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Let $\boldsymbol{\Sigma}:=(\Sigma,\mathbb{M},\mathbb{P})$ be a surface withmarked points $\mathbb{M}\subset\partial\Sigma\neq\varnothing$ on the boundary,and punctures $\mathbb{P}\subset\Sigma\setminus\partial\Sigma$, and $T$ anarbitrary tagged triangulation of $\boldsymbol{\Sigma}$ in the sense ofFomin-Shapiro-Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T), W(T))$corresponding to the non-degenerate potential $W(T)$ defined by Cerulli Irelliand the second author is tame, as shown by Schr\"{o}er and the first twoauthors. In this paper, we show that there is a natural isomorphism$\pi_T:\operatorname{Lam}(\boldsymbol{\Sigma})\rightarrow\operatorname{DecIrr}^\tau(A(T))$of tame partial KRS-monoids that intertwines dual shear coordinates withrespect to $T$, and generic $g$-vectors of irreducible components. Here,$\operatorname{Lam}(\boldsymbol{\Sigma})$ is the set of laminations of$\boldsymbol{\Sigma}$ considered by Musiker-Schiffler-Williams, with thedisjoint union of non-intersecting laminations as partial monoid operation. Onthe other hand, $\operatorname{DecIrr}^\tau(A(T))$ denotes the set ofgenerically $\tau$-reduced irreducible components of the decoratedrepresentation varieties of $A(T)$, with the direct sum of generically$E$-orthogonal irreducible components as partial monoid operation, where $E$ isthe symmetrized $E$-invariant of Derksen-Weyman-Zelevinsky,$E(-,\bullet)=\dim\operatorname{Hom}_{A(T)}(-,\tau(\bullet))+\dim\operatorname{Hom}_{A(T)}(\bullet,\tau(-))$.

Laminations of punctured surfaces as $\tau$-reduced irreducible components