Full abstract family of languages

A full abstract family of languages is an abstract family of languages closed under all word monoid homomorphisms, including those erasing letters. It retains the other closure requirements: finite union, concatenation of formal languages, positive closure of a formal language, inverse homomorphism, and intersection with regular languages.

For example, the homomorphism $h(a)=a$, $h(b)=\epsilon$ sends $\{a^n{b}^n:n\ge 0\}$ to $\{a^n:n\ge 0\}$. Such a map is excluded by a nonerasing-only homomorphism axiom. The adjective full says this additional erasure operation is allowed; it does not assert closure under every operation on languages.

Both regular languages and context-free languages form full abstract families. For the latter, intersection with a regular language is allowed, but intersection of two arbitrary context-free languages need not remain context-free. Erasing an arbitrarily long part of a representation can conceal unbounded intermediate work, so the distinction matters when comparing computational language classes.^[ref]