Decomposing a Go-Board into Cells

The articles [20], [23], [6], [22], [9], [2], [14], [17], [18], [24], [1], [5], [3], [4], [21], [10], [11], [16], [15], [7], [8], [12], [13], and [19] provide the terminology and notation for this paper. For simplicity we follow a convention: q will be a point of E 2 T, i, i1, i2, j, j1, j2, k will be natural numbers, r, s will be real numbers, and G will be a Go-board. We now state the proposition (1) Let M be a tabular finite sequence and given i, j. If 〈i, j〉 ∈ the indices of M , then 1 ≤ i and i ≤ lenM and 1 ≤ j and j ≤ widthM. Let us consider G, i. The functor vstrip(G, i) yielding a subset of the carrier of E T is defined as follows: (Def.1) (i) vstrip(G, i) = {[r, s] : (Gi,1)1 ≤ r ∧ r ≤ (Gi+1,1)1} if 1 ≤ i and i < lenG, (ii) vstrip(G, i) = {[r, s] : (Gi,1)1 ≤ r} if i ≥ lenG, (iii) vstrip(G, i) = {[r, s] : r ≤ (Gi+1,1)1}, otherwise. The functor hstrip(G, i) yields a subset of the carrier of E 2 T and is defined by: (Def.2) (i) hstrip(G, i) = {[r, s] : (G1,i)2 ≤ s ∧ s ≤ (G1,i+1)2} if 1 ≤ i and i < widthG, (ii) hstrip(G, i) = {[r, s] : (G1,i)2 ≤ s} if i ≥ widthG, (iii) hstrip(G, i) = {[r, s] : s ≤ (G1,i+1)2}, otherwise. We now state a number of propositions: (2) If 1 ≤ j and j ≤ widthG and 1 ≤ i and i ≤ lenG, then (Gi,j)2 = (G1,j)2. (3) If 1 ≤ j and j ≤ widthG and 1 ≤ i and i ≤ lenG, then (Gi,j)1 = (Gi,1)1. (4) If 1 ≤ j and j ≤ widthG and 1 ≤ i1 and i1 < i2 and i2 ≤ lenG, then (Gi1,j)1 < (Gi2,j)1.