It was shown by Fock and Goncharov (Dual Teichmüller and lamination spaces. Handbook of Teichmüller Theory, 2007), and Fomin et al. (Acta Math 201(1):83–146, 2008) that some cluster algebras arise from triangulated orientable surfaces. Subsequently, Dupont and Palesi (J Algebraic Combinatorics 42(2):429–472, 2015) generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasi-cluster algebras. In Wilson (Int Math Res Notices 341, 2017) we linked this framework to Lam and Pylyavskyy’s Laurent phenomenon algebras (J Math 4(1):121–162, 2016), showing that unpunctured surfaces admit an LP structure. In this paper we extend quasi-cluster algebras to include punctured surfaces. Moreover, by adding laminations to the surface we demonstrate that all punctured and unpunctured surfaces admit LP structures. In short, we link two constructions which arose as seemingly unrelated generalisations of cluster algebras—one of the generalisations (quasi-cluster algebras) being based on triangulated surfaces, and the other (Laurent phenomenon algebras) based on the Laurent phenomenon. We thus provide a rich class of geometric examples in which to help study Laurent phenomenon algebras.

Fock和Goncharov（双重Teichmüller和层压空间。Teichmüller理论手册，2007年）和Fomin等人对此进行了展示。（Acta Math 201（1）：83–146，2008）某些聚类代数是由三角可定向曲面产生的。随后，Dupont和Palesi（J Algebraic Combinatorics 42（2）：429–472，2015）将这种构造推广到包括未打孔的不可定向曲面，从而产生了准集群代数。在Wilson（Int Math Res Notices 341，2017）中，我们将此框架与Lam和Pylyavskyy的Laurent现象代数关联（J Math 4（1）：121–162，2016），表明未穿刺的表面承认LP结构。在本文中，我们将拟簇代数扩展为包括穿孔的表面。此外，通过在表面上添加层压板，我们证明了所有穿孔和未穿孔的表面均允许使用LP结构。简而言之，我们将两种构造联系在一起，这似乎是簇代数的看似无关的概括—一种是基于三角化曲面的（准簇代数），另一种是基于Laurent现象的（Laurent现象代数）。因此，我们提供了丰富的几何示例，可帮助研究洛朗现象代数。