The paper studies the behavior of control structures of algorithms designed for (partly) parallel execution. A generalization of Peterson’s computation sequence set, the partial language, is discussed, which reflects the concurrency of events. In particular, the families of partial languages definable by Petri nets and by safe Petri nets are investigated with respect to closedness under certain operations. Trace languages (Mazurkiewicz) and path expressions (Campbell and Habermann) are included in the considerations.

We introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous.In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables.We apply these results to show that the quantum Grassmannians Kq[Gr(k, n)] admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Geiß-Leclerc-Schröer and completes earlier work of the authors on the finite-type cases.

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In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study.We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification.Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics-namely tropical frieze patterns-on the Auslander-Reiten quivers of the categories.

Holm and Jørgensen have shown the existence of a cluster structure on a certain category D that shares many properties with finite type A cluster categories and that can be fruitfully considered as an infinite analogue of these. In this work we determine fully the combinatorics of this cluster structure and show that these are the cluster combinatorics of cluster algebras of infinite rank. That is, the clusters of these algebras contain infinitely many variables, although one is only permitted to make finite sequences of mutations.The cluster combinatorics of the category D are described by triangulations of an ∞-gon and we see that these have a natural correspondence with the behaviour of Plücker coordinates in the coordinate ring of a doubly-infinite Grassmannian, and hence the latter is where a concrete realization of these cluster algebra structures may be found. We also give the quantum analogue of these results, generalising work of the first author and Launois.An appendix by Michael Groechenig provides a construction of the coordinate ring of interest here, generalizing the well-known scheme-theoretic constructions for Grassmannians of finite-dimensional vector spaces. †