well. Suffix-convex languages include left ideals,
suffix-closed languages, and suffix-free languages, which were studied
previously. In this paper, we concentrate on suffix-convex languages that do
not belong to any one of these classes; we call such languages proper. In order
to study this language class, we define a structure called a suffix-convex
triple system that characterizes the automata recognizing suffix-convex
languages. We find tight upper bounds for reversal, star, product, and boolean
operations of proper suffix-convex languages, and we conjecture on the size of
the largest syntactic semigroup. We also prove that three witness streams are
required to meet all these bounds.
