Order 221 = 1 + 5*44, from a common submagma at infinity.
C(E, M, k, B) adjoins a submagma M (m elements) of a model E to every group of a transversal design TD(k, g), g = |E| - m, so that every enlarged group is a copy of E and all of them share M; every block carries B, a model of order k in which x*x = x. Two elements of M multiply in M; an element of M or of group i with one of group i, in group i's copy of E; elements of different groups, in their block. It satisfies Equation 677 whenever E and B do.
This magma is C(E, M, 5, B) with
E = P(11, 2, magma#e549b5f8492c9b6b5ad530e3aa4f39c6e23d08645ebd1b36a3c2de2a5a23bac5, 0),
M = {1} in E,
B = magma#e549b5f8492c9b6b5ad530e3aa4f39c6e23d08645ebd1b36a3c2de2a5a23bac5.
Labels: 0..m-1 are M, in increasing order of their labels in E; m + g*i + c (i < k, c < g) is point c of group i, which plays the c-th element of E outside M, in increasing order. A block's point in group i plays element i of B.
TD(5, 44): MacNeish's product over GF(4) x GF(11); GF(4) = F_2[x]/(x^2 + x + 1); GF(p^r) = F_p[x]/(f) labels c_0 + c_1 x + ... as c_0 + c_1 p + ..., and point c of a group is the c-th element of the product in lexicographic order. Block (a, b), a and b in the product, meets group i in the point whose coordinate in each field is a + s_i b, s_i the element of that field labelled i, or b when the field has exactly i elements.
P(l, q, S, v) is the Paley pencil over Z/l with multiplier q on the seed S of order K + 1 with vertex v: S is relabelled with v as 0 and its other elements as 1..K in increasing order; the points are 0 (the vertex) and 1 + K x + t for (x, t) in Z/l x Z/K; points of one line, or a point and the vertex, multiply as in S (0 the vertex, t + 1 the point (x, t)); and (x, s)*(y, t) for x != y is ((1-q)x + qy, t) when y - x is a nonzero square mod l, else ((1-q)x + qy, -s-t mod K).
Equation 255: satisfied. Idempotent elements: 221.
qawbecrdtey · 2026-10-01 22:08:20