By Gottfried Wilhelm Leibniz
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LL This means that ~ is the pattern. associated to the f_la p ' the (p+l)-level. which corresponds to the (given) pattern w at the p-level. Now an increment 6w. in a pattern w. may thereby be associated to such 48 a All. via In the applications of the idea of patterns we interpret an incremental change 6w as a force in the structure; if 6w speak of a t-force in the complex K. to a pattern (~~) = (6w)t then we This t-force is therefore associated defined at the (t+I)-level. It is in this sense that we interpret the incremental 6w in the abstract geometry of K.
X j = -x. J Xi; for all i. j. 51 It fol1ows that Ex (X UX) contains the sub-algebras Ax and AX, it is not the direct sum of these. but Furthermore, every oriented, augmented, complemented complex K (abbreviated to "OAC-complex") possesses an algebraic representation in Ex (K). 1T This representation is via a polynomial, ~ - i l Ei 0 p=O P o where Ei = 0, 1; m' n - 1 and where 0p i - i E K; each 0p being represented by a monomial of the form x Qs in Ex (K). When s =p xß xß 1 2 x ßp _S + l this; i E K, with a representation in A(K), and + p when x = 0 this 0 i E K, with a representation in A(K).
Indicating that White has placed p crosses and Black q noughts. Then r is a function of the mode. and so the structures Ks(L; r) and KL (S; r- l ) are also functions of the mode. We consider KL(S) in agame which White wins (by faulty play on Black's behalf) and represent the status of White's game (and of Black's) by a pattern polynomial on this KL(S). algebraic variables in AV will be SI' s2' •••• s9' The When White places a cross in square 5 i then this square has been removed from Baack's view of the board, and so the structure of the game (from Black's side) has altered.