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3 Definition. Given any homogeneous relation B : V −→ V , we may speak of a 1-graph, interpreting i) V as a set of vertices, ii) B as a relation providing arrows between vertices of V , the associated relation. As an example we present V = {a, b, c, d } B = {(a, a), (d, a), (d, c), (c, d)} ⊆ V × V which is also shown as a graph and as a matrix in Fig. 3. visualized as a 1-graph a b c d a d b a b c d 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 stored as a matrix c Fig. 3 Homogeneous relation as a 1-graph A little bit of caution is necessary when presenting a relation on some set without further context.

A reflexive, transitive, and antisymmetric relation; when arranged nicely, it immediately turns out to be. It is, thus, a specific task to identify the techniques according to which one may get a nice arrangement. For a given example, this will easily be achieved. But what about an arbitrary input? Can one design a general procedure? 17 1 2 0 19 0 9 0  3 0 7 0 4 0  8 0 6 0 5 0  1 0 10 0 15 0  11 0 12 0 13 0 14 0 16 0 18 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0  0 0 0  0 0 0  0 0 0  0 0 0 0 0 1 Fig.

2 The right residual describes how rows of the relation Q above the fraction slash contain rows of the second relation P below We will use the following identities that may be seen as a first test as to the quotient properties. 3 Proposition.

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