By Schmidt G.

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**Sample text**

Given any pair of heterogeneous relations A : P −→ V and E : P −→ V that are both mappings, we may speak of a directed graph, interpreting i) ii) iii) iv) V P A E as a set of vertices, as a set of arrows, as a relation assigning precisely one initial vertex to each arrow, as a relation assigning precisely one target vertex to each arrow. An example is provided with Fig. 5 in which one should have in particular a look on the parallel arrows s, t. They could not be distinguished by the concept of a 1-graph as presented in Def.

Observe that “∃” stands as the possibly infinite version of “∨” in much the same way as “ ” does for “+”. (R; S)xz = ∃i ∈ Y : Rxi ∧ Siz (R · S)xz = i∈Y Rxi · Siz For the presentation of the whole text of this book it is important to have some formulae available which in the strict line of development would be presented only later. We exhibit them here simply as observations. 3 Proposition. Let triples of relations Q, R, S or A, B, C be given, and assume that the constructs are well-formed. Then they will always satisfy the Schr¨ oder rule: and the A; B ⊆ C ⇐⇒ AT ; C ⊆ B ⇐⇒ C; B T ⊆ A 42 Algebraic Operations on Relations R; S ∩ Q ⊆ (R ∩ Q; S T ); (S ∩ RT; Q) Dedekind rule: Some other rules one might have expected follow from these, not least the wellknown (R; S)T = S T; RT for transposing products.

A nested recursive definition for this procedure is possible. Some more information on how P has been designed is contained in Appendix C. PART TWO OPERATIONS AND CONSTRUCTIONS At this point of the book, a major break of style may be observed. So far we have used free-hand formulations, not least in Haskell, and have presented basics of set theory stressing how to represent sets, subsets, elements, relations, and mappings. We have, however, so far not used relations in an algebraic form. From now on, we shall mainly concentrate on topics that inherently need some algebraic treatment.