A step towards cluster superalgebras

We introduce a class of commutative superalgebras generalizing cluster algebras. A cluster superalgebra is defined by a hypergraph called an "extended quiver", and transformations called mutations. We prove the super analog of the "Laurent phenomenon", i.e., that all elements of a given cluster superalgebra are Laurent polynomials in the initial variables, and find an invariant presymplectic form. Examples of cluster superalgebras are provided by superanalogs of Coxeter's frieze patterns. We apply the Laurent phenomenon to construct a new integer sequence extending the Somos-$4$ sequence.