Size 121 = 11² is among the most populated sizes in the DB (65 magmas as of writing). All 65 are idempotent and satisfy Equation 255. The cleanest sorting is by **# size-11 sub-magmas** (= # parallel classes of AG(2, 11) closed under the operation):
| sub-11 | classes | # | RC | family |
|---|---|---|---|---|
| 0 | 0 | 8 | mixed | F_121 / Zassenhaus / order-k multiplier |
| 12 | 11 fibers + 1 transversal | 1 | LC only | magma#ee53a960 fiber-bundle |
| 22 | 2 | 8 | yes | "2-class" twisted AG(2, 11) |
| 44 | 4 | 2 | no | "4-class" twisted AG(2, 11) |
| 88 | 8 | 4 | no | "8-class" twisted AG(2, 11) |
| 132 | all 12 | 42 | yes | full AG(2, 11) line family |
**Within sub-11 = 0**, the 8 magmas split by L_0/R_0 cycle:
- **1 + 15·8** (multiplier order 8, NEW family of 5): magma#70e5572a, magma#873c2695, magma#58669675, magma#af70b7bb, magma#fdc7e335. Uses (F_121*)^15 (order-8 subgroup) as multiplier; OUTSIDE Pace Nielsen's Type-1/Type-2 (which require multiplier order 10 or roots of Phi_2_5).
- **1 + 6·20** (multiplier order 20, 1 magma): magma#74c24c4d.
- **1 + 4·30** (multiplier order 30, 1 magma): magma#5ebfbb80, the **Zassenhaus exceptional near-field**. Multiplicative group Z_5 × SL(2, 3); Aut sharply 2-transitive of order 14,520; also satisfies Eq 3345.
- **Asymmetric L_0/R_0** (NON-RC, 1 magma): magma#3f8367e9.
**Construction families:**
1. **Direct products F_11 × F_11** (sub-11 = 132, 4 entries): (x_1, x_2) ◇ (y_1, y_2) componentwise. α ∈ {2, 6, 7, 8}: magma#d4b8c23f, magma#199764ee, magma#65eed598, magma#c3819eed.
2. **Full AG(2, 11) line family** (sub-11 = 132, 42 magmas). All 12 parallel classes preserved as F_11-line sub-magmas; non-isomorphic Phi_10-slope assignments give different iso classes (e.g. 12+0, 10+2, 8+4, 6+6 splits). Example: magma#83a10ff4, magma#270b7c37 (10+2 split).
3. **"k-class" twisted AG(2, 11)** (NEW). Exactly k ∈ {2, 4, 8} of 12 parallel classes preserved. k=2: 8 RC iso classes including magma#6bab1ff9. k=4 and k=8: 2 + 4 non-RC iso classes; see magma#014291bf and magma#c8dcedeb.
4. **Zassenhaus near-field II** (sub-11 = 0, magma#5ebfbb80; see above).
5. **"Order-8 multiplier" family** (NEW, sub-11 = 0, 5 magmas; see above).
6. **"Order-20 multiplier"** (sub-11 = 0, 1 magma): magma#74c24c4d.
7. **Twisted product fiber bundle** (sub-11 = 12, magma#ee53a960). 11 fibers + 1 transversal; LC but not RC.
8. **Non-RC outlier** (sub-11 = 0, magma#3f8367e9).
**Display convention**: magmas with sub-11 ≥ 22 use an F_11 × F_11 grid reorder where positions 11·r + c correspond to the intersection of P1-row r and P2-column c (using 2 preserved parallel classes as axes). The 5 "order-8" magmas use an F_11 × F_11 ADDITIVE grid (commuting order-11 translations as axes).
**Lower bound on iso classes**: at least 13,794 distinct Eq 677 magmas of size 121 exist (Burnside count over the AG(2, 11) line construction, varying Phi_10-slope assignments to the 12 parallel classes). The 65 currently in the DB are a small sample of this combinatorial space.
[text written by Claude]
dwrensha · 2026-05-18 17:01:20
Size 121 = 11² is among the most populated sizes in this database (65 magmas as of writing), reflecting the abundance of structures available on (Z_11)² ≅ AG(2, 11) ≅ the additive group of F_121.
All 65 are idempotent and satisfy Equation 255. The cleanest sorting is by **number of size-11 sub-magmas**, which directly counts how many "parallel classes" of the underlying AG(2, 11) affine plane are closed under the operation:
| sub-11 count | parallel classes closed | # magmas | RC? | family |
|---|---|---|---|---|
| 0 | 0 | 8 | yes | F_121 affine line + Zassenhaus near-field |
| 12 | (11 fibers + 1 transversal) | 1 | no (LC only) | magma#ee53a960 fiber-bundle |
| 22 | 2 of 12 | 8 | yes | "2-class" twisted AG(2, 11) — new family |
| 44 | 4 of 12 | 2 | no | "4-class" twisted AG(2, 11) |
| 88 | 8 of 12 | 4 | no | "8-class" twisted AG(2, 11) |
| 132 | all 12 | 42 | yes | full AG(2, 11) line family (132 = 12·11) |
**Construction families currently represented:**
1. **F_121 affine line** (sub-11 = 0): Linear magma over the extension field F_121 = F_11[α]/⟨α²-r⟩. Operation x ◇ y = a*x + b*y with (a, b) ∈ F_121² satisfying Phi_10(b) = 0, a = 1 - b. Phi_10 has 4 roots in F_121 (since 10 | 120); up to Galois orbit, 2 iso classes. F_121 has unique F_11 prime subfield but the linear operation does NOT preserve it (multiplication by α ∈ F_121 \ F_11 maps F_11 outside F_11), so no size-11 sub-magmas.
2. **Direct products F_11 × F_11** (sub-11 = 132, 4 of the 42 entries). (x_1, x_2) ◇ (y_1, y_2) = ((1-α)x_i + α y_i mod 11) coordinate-wise. The 4 entries use α ∈ {2, 6, 7, 8}: magma#d4b8c23f, magma#199764ee, magma#65eed598, magma#c3819eed. In a direct product every parallel class is a sub-magma, putting these in the 132 bin.
3. **Full AG(2, 11) line magmas** (sub-11 = 132, 42 magmas total). All 12 parallel classes preserved as F_11-line sub-magmas. Each parallel class can carry a different slope α ∈ Phi_10 roots; non-isomorphic assignments give many iso classes. Example: magma#83a10ff4.
4. **NEW: "2-class" twisted AG(2, 11)** (sub-11 = 22, 8 magmas, RC). Exactly 2 of the 12 parallel classes are preserved as sub-magmas; the remaining 10 classes have lines that do not close. Every pair of points (x, y) not in the same preserved-class generates the whole 121-element magma. The 8 magmas: magma#18db3d82, magma#8edce4bd, magma#e2b4b45d, magma#ef441533, magma#0d2a7fdd, magma#6bab1ff9, magma#12644a1d, magma#38c42b28.
5. **"4-class" and "8-class" twisted AG(2, 11)** (sub-11 = 44 or 88, NON-right-cancellative). Intermediate cases: 4 classes preserved (2 magmas: magma#014291bf, magma#6c5e9b39) or 8 classes preserved (4 magmas: magma#c8dcedeb, magma#fdc3bc3f, magma#d7e85ead, magma#8b0e368b). All are non-RC, distinguishing them from the full AG(2, 11) line family.
6. **Zassenhaus exceptional near-field of order 121** (1 magma, magma#5ebfbb80, sub-11 = 0). Built on the Zassenhaus exceptional near-field of order 121; multiplicative group is Z_5 × SL(2, 3); Aut is the sharply 2-transitive (Z_11)² ⋊ (Z_5 × SL(2, 3)) of order 14,520. Also satisfies Eq 3345.
7. **Twisted product F_11 × F_11 fiber bundle** (1 magma, magma#ee53a960, sub-11 = 12, NON-RC). Decomposes as 11 fibers (one congruence) + 1 transversal sub-magma (12 size-11 sub-magmas in an AG(2, 11)-pencil configuration); left-cancellative but not right-cancellative.
**Display convention**: all magmas with sub-11 ≥ 22 have their reorder set to a F_11 × F_11 grid layout, with positions 11·r + c corresponding to the unique point at the intersection of P1-row r and P2-column c (using two of the preserved parallel classes as the grid axes). Diagonal blocks of the Cayley table then show the row sub-magma operations.
[text written by Claude]
dwrensha · 2026-05-18 16:57:59
Size 121 = 11² is among the most populated sizes in this database (65 magmas as of writing), reflecting the abundance of structures available on (Z_11)² ≅ AG(2, 11) ≅ the additive group of F_121.
All 65 are idempotent (x ◇ x = x) and satisfy Equation 255. Most (~58) are right-cancellative. The cleanest sorting is by **number of size-11 sub-magmas**, which directly counts how many "parallel classes" of the underlying AG(2, 11) affine plane are closed under the operation:
| sub-11 count | parallel classes preserved | # magmas | RC | family |
|---|---|---|---|---|
| 0 | none | 8 | mixed | F_121 affine line + Zassenhaus near-field |
| 12 | none (12 = 11 fibers + 1 transversal) | 1 | LC, not RC | magma#ee53a960 fiber-bundle |
| 22 | 2 of 12 | 8 | yes | "2-class" twisted AG(2, 11) — new family |
| 44 | 4 of 12 | 2 | yes | "4-class" twisted AG(2, 11) |
| 88 | 8 of 12 | 4 | yes | "8-class" twisted AG(2, 11) |
| 132 | all 12 | 42 | yes | AG(2, 11) line family (132 = 12*11 sub-magmas) |
**Construction families currently represented:**
1. **F_121 affine line** (sub-11 = 0): Linear magma over the extension field F_121 = F_11[α]/⟨α²-r⟩. Operation x ◇ y = a*x + b*y with (a, b) ∈ F_121² satisfying Phi_10(b) = 0, a = 1 - b. There are 4 Phi_10 roots in F_121 (since 10 | 120), giving 4 raw constructions; up to Galois iso, **2 iso classes**.
2. **Direct products F_11 × F_11** (sub-11 = ?): (x_1, x_2) ◇ (y_1, y_2) = ((1-α)x_i + α y_i mod 11) coordinate-wise. The 4 entries use α ∈ {2, 6, 7, 8}: magma#d4b8c23f, magma#199764ee, magma#65eed598, magma#c3819eed. (These appear in the sub-11 = 132 group, since every parallel class is a sub-magma in a direct product.)
3. **AG(2, 11) line magmas** (sub-11 = 132, 42 magmas). All 12 parallel classes preserved as F_11-line sub-magmas. Each parallel class can carry a different slope α ∈ Phi_10 roots; non-isomorphic assignments give many iso classes. Example: magma#83a10ff4.
4. **Twisted AG(2, 11) with k parallel classes preserved** (NEW). Only k ∈ {2, 4, 8} of the 12 parallel classes are closed under the operation. The remaining 12-k classes have lines that don't close — generating these pairs spans the whole magma. Three sub-families:
- **k = 2** (8 magmas): sub-11 = 22. Includes magma#6bab1ff9. The "minimum AG(2, 11) structure" — only 2 parallel classes give sub-magmas.
- **k = 4** (2 magmas): sub-11 = 44.
- **k = 8** (4 magmas): sub-11 = 88. (The earlier commentary placed this in a non-RC family; the current 4 RC entries show the same "8 closed classes" structure is achievable while remaining a quasigroup.)
These differ from the full AG(2, 11)-line family by an *intentional* breaking of some parallel classes.
5. **Zassenhaus exceptional near-field of order 121** (1 magma, magma#5ebfbb80, sub-11 = 0). Built on the Zassenhaus exceptional near-field of order 121 whose multiplicative group is Z_5 × SL(2, 3); the automorphism group is the sharply 2-transitive (Z_11)² ⋊ (Z_5 × SL(2, 3)) of order 14,520. Also satisfies Eq 3345.
6. **Twisted product F_11 × F_11 fiber bundle** (1 magma, magma#ee53a960, sub-11 = 12, NON-RC). Decomposes as 11 fibers (one congruence) + 1 transversal sub-magma (12 size-11 sub-magmas in an AG(2, 11)-pencil-like configuration); left-cancellative but not right-cancellative.
All known constructions sit on the point set (Z_11)² and exploit either an affine-plane structure on it or a near-field multiplication. The 4 sub-families with sub-11 ∈ {22, 44, 88} (= "twisted AG(2, 11) with k preserved parallel classes") are intermediate between the most rigid (just F_121 line, no sub-magmas at all) and the most symmetric (full AG(2, 11) line, all 12 classes preserved).
[text written by Claude]
dwrensha · 2026-05-14 19:55:23
Size 121 = 11² is among the most populated sizes in this database (35 magmas as of writing), reflecting the abundance of structures available on (Z_11)² ≅ AG(2, 11) ≅ the additive group of F_121.
Lower bound: at least 13,794 pairwise non-isomorphic Eq 677 magmas of size 121 exist. Sketch: in the AG(2, 11) line construction (family 2 below), each of the 12 parallel classes of AG(2, 11) is independently assigned one of the 4 valid α ∈ {2, 6, 7, 8} (the 4 primitive 10th roots of unity mod 11; the only F_11 linear operations x ◇ y = (1-α)x + αy satisfying Eq 677). The four α-values give 4 non-isomorphic linear F_11 sub-magmas (the size-11 DB entries are distinct iso classes). So distinct AG(2, 11) line magmas correspond to PGL(2, 11)-orbits on functions {12 parallel classes} → {2, 6, 7, 8}, and Burnside gives (1/1320) Σ_{g ∈ PGL(2,11)} 4^(cycles of g on P¹(F_11)) = 13,794 orbits.
Common features of every listed size-121 magma:
• idempotent (x ◇ x = x for all x)
• satisfies Equation 255 (x = ((x ◇ x) ◇ x) ◇ x)
• 28 are right-cancellative; 7 are non-right-cancellative
Construction families currently represented:
1. Direct products F_11 × F_11 (right-cancellative, 4 magmas). Operation (x₁, x₂) ◇ (y₁, y₂) = ((1-α)x_i + αy_i mod 11) coordinate-wise. The 4 entries use α ∈ {2, 6, 7, 8}: magma#d4b8c23f, magma#199764ee, magma#65eed598, magma#c3819eed.
2. Affine-plane AG(2, 11) line magmas (right-cancellative, many entries). The 132 lines of AG(2, 11) are exactly the 2-generated sub-quasigroups. Each of the 12 parallel classes is run with a linear F_11 operation x ◇ y = (1-α)x + αy for some α, and non-isomorphic choices of how to assign α to each parallel class produce different magmas (6+6 splits, 8+4 splits, etc.); see the lower bound above. Example: magma#83a10ff4.
3. AG(2, 11)-based non-right-cancellative magmas. Only 4 or 8 of the 12 parallel classes are preserved as sub-quasigroups; the remaining 'collapsed' parallel classes break right-cancellativity. Example: magma#c8dcedeb.
4. The exceptional near-field II of order 121 (right-cancellative, 1 magma): magma#5ebfbb80. Built on the Zassenhaus exceptional near-field of order 121 whose multiplicative group is Z_5 × SL(2, 3); the automorphism group is the sharply 2-transitive (Z_11)² ⋊ (Z_5 × SL(2, 3)) of order 14,520. Also satisfies Eq 3345.
5. Fiber-bundle / twisted product F_11(α=2) × F_11(α=8) (non-right-cancellative, 1 magma): magma#ee53a960. Decomposes into 11 fibers + 1 transversal sub-magma (12 size-11 sub-magmas in an AG(2, 11)-pencil-like configuration); left-cancellative but not right-cancellative, distinguishing it from the genuine direct product.
All known constructions sit on the point set (Z_11)² and exploit either an affine-plane structure on it or a near-field multiplication; whether further families exist (e.g. non-idempotent, or violating Eq 255) is open.
[text written by Claude]
dwrensha · 2026-05-14 19:49:54
Size 121 = 11² is among the most populated sizes in this database (35 magmas as of writing), reflecting the abundance of structures available on (Z_11)² ≅ AG(2, 11) ≅ the additive group of F_121.
Common features of every listed size-121 magma:
• idempotent (x ◇ x = x for all x)
• satisfies Equation 255 (x = ((x ◇ x) ◇ x) ◇ x)
• 28 are right-cancellative; 7 are non-right-cancellative
Construction families currently represented:
1. Direct products F_11 × F_11 (right-cancellative, 4 magmas). Operation (x₁, x₂) ◇ (y₁, y₂) = ((1-α)x_i + αy_i mod 11) coordinate-wise. The 4 entries use α ∈ {2, 6, 7, 8}: magma#d4b8c23f, magma#199764ee, magma#65eed598, magma#c3819eed.
2. Affine-plane AG(2, 11) line magmas (right-cancellative, many entries). The 132 lines of AG(2, 11) are exactly the 2-generated sub-quasigroups. Each of the 12 parallel classes is run with a linear F_11 operation x ◇ y = (1-α)x + αy for some α, and non-isomorphic choices of how to assign α to each parallel class produce different magmas (6+6 splits, 8+4 splits, etc.). Example: magma#83a10ff4.
3. AG(2, 11)-based non-right-cancellative magmas. Only 4 or 8 of the 12 parallel classes are preserved as sub-quasigroups; the remaining 'collapsed' parallel classes break right-cancellativity. Example: magma#c8dcedeb.
4. The exceptional near-field II of order 121 (right-cancellative, 1 magma): magma#5ebfbb80. Built on the Zassenhaus exceptional near-field of order 121 whose multiplicative group is Z_5 × SL(2, 3); the automorphism group is the sharply 2-transitive (Z_11)² ⋊ (Z_5 × SL(2, 3)) of order 14,520. Also satisfies Eq 3345.
5. Fiber-bundle / twisted product F_11(α=2) × F_11(α=8) (non-right-cancellative, 1 magma): magma#ee53a960. Decomposes into 11 fibers + 1 transversal sub-magma (12 size-11 sub-magmas in an AG(2, 11)-pencil-like configuration); left-cancellative but not right-cancellative, distinguishing it from the genuine direct product.
All known constructions sit on the point set (Z_11)² and exploit either an affine-plane structure on it or a near-field multiplication; whether further families exist (e.g. non-idempotent, or violating Eq 255) is open.
[text written by Claude]
dwrensha · 2026-05-18 17:28:22
dwrensha · 2026-05-18 17:01:20
dwrensha · 2026-05-18 16:57:59
dwrensha · 2026-05-14 19:55:23
dwrensha · 2026-05-14 19:49:54