By Fomin A.A.

Each τ -adic matrix represents either a quotient divisible workforce and a torsion-free, finite-rank workforce. those representations are an equivalence and a duality of different types, respectively.

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**Sample text**

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 only be presented later. We exhibit them here simply as observations. 4. 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 Algebraic operations on relations 42 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.

4 Equivalence Ξ with simultaneous permutation P to a block-diagonal form When permuting only the rows of a homogeneous relation, it will become a heterogeneous relation, albeit with square matrix. 2 Relations describing graphs Relational considerations may very often be visualized with graphs. , relations. Usually, graph theory stresses other aspects of graphs than our relational approach. So we will present what we consider relational graph theory. This will mean, not least, making visible the diﬀerences between the various forms in which graphs are used with relations.

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