Quasi-cluster algebras from non-orientable surfaces

With any non-necessarily orientable unpunctured marked surface , we associate a commutative algebra , called quasi-cluster algebra, equipped with a distinguished set of generators, called quasi-cluster variables, in bijection with the set of arcs and one-sided simple closed curves in . Quasi-cluster variables are naturally gathered into possibly overlapping sets of fixed cardinality, called quasi-clusters, corresponding to maximal non-intersecting families of arcs and one-sided simple closed curves in . If the surface is orientable, then is the cluster algebra associated with the marked surface in the sense of Fomin, Shapiro and Thurston. We classify quasi-cluster algebras with finitely many quasi-cluster variables and prove that for these quasi-cluster algebras, quasi-cluster monomials form a linear basis. Finally, we attach to a family of discrete integrable systems satisfied by quasi-cluster variables associated to arcs in and we prove that solutions of these systems can be expressed in terms of cluster variables of type A.