# Extensions of Families of Languages: Lattices of Full X-AFL's

P.R.J. Asveld

Research output: Book/ReportReportOther research output

### 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 Enschede University of Twente, Department of Applied Mathematics 34 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.
Asveld, P.R.J. / Extensions of Families of Languages: Lattices of Full X-AFL's. Enschede : University of Twente, Department of Applied Mathematics, 1976. 34 p.
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title = "Extensions of Families of Languages: Lattices of Full X-AFL's",
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.",
keywords = "EWI-3696, HMI-SLT: Speech and Language Technology",
author = "P.R.J. Asveld",
note = "Research supported by Netherlands Organization for the Advancement of Pure Research (ZWO).",
year = "1976",
language = "Undefined",
publisher = "University of Twente, Department of Applied Mathematics",

}

Asveld, PRJ 1976, Extensions of Families of Languages: Lattices of Full X-AFL's. University of Twente, Department of Applied Mathematics, Enschede.

Extensions of Families of Languages: Lattices of Full X-AFL's. / Asveld, P.R.J.

Enschede : University of Twente, Department of Applied Mathematics, 1976. 34 p.

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - Extensions of Families of Languages: Lattices of Full X-AFL's

AU - Asveld, P.R.J.

N1 - Research supported by Netherlands Organization for the Advancement of Pure Research (ZWO).

PY - 1976

Y1 - 1976

N2 - 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.

AB - 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.

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KW - HMI-SLT: Speech and Language Technology

M3 - Report

BT - Extensions of Families of Languages: Lattices of Full X-AFL's

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Asveld PRJ. Extensions of Families of Languages: Lattices of Full X-AFL's. Enschede: University of Twente, Department of Applied Mathematics, 1976. 34 p.