Cluster algebras were conceived by Fomin and Zelevinsky in 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups and their quantum analogs. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, Poisson geometry, string theory, discrete dynamical systems and integrability, and combinatorics. Gradings for cluster algebras have been introduced in various ways by a number of authors and for a number of purposes. Recently, Booker-Price T in [J. Algebra, 2020, 560: 89-113] study the gradings arising from triangulations of marked bordered 2-dimensional surfaces without puncture. In this paper, we mainly study the degree of triangulation of marked bordered 2-dimensional surfaces with puncture. We give a valuation functions on such a surface by using the methods of [Acta Math., 2008, 21:83-146] and [J. Algebr. Comb., 2015, 42: 1111-1134]. We give two examples to illustrate the theory and show that each homogeneous space has finitely many variables. This work is jont with Dr. Yongyue Zhong.

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