Generative Structural Theory v1.21 — Closure-Preserving Sub-relations: Structural Conditions and Counting

GST v1.20 explained where the number 59 reported for C_2 comes from. It characterised the set H_lt(z_t) of proper sub-relations of one cut, z_t, that keep its reflexive transitive closure, and counted 59 of them; the identification of that set with C_2 rests on the cited lemma D2 and condition C-cone. This manuscript states the characterisation for any finite loop-free relation z and separates what is general from what is particular to z_t. A sub-relation w of z has the closure of z if and only if w is strongly connected inside each strongly connected component of z and, for each link of the transitive reduction of the condensation of z, w keeps at least one pair of z along that link (Theorem A2). The two conditions fall on disjoint sets of pairs, which gives a product formula for the number of such sub-relations (Corollary A3); the factor for the inside of a component is left as it stands. No proper sub-relation of this kind is closed (Proposition A4). With the code definitions used since v1.18, for loop-free relations on one node set with w in H_lt(z) and z a subset of z', the letter is letter(w, z') = [cl(z) != cl(z')] [h(z')] c3(z, z'), and [cl(z) != cl(z')] [h(z')] 0 when z is persistent (Proposition A5). For z_t the formula gives |H_le(z_t)| = 60 = 5 x 1 x 1 x 1 x 3 x 4 and |H_lt(z_t)| = 59, the factor 5 being argued inside the strongly connected part {0, 2, 3, 4}; it recovers the four groups of v1.20 and the letters 000 and 100. The decomposition is related to the known reduction of the minimum equivalent digraph problem; no novelty is claimed. All results are argued; no computation was made. The general results are not applied to, or evaluated on, any other particular cut; nothing is claimed about the member list of the class C evaluation, acceptance at a root, or why the stored answer was chosen.