Brzozowski, JanuszSinnamon, Corwin2020-03-182020-03-182019-10-01https://doi.org/10.1016/j.tcs.2018.07.015http://hdl.handle.net/10012/15697The final publication is available at Elsevier via https://doi.org/10.1016/j.tcs.2018.07.015. © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A language L over an alphabet Σ is prefix-convex if, for any words x,y,z ∈ Σ*, whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages, which were studied elsewhere. Here we concentrate on prefix-convex languages that do not belong to any one of these classes; we call such languages proper. We exhibit most complex proper prefix-convex languages, which meet the bounds for the size of the syntactic semigroup, reversal, complexity of atoms, star, product, and boolean operations.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/atommost complexprefix-convexproperquotient complexityregular languagestate complexitysyntactic semigroupArticle