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We describe a construction of ordered algebraic structures (ordered abeliansemigroups, ordered commutative semirings, etc.) and describe applications tocodimension-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 admitstransverse $\mathbb L$-measures for some $\mathbb L$. Transverse $\mathbbL$-measures can be used to understand classes of laminations much larger thanthe class of laminations admitting transverse positive $\mathbb R$-measures. Inparticular, we show that "finite or infinite depth measured laminations" arelaminations admitting transverse measures with values in a certain orderedsemiring $\bar{\mathbb O}$ satisfying the additional property that locally thevalues lie in a smaller semiring $\mathbb P$. We consider the "realizationproblem:" In one version, this deals with the problem whether an $\mathbbP$-invariant weight vector assigned to a branched manifold $B$ (satisfyingcertain branch equations) determines a lamination $L$ carried by $B$ with atransverse $\bar{\mathbb O}$-measure inducing the weights on $B$. We describefurther laminations which may not be $\mathbb L$-measured, but are"well-covered" by laminations with transverse $\mathbb L$-measures. We alsoinvestigate actions on $\mathbb L$-trees which are associated to essentiallaminations 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 ofprobability measure.

Laminations with transverse measures in ordered abelian semigroups