  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 $\nu$ 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.
