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...tropy of a source. These entropies give a real number quantifying, loosely speaking, the randomness of the source. Such functions provide insight into whether one source might be simulated by another =-=[10]-=-. In terms of partitioned process theories (C,Cfree), this suggests we might look at order-preserving functions from the ordered monoid derived from PCD(C,Cfree) to the real numbers. We call such func...

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...al subcategory Cfree. With this definition we see that examples of partitioned process theories abound, not only describing structures such as entanglement and athermality arising in applied sciences =-=[1, 6]-=-, but also simply natural ideas within mathematics itself. For example, any category C with products can be considered a symmetric monoidal category with monoidal product given by the categorical prod...

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...al subcategory Cfree. With this definition we see that examples of partitioned process theories abound, not only describing structures such as entanglement and athermality arising in applied sciences =-=[1, 6]-=-, but also simply natural ideas within mathematics itself. For example, any category C with products can be considered a symmetric monoidal category with monoidal product given by the categorical prod...

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... definitions coincide, although in general they do not. We also note that it is possible to interpret these different constructions as different methods of constructing an operad of ‘wiring diagrams’ =-=[9]-=- from the set of free processes. In this paper we are interested in understanding the ordered monoid corresponding to this resource theory. Recall that an ordered monoid (X ,⪰, ⋅) is a set X together ...

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...ories of parallel-combinable processes with discarding We formalise our ideas about resources and free processes using symmetric monoidal categories, following the work of Coecke, Fritz, and Spekkens =-=[2, 3]-=-. We say that a subcategory D of a category C is a wide subcategory if it includes all the objects of C. Definition 1. A partitioned process theory (C,Cfree) consists of a symmetric monoidal category ...

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...ete families of additive monotones for various partitioned process theories based on the category of finite sets, in order to shed light on the way such families can be constructed. 1 Introduction In =-=[3]-=-, Coecke, Fritz, and Spekkens make a well-illustrated case for viewing symmetric monoidal categories as theories of resources: the objects of the category are interpreted as resources, the morphisms m...

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...y each object is equipped with a commutative comonoid structure. We may generate a wide symmetric monoidal subcategory from these comonoid morphisms to serve as our category Cfree. A similar, topical =-=[7, 4, 8]-=-, example arises from the analogous construction on so-called multigraph categories—categories in which every object is equipped with a special commutative Frobenius monoid. We caution that a partitio...

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...y each object is equipped with a commutative comonoid structure. We may generate a wide symmetric monoidal subcategory from these comonoid morphisms to serve as our category Cfree. A similar, topical =-=[7, 4, 8]-=-, example arises from the analogous construction on so-called multigraph categories—categories in which every object is equipped with a special commutative Frobenius monoid. We caution that a partitio...

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...y each object is equipped with a commutative comonoid structure. We may generate a wide symmetric monoidal subcategory from these comonoid morphisms to serve as our category Cfree. A similar, topical =-=[7, 4, 8]-=-, example arises from the analogous construction on so-called multigraph categories—categories in which every object is equipped with a special commutative Frobenius monoid. We caution that a partitio...