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By Martin Heidegger (Trans. H. Robbins)

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For vectors and matrices we define addition and multiplication by scalars as componentwise addition and multiplication by scalars in M. For v, w:X ~M, F, G:XxY ~M and i E R we have (v+w):x ...... v(x)+w(x) and (iv):x (F+G):(x,y) f-t f-t iv(x) for x E X, F(x,y)+G(x,y) and (iF):(x,y) f-t iF(x,y) for x E X,y E Y. 6). 6 Let R be ring, let M be a R-module, and let X, Y be arbitrary sets. Under addition and multiplication by scalars, the set of X-vectors over M and the set of X,Y-matrices over M are R-modules.

While the scalar product v·w is an integer, v·w = -6 - 2 + 0 = -8, we now wish to extend C(X) to the ring of formal polynomials in X such that the ring product of v and w is the polynomial vw = -6a 2 + tab - 2ac - 2b 2 + bc. 15 Let X be a finite set, X = {Xl, ••• ,X n }. The set of polynomials in X over Z is designated as A(X) (A because we get a linear algebra over X). It is defined as A(X) = [[X -+N]-+Z]Jin' The elements of [X -+N] are called the monomials. The monomials form a commutative monoid which we write with multiplication.

We do not suggest, however, to always translate invariants into dead transitions. Our main reason for assigning a net theoretic interpretation to every formula prefixed by the modal symbol ILl was to make sure that it can be used safely, beyond any doubt about its meaning, for expressing an invariant property of a PrT-net. In addition, the translation into dead transition yields a decomposition of an arbitrary logical invariant into a set of elementary net theoretic system invariants. 28 233 5 Linear Algebraic Analysis of PrT-nets The transition rules for ordinary PT-nets as well as for PrT-nets show that the occurrences of transitions have the following linearity property.

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