Abstract. A categoria! semantic domain for general (discrete event) systems based on labeled transition systems with full concurrency is constructed, where synchronization and hiding are functorial. Moreover, we claim that, within the proposed framework , a class of mappings stands for refinement. 1l1en we prove that refinement satisfies the diagonal compositionality requirement, i.e. , refinements compose (vertical) and distribute over system composition (horizontal). IntroductionWe construct a semantic domain for interacting systems which satisfies the diagonal compositionality requirement, i.e., refmements compose (vettically) , reflecting the stepwise description of systems, involving severa! leveis of abstraction, and distributes through combinators (horizontally), meaning that the refinement of a composite system is the composition of the refmement of its parts. Taking into consideration the developments in Petri net theory (mainly with seminal papers like [17], [11] and [15]) it was clear that nets rnight be good candidates. However, most of net-based models such as Petri nets in the sense of [14] and labeled n·ansition systems (see [12]) lack composition operations (modularity) and absn·action mechanisms in their original defmitions. This motivate the use of the category theory: the approach in [17] provides the former, where categorical constructions such as product and coproduct stand for system composition, and the approach in [11] provides the later for Pen·i nets where a special kind of net morphism conesponds to the notion of irnplementation. Also, category theory provides powelful techniques to unify different categories of models (i.e., classes of models categorically sn·uctured) through adjunctions (usually reflections and coreflections) expressing the relation of their semantics as in [15].We introduce the concept of (nonsequential) automaton as a kind of automaton structured on states and n·ansitions. Structured states are "bags" of local states like tokens in Pen·i nets and structured transitions specify a concwTency relationship between component transitions in the sense of [3] and [7]. In [9] we show that nonsequential automata are more concrete then Petri nets (in fact, categories of Petri nets are isomorphic to subcategories of nonsequential automata) extending the approach in [15], where a formal framework for classification of models for concunency is set.The resulting category is bicomplete where the categoria! product and coproduct stand for (system) composition. Synchronization and hiding are functorial operations. A Refinernent Mapping for General (Discrete E vent) Systems Theory

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A categorical semantic domain is constructed for Petri nets which satisfies the diagonal compositionality requirement with respect to anticipations, i.e., Petri nets are equipped with a compositional anticipation mechanism (vertical compositionality) that distributes through net combinators (horizontal compositionality). The anticipation mechanism is based on graph transformations (single pushout approach). A finitely bicomplete category of partia! Petri nets and partia! morphisms is introduced. Classes of transformations stand for anticipations. The composition of anticipations (i.e., composition of pushouts) is defined, leading to a category of nets and anticipations which is also complete and cocomplete. Since the anticipation operation composes, the vertical compositionality requirement of Petri nets is achieved. Then, it is proven that the anticipation also satisfies the horizontal compositionality requirement. A specification grammar stands for a system specification and the corresponding induced subcategory of nets and anticipation's stands for ali possible dynamic anticipation's of the system (objects) and their relationship (morphims).

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