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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 automataarXiv:1805.03375v2 fatcat:eobdtd4nvvf45l26fqnsukrw5e