2010/02/08 by Stephen L. Bloom, Bloom, Stephen L., Zoltán Ésik +2
Computer Science · #Advanced Algebra and Logic #Logic, programming, and type systems #cs.FL #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1002.1624
arxiv created 2010/02/10 · arxiv updated 2010/02/26
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.