序交换半群中的横测度层
Laminations with transverse measures in ordered abelian semigroups
摘要：
我们描述了序代数结构（序交换半群、序交换半环等）的一种构造方法。并描述了余维1叠层的应用。对于适当的序半代数结构$\mathbb L$和可测空间$X$，我们定义了$X$上的$\mathbb L$-测度$ν$.如果$L$是流形中的余维1叠层，则对于某些$\mathbb L$，它经常允许横向$\mathbb L$-测度。横向$\mathbb L$-测度可以用来理解比允许横向正$\mathbb R$-测度的叠层类大得多的叠层类。特别是，我们表明，"有限或无限深度测量叠层"是叠层承认横向措施的值在一定的有序半环$\bar{\mathbb O}$满足的附加属性，当地的值位于一个较小的半环$\mathbb P$。我们认为"实现问题"：在一个版本中，这涉及的问题是否$\mathbb P$不变的权重向量分配给一个分支流形$B$（满足某些分支方程）确定一个层$L$进行$B$与横向$\bar{\mathbb O}$-措施诱导$B$的权重。我们进一步描述了可能未进行$\mathbb L$测量，但被具有横向$\mathbb L$测量的叠层"良好覆盖"的叠层。我们还调查行动的$\mathbb L$树与横向$\mathbb L$措施的基本层压。在附录中，我们进一步发展了关于$\mathbb L$-度量的思想，例如，展示了$\mathbb P$-度量可以被解释为一种概率测度。
英文摘要：
We describe a construction of ordered algebraic structures (ordered abelian semigroups, ordered commutative semirings, etc.) and describe applications to codimension-1 laminations. For a suitable ordered semi- algebraic structure $\mathbb L$ and measurable space $X$ we define $\mathbb L$-measures $ν$ on $X$. If $L$ is a codimension-1 lamination in a manifold, it often admits transverse $\mathbb L$-measures for some $\mathbb L$. Transverse $\mathbb L$-measures can be used to understand classes of laminations much larger than the class of laminations admitting transverse positive $\mathbb R$-measures. In particular, we show that "finite or infinite depth measured laminations" are laminations admitting transverse measures with values in a certain ordered semiring $\bar{\mathbb O}$ satisfying the additional property that locally the values lie in a smaller semiring $\mathbb P$. We consider the "realization problem:" In one version, this deals with the problem whether an $\mathbb P$-invariant weight vector assigned to a branched manifold $B$ (satisfying certain branch equations) determines a lamination $L$ carried by $B$ with a transverse $\bar{\mathbb O}$-measure inducing the weights on $B$. We describe further laminations which may not be $\mathbb L$-measured, but are "well-covered" by laminations with transverse $\mathbb L$-measures. We also investigate actions on $\mathbb L$-trees which are associated to essential laminations with transverse $\mathbb L$-measures. In appendices, we develop ideas about $\mathbb L$-measures a little further, for example showing that a $\mathbb P$-measure can be interpreted as a kind of probability measure.
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