On Operations over Language Families

Abstract

Let $O$ and $\F$ be an operation and a language family, respectively. So far, in terms of closure properties, the classical language theory has only investigated whether $O(\F) \subseteq \F$, where $O(\F)$ is the family resulting from $O$ applied to all members of $\F$. If $O(\F) \subseteq \F$, $\F$ is closed under $O$; otherwise, it is not. This paper proposes a finer and wider approach to this investigation. Indeed, it studies almost all possible set-based relations between $\F$ and $O(\F)$, including $O(\F) = \emptyset$; $F \not\subset O(\F)$, $O(\F) \not\subset \F$, $\F \cap O(\F) \neq \emptyset$; $\F \cap O(\F) = \emptyset$, $O(\F) \neq \emptyset$; $O(\F) = \F$; and $\F \subset O(\F)$. Many operations are studied in this way. A sketch of application perspectives and open problems closes the paper.

Keywords

• language operations
• language families
• closure properties
• finer approach
• new trend
• set theory