Continuous languages

Abstract

A language is prefix-continuous if it satisfies the condition that, if a word w and its prefix u are in the language, then so is every prefix of w that has u as a prefix. Prefix-continuous languages include prefix-closed languages at one end of the spectrum, which include prefix codes, at the other. In a similar way, we define suffix-, bifix-, factor-, and subword- continuous languages and their closed and free counterparts. We generalize these notions to arbitrary binary relations on the set of all words over a finite alphabet. This provides a common framework for diverse languages such as codes, factorial languages and ideals. We examine the relationships among these languages and their closure properties.