On the classification of distributive (medial, really) Mendelsohn triple systems (semisymmetric Latin quandles)
Alex W. Nowak Iowa State University
Loops 2019 Budapest University of Technology and Economics
July 9, 2019
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Summary
1 Mendelsohn triple systems
2 Distributivity & linearity
3 The Eisenstein integers
4 Classification of distributive, “non-ramified” MTS
5 Ramified MTS
6 From polynomial identities to quasigroup varieties
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Mendelsohn triple systems Mendelsohn triple systems
Let Q be a set of points and B, a set of blocks, consist of cyclic triples of points (x y z) ≡ (z x y) ≡ (y z x)
Defintion: Mendelsohn triple system (Mendelsohn, 1978) The pair (Q, B) is a Mendelsohn triple system (MTS) if each ordered pair of distinct points appears in exactly one block.
An MTS exists on Q if and only if |Q| ≡ 0, 1 mod 3 (except for n = 1, 6) (Mendelsohn, 1978).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Mendelsohn triple systems Mendelsohn triple systems
Let Q be a set of points and B, a set of blocks, consist of cyclic triples of points (x y z) ≡ (z x y) ≡ (y z x)
Defintion: Mendelsohn triple system (Mendelsohn, 1978) The pair (Q, B) is a Mendelsohn triple system (MTS) if each ordered pair of distinct points appears in exactly one block.
An MTS exists on Q if and only if |Q| ≡ 0, 1 mod 3 (except for n = 1, 6) (Mendelsohn, 1978).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Mendelsohn triple systems Mendelsohn triple systems
Let Q be a set of points and B, a set of blocks, consist of cyclic triples of points (x y z) ≡ (z x y) ≡ (y z x)
Defintion: Mendelsohn triple system (Mendelsohn, 1978) The pair (Q, B) is a Mendelsohn triple system (MTS) if each ordered pair of distinct points appears in exactly one block.
An MTS exists on Q if and only if |Q| ≡ 0, 1 mod 3 (except for n = 1, 6) (Mendelsohn, 1978).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Mendelsohn triple systems MTS as quasigroups
Define a quasigroup multiplication on the point set Q of an MTS by x · y = z if x 6= y and (x, y, z) ∈ B, and x2 = x. The quasigroup (Q, ·) satisfies
x2 = x (I) xy · x = y (S)
Conversely, any quasigroup (Q, ·) satisfying (I) and (S) may be interpreted as an MTS (Q, B), where B := {(x, y, xy) | (x, y) ∈ Q2 \ Qb}.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Mendelsohn triple systems MTS as quasigroups
Define a quasigroup multiplication on the point set Q of an MTS by x · y = z if x 6= y and (x, y, z) ∈ B, and x2 = x. The quasigroup (Q, ·) satisfies
x2 = x (I) xy · x = y (S)
Conversely, any quasigroup (Q, ·) satisfying (I) and (S) may be interpreted as an MTS (Q, B), where B := {(x, y, xy) | (x, y) ∈ Q2 \ Qb}.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Mendelsohn triple systems MTS as quasigroups
Define a quasigroup multiplication on the point set Q of an MTS by x · y = z if x 6= y and (x, y, z) ∈ B, and x2 = x. The quasigroup (Q, ·) satisfies
x2 = x (I) xy · x = y (S)
Conversely, any quasigroup (Q, ·) satisfying (I) and (S) may be interpreted as an MTS (Q, B), where B := {(x, y, xy) | (x, y) ∈ Q2 \ Qb}.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Distributivity & linearity The Point, broadly
We want to understand MTS satisfying
xy · uv = xu · yv (M)
and more generally,
x · yz = xy · xz. (LD)
We will give a direct product decomposition theorem for and enumerate isomorphism classes of finite LD Mendelsohn quasigroups of order coprime to 3.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Distributivity & linearity The Point, broadly
We want to understand MTS satisfying
xy · uv = xu · yv (M)
and more generally,
x · yz = xy · xz. (LD)
We will give a direct product decomposition theorem for and enumerate isomorphism classes of finite LD Mendelsohn quasigroups of order coprime to 3.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Distributivity & linearity Linear representations for (LD) and (M)
Recall that a quasigroup (M, ·) is distributive if and only it admits an arithmetic form Lin(M,R) := (M, +,R, 1 − R, 0), where (M, +) is a CML, and R is a nuclear loop automorphism (Belousov, 1960):
x · y = xR + y(1 − R).
Medial Mendelsohn quasigroups are precisely the distributive MTS for which (Q, +) is an abelian group. Semisymmetry forces 1 − R = R−1, or R2 − R + 1 = 0.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Distributivity & linearity Linear representations for (LD) and (M)
Recall that a quasigroup (M, ·) is distributive if and only it admits an arithmetic form Lin(M,R) := (M, +,R, 1 − R, 0), where (M, +) is a CML, and R is a nuclear loop automorphism (Belousov, 1960):
x · y = xR + y(1 − R).
Medial Mendelsohn quasigroups are precisely the distributive MTS for which (Q, +) is an abelian group. Semisymmetry forces 1 − R = R−1, or R2 − R + 1 = 0.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Distributivity & linearity
Theorem: Fischer-Galkin-Smith, 1970s
r1 rk Let Q be a finite distributive quasigroup of order p1 ··· pk . Then ∼ ri Q = Q1 × · · · Qk, where |Qi| = pi , and if Qi is not linear over an abelian group, then pi = 3 and ri ≥ 4.
Theorem: Kepka-Nemec, 1981
The LD quasigroups Lin(M1,R1) and Lin(M2,R2) are isomorphic if and only if there exists a CML-isomorphism ϕ : M1 → M2 such that −1 ϕ R1ϕ = R2.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Distributivity & linearity
Theorem: Fischer-Galkin-Smith, 1970s
r1 rk Let Q be a finite distributive quasigroup of order p1 ··· pk . Then ∼ ri Q = Q1 × · · · Qk, where |Qi| = pi , and if Qi is not linear over an abelian group, then pi = 3 and ri ≥ 4.
Theorem: Kepka-Nemec, 1981
The LD quasigroups Lin(M1,R1) and Lin(M2,R2) are isomorphic if and only if there exists a CML-isomorphism ϕ : M1 → M2 such that −1 ϕ R1ϕ = R2.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Distributivity & linearity Refining the classification program
In order to classify distributive MTS, it suffices to describe the conjugacy classes of automorphisms annihilated by X2 − X + 1 in Aut(M) for 1 M an abelian group of order pn (medial MTS); 2 M a CML of order 3n, n ≥ 4 (distributive, non-medial). This is the framework set up by Donovan, Griggs, McCourt, Opršal, and Stanovsk´y in their 2015 paper. They classify/enumerate all dist. MTS of order p and p2, enumerate medial MTS for pn ≤ 1000.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Distributivity & linearity Refining the classification program
In order to classify distributive MTS, it suffices to describe the conjugacy classes of automorphisms annihilated by X2 − X + 1 in Aut(M) for 1 M an abelian group of order pn (medial MTS); 2 M a CML of order 3n, n ≥ 4 (distributive, non-medial). This is the framework set up by Donovan, Griggs, McCourt, Opršal, and Stanovsk´y in their 2015 paper. They classify/enumerate all dist. MTS of order p and p2, enumerate medial MTS for pn ≤ 1000.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 The Eisenstein integers The Eisenstein integers
The Eisenstein integers have presentation 2 Z[X]/(X − X + 1) = √Z[ζ] = {a + bζ | a, b ∈ Z}, πi/3 1 3 where ζ = e = 2 + 2 i. Proposition, 2019 Let Z ⊗ MTS denote the category of medial Mendelsohn quasigroups. There is a faithful, dense functor F : Z[ζ]-Mod → Z ⊗ MTS defined on objects via (M,R) 7→ Lin(M,R) (R describes the action of ζ), and inserting morphisms f 7→ f.
2 2 Under N : a + bζ 7→ a + ab + b , Z[ζ] is a Euclidean domain (PID. . . nice!) Now, it suffices to describe conjugacy classes for automorphisms annihilated by X2 − X + 1 in abelian groups of the form Z[ζ]/(πn) for π a prime in Z[ζ].
A. W. Nowak (ISU) Distributive MTS July 9, 2019 The Eisenstein integers The Eisenstein integers
The Eisenstein integers have presentation 2 Z[X]/(X − X + 1) = √Z[ζ] = {a + bζ | a, b ∈ Z}, πi/3 1 3 where ζ = e = 2 + 2 i. Proposition, 2019 Let Z ⊗ MTS denote the category of medial Mendelsohn quasigroups. There is a faithful, dense functor F : Z[ζ]-Mod → Z ⊗ MTS defined on objects via (M,R) 7→ Lin(M,R) (R describes the action of ζ), and inserting morphisms f 7→ f.
2 2 Under N : a + bζ 7→ a + ab + b , Z[ζ] is a Euclidean domain (PID. . . nice!) Now, it suffices to describe conjugacy classes for automorphisms annihilated by X2 − X + 1 in abelian groups of the form Z[ζ]/(πn) for π a prime in Z[ζ].
A. W. Nowak (ISU) Distributive MTS July 9, 2019 The Eisenstein integers The Eisenstein integers
The Eisenstein integers have presentation 2 Z[X]/(X − X + 1) = √Z[ζ] = {a + bζ | a, b ∈ Z}, πi/3 1 3 where ζ = e = 2 + 2 i. Proposition, 2019 Let Z ⊗ MTS denote the category of medial Mendelsohn quasigroups. There is a faithful, dense functor F : Z[ζ]-Mod → Z ⊗ MTS defined on objects via (M,R) 7→ Lin(M,R) (R describes the action of ζ), and inserting morphisms f 7→ f.
2 2 Under N : a + bζ 7→ a + ab + b , Z[ζ] is a Euclidean domain (PID. . . nice!) Now, it suffices to describe conjugacy classes for automorphisms annihilated by X2 − X + 1 in abelian groups of the form Z[ζ]/(πn) for π a prime in Z[ζ].
A. W. Nowak (ISU) Distributive MTS July 9, 2019 The Eisenstein integers The Eisenstein integers
The Eisenstein integers have presentation 2 Z[X]/(X − X + 1) = √Z[ζ] = {a + bζ | a, b ∈ Z}, πi/3 1 3 where ζ = e = 2 + 2 i. Proposition, 2019 Let Z ⊗ MTS denote the category of medial Mendelsohn quasigroups. There is a faithful, dense functor F : Z[ζ]-Mod → Z ⊗ MTS defined on objects via (M,R) 7→ Lin(M,R) (R describes the action of ζ), and inserting morphisms f 7→ f.
2 2 Under N : a + bζ 7→ a + ab + b , Z[ζ] is a Euclidean domain (PID. . . nice!) Now, it suffices to describe conjugacy classes for automorphisms annihilated by X2 − X + 1 in abelian groups of the form Z[ζ]/(πn) for π a prime in Z[ζ].
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Eisenstein primes
There are three classes of Eisenstein primes. Up to association by units {±1, ±ζ, ±ζ}, they take the forms 1 π, where πN ≡ 1 mod 3 is a split prime in Z; 2 p ∈ Z, with p ≡ 2 mod 3, is prime in Z and Z[ζ]; call these inert primes; 3 1 + ζ makes 3 = (1 + ζ)(1 + ζ) ramified over Z[ζ]. Call distributive MTS of order coprime to 3 distributive, non-ramified (DNR).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Eisenstein primes
There are three classes of Eisenstein primes. Up to association by units {±1, ±ζ, ±ζ}, they take the forms 1 π, where πN ≡ 1 mod 3 is a split prime in Z; 2 p ∈ Z, with p ≡ 2 mod 3, is prime in Z and Z[ζ]; call these inert primes; 3 1 + ζ makes 3 = (1 + ζ)(1 + ζ) ramified over Z[ζ]. Call distributive MTS of order coprime to 3 distributive, non-ramified (DNR).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Split primes: 1 mod 3
Let π ∈ Z[ζ] with p := πN ≡ 1 mod 3. n ∼ n ∼ × Then Z[ζ]/(π ) = Z/pn , so Aut(Z[ζ]/(π )) = (Z/pn ) X2 − X + 1 has two roots modulo pn (Donovan et. al., 2015); call them a±1. n ±1 n (Z[ζ]/(π ), a ) are possible MTS isomorphism classes on Z[ζ]/(π ).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Split primes: 1 mod 3
Let π ∈ Z[ζ] with p := πN ≡ 1 mod 3. n ∼ n ∼ × Then Z[ζ]/(π ) = Z/pn , so Aut(Z[ζ]/(π )) = (Z/pn ) X2 − X + 1 has two roots modulo pn (Donovan et. al., 2015); call them a±1. n ±1 n (Z[ζ]/(π ), a ) are possible MTS isomorphism classes on Z[ζ]/(π ).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Split primes: 1 mod 3
Let π ∈ Z[ζ] with p := πN ≡ 1 mod 3. n ∼ n ∼ × Then Z[ζ]/(π ) = Z/pn , so Aut(Z[ζ]/(π )) = (Z/pn ) X2 − X + 1 has two roots modulo pn (Donovan et. al., 2015); call them a±1. n ±1 n (Z[ζ]/(π ), a ) are possible MTS isomorphism classes on Z[ζ]/(π ).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Split primes: 1 mod 3
Let π ∈ Z[ζ] with p := πN ≡ 1 mod 3. n ∼ n ∼ × Then Z[ζ]/(π ) = Z/pn , so Aut(Z[ζ]/(π )) = (Z/pn ) X2 − X + 1 has two roots modulo pn (Donovan et. al., 2015); call them a±1. n ±1 n (Z[ζ]/(π ), a ) are possible MTS isomorphism classes on Z[ζ]/(π ).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Inert primes: 2 mod 3
Let p be a rational prime congruent to 2 mod 3. n ∼ n ∼ Then Z[ζ]/(p ) = Z/pn [ζ], so Aut(Z[ζ]/(p )) = GL2(Z/pn ). n One isomorphism class on Z[ζ]/(p ); it is given by n 2 Lin(Z/pn [ζ]) := Lin((Z/p ) ,T ), where T is the companion matrix of X2 − X + 1. Proof Outline: 2 > Suffices to show ∃v ∈ (Z/pn ) so that (v vA) ∈ GL2(Z/pn ) (Prokip, 2005) (∗). 2 Take the entries of A modulo p, and act on (Z/p) . Because X2 − X + 1 does not split modulo p, (∗) holds in the quotient. Z/pn is a local ring, so we can use Nakayama’s Lemma to lift our basis modulo p to one modulo pn.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Inert primes: 2 mod 3
Let p be a rational prime congruent to 2 mod 3. n ∼ n ∼ Then Z[ζ]/(p ) = Z/pn [ζ], so Aut(Z[ζ]/(p )) = GL2(Z/pn ). n One isomorphism class on Z[ζ]/(p ); it is given by n 2 Lin(Z/pn [ζ]) := Lin((Z/p ) ,T ), where T is the companion matrix of X2 − X + 1. Proof Outline: 2 > Suffices to show ∃v ∈ (Z/pn ) so that (v vA) ∈ GL2(Z/pn ) (Prokip, 2005) (∗). 2 Take the entries of A modulo p, and act on (Z/p) . Because X2 − X + 1 does not split modulo p, (∗) holds in the quotient. Z/pn is a local ring, so we can use Nakayama’s Lemma to lift our basis modulo p to one modulo pn.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Inert primes: 2 mod 3
Let p be a rational prime congruent to 2 mod 3. n ∼ n ∼ Then Z[ζ]/(p ) = Z/pn [ζ], so Aut(Z[ζ]/(p )) = GL2(Z/pn ). n One isomorphism class on Z[ζ]/(p ); it is given by n 2 Lin(Z/pn [ζ]) := Lin((Z/p ) ,T ), where T is the companion matrix of X2 − X + 1. Proof Outline: 2 > Suffices to show ∃v ∈ (Z/pn ) so that (v vA) ∈ GL2(Z/pn ) (Prokip, 2005) (∗). 2 Take the entries of A modulo p, and act on (Z/p) . Because X2 − X + 1 does not split modulo p, (∗) holds in the quotient. Z/pn is a local ring, so we can use Nakayama’s Lemma to lift our basis modulo p to one modulo pn.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Inert primes: 2 mod 3
Let p be a rational prime congruent to 2 mod 3. n ∼ n ∼ Then Z[ζ]/(p ) = Z/pn [ζ], so Aut(Z[ζ]/(p )) = GL2(Z/pn ). n One isomorphism class on Z[ζ]/(p ); it is given by n 2 Lin(Z/pn [ζ]) := Lin((Z/p ) ,T ), where T is the companion matrix of X2 − X + 1. Proof Outline: 2 > Suffices to show ∃v ∈ (Z/pn ) so that (v vA) ∈ GL2(Z/pn ) (Prokip, 2005) (∗). 2 Take the entries of A modulo p, and act on (Z/p) . Because X2 − X + 1 does not split modulo p, (∗) holds in the quotient. Z/pn is a local ring, so we can use Nakayama’s Lemma to lift our basis modulo p to one modulo pn.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Inert primes: 2 mod 3
Let p be a rational prime congruent to 2 mod 3. n ∼ n ∼ Then Z[ζ]/(p ) = Z/pn [ζ], so Aut(Z[ζ]/(p )) = GL2(Z/pn ). n One isomorphism class on Z[ζ]/(p ); it is given by n 2 Lin(Z/pn [ζ]) := Lin((Z/p ) ,T ), where T is the companion matrix of X2 − X + 1. Proof Outline: 2 > Suffices to show ∃v ∈ (Z/pn ) so that (v vA) ∈ GL2(Z/pn ) (Prokip, 2005) (∗). 2 Take the entries of A modulo p, and act on (Z/p) . Because X2 − X + 1 does not split modulo p, (∗) holds in the quotient. Z/pn is a local ring, so we can use Nakayama’s Lemma to lift our basis modulo p to one modulo pn.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS A direct product decomposition theorem
Theorem Every DNR MTS is isomorphic to a direct product of quasigroups of the ±1 form Lin( / n , a ) and Lin( / n [ζ]) for p ≡ 1 mod 3 and p ≡ 2 Z p1 Z p2 1 2 mod 3.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Enumeration of DNR MTS
Denote integer partitions via multisets (X, µ). P (n) = number of partitions of n
PE(n) = number of partitions consisting of even parts.
Theorem Let p 6= 3 be prime. Then, up to isomorphism, the number of distributive MTS of order pn is given by ! X X a.) µ(r) + 1 whenever p ≡ 1 mod 3; (X,µ)`n r∈X
b.) PE(n) whenever p ≡ 2 mod 3.
2+µ(r)−1 a.) comes from the fact that µ(r) = µ(r) + 1.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Enumeration of DNR MTS
Denote integer partitions via multisets (X, µ). P (n) = number of partitions of n
PE(n) = number of partitions consisting of even parts.
Theorem Let p 6= 3 be prime. Then, up to isomorphism, the number of distributive MTS of order pn is given by ! X X a.) µ(r) + 1 whenever p ≡ 1 mod 3; (X,µ)`n r∈X
b.) PE(n) whenever p ≡ 2 mod 3.
2+µ(r)−1 a.) comes from the fact that µ(r) = µ(r) + 1.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Classification of distributive, “non-ramified” MTS Enumeration of DNR MTS
Denote integer partitions via multisets (X, µ). P (n) = number of partitions of n
PE(n) = number of partitions consisting of even parts.
Theorem Let p 6= 3 be prime. Then, up to isomorphism, the number of distributive MTS of order pn is given by ! X X a.) µ(r) + 1 whenever p ≡ 1 mod 3; (X,µ)`n r∈X
b.) PE(n) whenever p ≡ 2 mod 3.
2+µ(r)−1 a.) comes from the fact that µ(r) = µ(r) + 1.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Ramified MTS The ramified case
2k ∼ Z[ζ]/(1 + ζ) = Z/3k [ζ], so even powers work just like inert primes. However, 2k+1 ∼ 2 k k+1 ∼ Z[ζ]/(1+ζ) = Z[X]/(X −X +1, 3 X, 3 ) =Ab Z/3k ⊕Z/3k+1 .
Leads to representation theory of mixed congruence classes. However, numerical evidence from the paper of Donovan et. al. seems to indicate that there is only one isomorphism class on each Z[ζ]/(1 + ζ)2k+1. If this is true, then the number of medial MTS of order 3n is P (n).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Ramified MTS The ramified case
2k ∼ Z[ζ]/(1 + ζ) = Z/3k [ζ], so even powers work just like inert primes. However, 2k+1 ∼ 2 k k+1 ∼ Z[ζ]/(1+ζ) = Z[X]/(X −X +1, 3 X, 3 ) =Ab Z/3k ⊕Z/3k+1 .
Leads to representation theory of mixed congruence classes. However, numerical evidence from the paper of Donovan et. al. seems to indicate that there is only one isomorphism class on each Z[ζ]/(1 + ζ)2k+1. If this is true, then the number of medial MTS of order 3n is P (n).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Ramified MTS The ramified case
2k ∼ Z[ζ]/(1 + ζ) = Z/3k [ζ], so even powers work just like inert primes. However, 2k+1 ∼ 2 k k+1 ∼ Z[ζ]/(1+ζ) = Z[X]/(X −X +1, 3 X, 3 ) =Ab Z/3k ⊕Z/3k+1 .
Leads to representation theory of mixed congruence classes. However, numerical evidence from the paper of Donovan et. al. seems to indicate that there is only one isomorphism class on each Z[ζ]/(1 + ζ)2k+1. If this is true, then the number of medial MTS of order 3n is P (n).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Ramified MTS The ramified case
2k ∼ Z[ζ]/(1 + ζ) = Z/3k [ζ], so even powers work just like inert primes. However, 2k+1 ∼ 2 k k+1 ∼ Z[ζ]/(1+ζ) = Z[X]/(X −X +1, 3 X, 3 ) =Ab Z/3k ⊕Z/3k+1 .
Leads to representation theory of mixed congruence classes. However, numerical evidence from the paper of Donovan et. al. seems to indicate that there is only one isomorphism class on each Z[ζ]/(1 + ζ)2k+1. If this is true, then the number of medial MTS of order 3n is P (n).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Ramified MTS A design-theoretic characterization of DNR MTS
Theorem A medial MTS is pure (totally anticommutative) if and only if it is self-orthogonal (orthogonal to its opposite) if and only if it is distributive, non-ramified.
In general, self-orthogonal =⇒ totally anticommutative, but not the other way round. Even though we don’t have a complete characterization of medial MTS (of order divisible by 3), we were still able to include them in this theorem.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Ramified MTS A design-theoretic characterization of DNR MTS
Theorem A medial MTS is pure (totally anticommutative) if and only if it is self-orthogonal (orthogonal to its opposite) if and only if it is distributive, non-ramified.
In general, self-orthogonal =⇒ totally anticommutative, but not the other way round. Even though we don’t have a complete characterization of medial MTS (of order divisible by 3), we were still able to include them in this theorem.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 Ramified MTS A design-theoretic characterization of DNR MTS
Theorem A medial MTS is pure (totally anticommutative) if and only if it is self-orthogonal (orthogonal to its opposite) if and only if it is distributive, non-ramified.
In general, self-orthogonal =⇒ totally anticommutative, but not the other way round. Even though we don’t have a complete characterization of medial MTS (of order divisible by 3), we were still able to include them in this theorem.
A. W. Nowak (ISU) Distributive MTS July 9, 2019 From polynomial identities to quasigroup varieties Reverse linearization?
The Eisenstein integers have an alternative presentation: 2 ∼ 2πi/3 Z[X]/(X + X + 1) = Z[e ]. The variety of left Eisenstein quasigroups, LE, is defined via
x(x · xy) = y.
Proposition A CML-linear quasigroup Lin(M,R,L) belongs to LE if and only of L2 + L + 1 = 0.
Theorem Every left distributive MTS is principally isotopic to a left Eisenstein quasigroup of the form Lin(M,R,R−1).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 From polynomial identities to quasigroup varieties Reverse linearization?
The Eisenstein integers have an alternative presentation: 2 ∼ 2πi/3 Z[X]/(X + X + 1) = Z[e ]. The variety of left Eisenstein quasigroups, LE, is defined via
x(x · xy) = y.
Proposition A CML-linear quasigroup Lin(M,R,L) belongs to LE if and only of L2 + L + 1 = 0.
Theorem Every left distributive MTS is principally isotopic to a left Eisenstein quasigroup of the form Lin(M,R,R−1).
A. W. Nowak (ISU) Distributive MTS July 9, 2019 The End
A. W. Nowak (ISU) Distributive MTS July 9, 2019