代数线性序

摘要：

代数线性序是可数线性序的连续范畴代数上的一阶递归方案的初始解的一个组成部分，配备了求和运算和常数1。由于一般Mezei-Wright型结果，代数线性序恰好同构于代数树的叶子的线性序。使用Courcelle的代数树的特征，我们得到的事实是，一个线性序是代数的当且仅当它可以表示为字典序的确定性上下文无关的语言。当代数线性序是良序时，它的序类型是代数序数。证明了任意离散代数线性序的Hausdorff秩小于$ω^ω$ .因此，代数序数恰好是那些小于$ω^{ω^ω}$的序数。

英文摘要：

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 $ω^{ω^ω}$.

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