Equation 677 Database

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Magma 59b541d0a50d…

magma 59b541d0a50d
Size
539
Isomorphism class hash
59b541d0a50d82cdf4265a30a7b4ff6de728ad6148de2b384e4bd287ef65f297
Satisfies Equation 255
yes
Right-cancellative
no
Idempotent
no
Fiber matrix
symmetric: no · normal: no · rank: 529 (nullity 10) what is this?
Submitted by
dwrensha
Submitted at
2026-09-05 00:11:02
Display reorder
17,23,12,13,14,15,16,39,40,41,36,37,38,7,8,9,10,11,6,31,32,33,34,35,30,29,24,25,26,27,28,1,2,3,4,5,0,19,20,21,22,18,522,523,524,525,526,527,528,534,234,224,229,244,239,249,286,407,183,360,482,100,112,306,473,427,90,173,104,350,311,95,355,402,109,477,178,296,162,463,124,417,370,80,291,365,119,188,75,412,487,301,129,85,468,345,168,422,531,271,261,266,281,276,256,319,445,217,392,517,62,149,339,507,435,52,207,139,382,314,57,387,440,144,512,212,329,197,497,159,455,372,46,324,397,154,192,67,450,492,334,134,49,502,377,202,460,537,232,222,227,242,237,247,285,409,186,361,485,101,116,305,475,429,91,176,106,351,310,96,356,404,111,480,181,295,166,465,126,419,371,81,290,366,121,191,76,414,490,300,131,86,470,346,171,424,535,235,225,230,245,240,250,282,411,184,357,483,98,115,302,472,431,88,174,103,347,307,93,352,406,107,478,179,292,164,462,123,421,367,78,287,362,117,189,73,416,488,297,128,83,467,342,169,426,536,236,226,231,246,241,251,283,410,185,359,484,99,114,303,474,430,89,175,105,349,308,94,354,405,110,479,180,293,165,464,125,420,369,79,288,364,120,190,74,415,489,298,130,84,469,344,170,425,530,269,259,264,279,274,254,320,443,220,393,521,65,151,340,511,433,54,210,141,383,315,60,388,438,146,516,215,330,200,501,161,453,373,44,325,398,156,195,70,448,496,335,136,50,506,378,205,458,532,267,257,262,277,272,252,318,442,219,395,520,66,147,338,510,432,56,209,137,385,313,61,390,437,142,515,214,328,199,500,157,452,375,45,323,400,152,194,71,447,495,333,132,51,505,380,204,457,529,270,260,265,280,275,255,321,444,221,394,518,63,148,341,508,434,55,211,138,384,316,58,389,439,143,513,216,331,201,498,158,454,374,43,326,399,153,196,68,449,493,336,133,47,503,379,206,459,538,233,223,228,243,238,248,284,408,182,358,486,97,113,304,476,428,87,172,102,348,309,92,353,403,108,481,177,294,163,466,122,418,368,77,289,363,118,187,72,413,491,299,127,82,471,343,167,423,533,268,258,263,278,273,253,317,446,218,396,519,64,150,337,509,436,53,208,140,386,312,59,391,441,145,514,213,327,198,499,160,456,376,42,322,401,155,193,69,451,494,332,135,48,504,381,203,461 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

A NON-IDEMPOTENT Eq677 magma with a NONNORMAL fiber matrix, of order 539=11*49. Exactly 11 elements are idempotent; the other 528 are not. It still satisfies Eq255, so this is not a counterexample to the main implication. Carrier F_11 x F_7^2. Pack a fiber vector (u,v) as 7u+v. Define (i,s)*(j,t) = (6j-5i mod 11, H_ij(s,t)). For i!=j, take H_ij(s,t)=t if j-i is a nonzero square in F11, and H_ij(s,t)=-s-t otherwise, with vector arithmetic in F7^2. For diagonal i=j!=0 take H_ii(s,t)=4s+3t. At i=j=0, use the canonical table of the non-idempotent, non-right-cancellative order-49 source magma#d50b3565f55696fb14ff77cb5b87148a39c99833e06e009f0d6349acb1e2c81d. Why Eq677 holds: distinct base inputs make all four products stay off the diagonal. The four fiber operations are A,A,B,B or B,B,A,A, where A(s,t)=t and B(s,t)=-s-t; both chains return the required input over any abelian group. Equal base inputs stay in a single diagonal fiber. Each diagonal law satisfies Eq677 independently; 4s+3t is the product of two standard non-idempotent seven-element models. Matrix explanation: let D be the fiber matrix of the order-49 source, K=D-J_49, B[z,j]=1 iff j-z is a nonzero nonsquare in F11, and E have its sole nonzero entry E[0,0]=1. Then F = J_539 + B tensor (49I_49-J_49) + E tensor K, FF^T-F^TF = 49*((B-B^T)E-E(B-B^T)) tensor K. D is symmetric, and K is nonzero, so the commutator is nonzero. These two complete matrix identities were also verified using integer arithmetic. In construction labels (i,s)->49i+s, 0*0=47, witnessing non-idempotence. Nonnormality witness: (FF^T)[0,53]=735, while (F^TF)[0,53]=441. Canonical server labels can differ. All 290,521 Eq677 instances, Eq255, the 11-element idempotent set, and both witnesses were independently verified by a separate scalar checker before submission. Corresponding rows and columns of F still have identical multisets of entries. Constructed and verified with OpenAI Codex, 2026-09-04. This adapts the independent-diagonal mechanism of magma#dc2f657e34e8de11126c640cda74edb6 using a non-idempotent core.

last edited by dwrensha at 2026-09-05 00:11:16 · history