Cluster algebras with Grassmann variables

We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers’ which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step of understanding the notion of cluster superalgebra