# Algebraic linear orderings

S. L. Bloom, Z. Ésik

Research output: Contribution to journalArticle

12 Citations (Scopus)

### 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 ωω. It follows that the algebraic ordinals are exactly those less than ωωω.

Original language English 491-515 25 International Journal of Foundations of Computer Science 22 2 https://doi.org/10.1142/S0129054111008155 Published - Feb 2011

### Fingerprint

Context free languages
Algebra

### Keywords

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

### ASJC Scopus subject areas

• Computer Science (miscellaneous)

### Cite this

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

In: International Journal of Foundations of Computer Science, Vol. 22, No. 2, 02.2011, p. 491-515.

Research output: Contribution to journalArticle

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