This chapter refines the structural analysis of concrete bigraphs. In Section 5.1 we establish some properties for concrete bigraphs, including RPOs. In Section 5.2 we enumerate all IPOs for a given span. Finally, in Section 5.3 we show that RPOs do not exist in general for abstract bigraphs.

RPOs for bigraphs

We begin with a characterisation of epimorphisms (epis) and monomorphisms (monos) in bigraphs. These notions are defined in a precategory just as in a category, as follows:

Definition 5.1 (epi, mono) An arrow f in a precategory is epi if g ° f = h ° f implies g = h. It is mono if f ° g = f ° h implies g = h. 〉

Proposition 5.2 (epis and monos in concrete bigraphs)A concrete place graph is epi iff no root is idle; it is mono iff no two sites are siblings. A concrete link graph is epi iff no outer name is idle; it is mono iff no two inner names are siblings.

A concrete bigraph G is an epi (resp. mono) iff its place graph GPand its link graph GLare so.

EXERCISE 5.1 Prove the above proposition, at least for the case of epi link graphs. Hint: Make the following intuition precise: if G and H differ then, when composed with F, the difference can be hidden if and only if F has an idle name. 〉

The proposition fails for abstract bigraphs, suggesting that concrete bigraphs have more tractable structure. We shall now provide further evidence for this by constructing RPOs for them.