### Abstract

This paper is a continuation of our research on $X$-extensions and full $X$-AFL's. We discuss the algebraic structure of the class of all full $X$-AFL's and establish a Canonical Form Theorem for full $X$-AFL's (similar to a theorem for full AFL's obtained earlier by Ginsburg & Spanier (1971)) i.e. we prove $\hat{\cal X}(K) = X\Pi(K)$, where $K$ is an arbitrary family, $\hat{\cal X}(K)$ [$\Pi(K)$] is the least full $X$-AFL [pre-quasoid] containing $K$, and $X$ is the $X$-extension. We consider full ($X$-)principal $X$-AFL's and full sub-$X$-AFL's of a given full $X$-AFL from the point of view of universal algebra and lattice theory. In order to fit some well-known results on full (semi-)AFL's in our framework of extensions, we introduce a few subrational extensions and establish the corresponding Canonical Form Theorems for full (semi-)AFL's. For a restricted class of language families we show another Canonical Form Theorem and finally we discuss some semigroups consisting of operators on families of languages.

Original language | Undefined |
---|---|

Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 34 |

Publication status | Published - 1976 |

### Keywords

- EWI-3696
- HMI-SLT: Speech and Language Technology

## Cite this

Asveld, P. R. J. (1976).

*Extensions of Families of Languages: Lattices of Full X-AFL's*. Enschede: University of Twente, Department of Applied Mathematics.