### Abstract

An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wright type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelle's characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than ω^{ω}. It follows that the algebraic ordinals are exactly those less than ωω^{ω}.

Original language | English |
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Pages (from-to) | 491-515 |

Number of pages | 25 |

Journal | International Journal of Foundations of Computer Science |

Volume | 22 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2011 |

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### Keywords

- context-free linear and well-orderings
- fixed point equations
- Linear orderings

### ASJC Scopus subject areas

- Computer Science (miscellaneous)

### Cite this

*International Journal of Foundations of Computer Science*,

*22*(2), 491-515. https://doi.org/10.1142/S0129054111008155

**Algebraic linear orderings.** / Bloom, S. L.; Ésik, Z.

Research output: Contribution to journal › Article

*International Journal of Foundations of Computer Science*, vol. 22, no. 2, pp. 491-515. https://doi.org/10.1142/S0129054111008155

}

TY - JOUR

T1 - Algebraic linear orderings

AU - Bloom, S. L.

AU - Ésik, Z.

PY - 2011/2

Y1 - 2011/2

N2 - An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wright type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelle's characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than ωω. It follows that the algebraic ordinals are exactly those less than ωωω.

AB - An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wright type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelle's characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than ωω. It follows that the algebraic ordinals are exactly those less than ωωω.

KW - context-free linear and well-orderings

KW - fixed point equations

KW - Linear orderings

UR - http://www.scopus.com/inward/record.url?scp=78650307134&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650307134&partnerID=8YFLogxK

U2 - 10.1142/S0129054111008155

DO - 10.1142/S0129054111008155

M3 - Article

VL - 22

SP - 491

EP - 515

JO - International Journal of Foundations of Computer Science

JF - International Journal of Foundations of Computer Science

SN - 0129-0541

IS - 2

ER -