By Rudenskaya O. G.
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Additional info for 3-Quasiperiodic functions on graphs and hypergraphs
2) Every finitely presented R-module is isomorphic to EB�= l Rj Rai for some n � 0 and ai E R (1 :S i :S n). Proof (1) Let Xl , . . , Xn be a basis for L and let Yl,' . ,Ym be a finite set of generators for M. Since Yi E L, we can write Yi = 2:: 7=1 aijXj for 1 :S i :S m, where aij E R. 4, there exist P E GLm (R) and Q E GLn (R) such that PAQ = D , where D is a diagonal matrix over R. Write P = (Pij) , p - l = (Pij), Q = (qij) and Q - l = (iiij). Put ei = 2::7= 1 iiijXj and fi = 2::� 1 PijYi· Then fi E M, and M is generated by h , .
El , va) E J(y, x) , and it follows easily that dr (x, y) = dr(y, x) . Clearly dr (x, y) 2 0, and dr (x, x) = O. Let JI(X, y) be the set of v = (vo, e l , . . , en, vn) E J(x, y) such that (1) for 1 ::; i ::; n - 1 , real (ei) i- real (ei+ d j (2) e l , . . , en is a reduced path in f. If v = (vo , e l , . . , en, vn) E J(x, y) and real ( ei ) = real(ei+ l ) for some i, then we can omit Vi and ei+ 1 from v to obtain a new element of J(x, y), without increasing Iv (since de satisfies the triangle inequality, for every edge e).
It is transitive because d(v, w) = Cor. 6. x, y, z Definition. The equivalence classes are called directions (or direction germs) at v. The degree of v, denoted by deg(v) , is defined to be the cardinal number of the set of directions at v. If deg(v) � 1 , v is called an endpoint of X , if deg(v) = 2 then v is called an edge point of X, and if deg(v) � 3 then v is called a branch point of X. Note that, with this definition, the endpoints of a segment which is consistent with earlier usage. [x, y] are x, y, Exercise.