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Computing the sets of totally symmetric and totally conjugate orthogonal partial Latin squares by means of a SAT solver

Ra´ ul M. Falc´

• n

Department of Applied Mathematics I. University of Seville. (Joint work with ´ Oscar J. Falc´

• n and Juan N´

u˜ nez).

• Rota. July 4, 2017.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

CONTENTS

1 Preliminaries. 2 Binary constraints for T SPLSn and

T COPLSn.

3 Lie partial quasigroup rings. Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

• I. Preliminaries.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Partial Latin squares.

Partial Latin square: n × n array where

1

each cell is empty or contains one symbol of [n] := {1, . . . , n}.

2

each symbol occurs at most once in each row and each column.

Weight: Number of non-empty cells. PLSn;m: Partial Latin squares of order n and weight m. P ≡ 2 1 1 3 ∈ PLS3;4. Entry set: {(row, column, symbol)}. Ent(P) = {(1, 1, 2), (1, 2, 1), (2, 1, 1), (3, 3, 3)}. LSn=PLSn;n2: Latin squares of order n. Any (P)LS is the multiplication table of a (partial) quasigroup.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Totally symmetric partial Latin squares.

Let π ∈ S3, the symmetric group of {1, 2, 3}. π-conjugate of P ∈ PLSn;m: Pπ ∈ PLSn;m such that E(Pπ){(pπ(1), pπ(2), pπ(3)): (p1, p2, p3) ∈ E(P)}.

P ≡ 1 2 3 1 P(12) ≡ 1 2 3 1 P(13) ≡ 1 3 1 2 P(23) ≡ 1 2 2 3 P(123) ≡ 1 3 2 2 P(132) ≡ 1 1 2 3

Totally symmetric (T SPLSn;m): The six conjugates coin- cide.

2 1 1 3

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Orthogonal partial Latin squares.

P = (pij), Q = (qij) ∈ PLSn;m. Orthogonal: All the ordered pairs (pij, qij) on non-empty en- tries are distinct. P =

1 2 3 1

and P(13) ≡

1 3 1 2

are orthogonal. P =

1 2 3 1

and P(12) ≡

1 2 3 1

are not orthogonal. π-orthogonal: Orthogonal to its π-conjugate.

π = (12) → self-orthogonal.

P(23) ≡

1 2 2 3

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Totally conjugate orthogonal partial Latin squares.

Totally conjugate orthogonal (T COPLSn;m): The six con- jugates are distinct and pairwise orthogonal.

P ≡ 3 2 1 3 P(12) ≡ 1 3 3 2 P(13) ≡ 3 2 3 1 P(23) ≡ 3 3 1 2 P(123) ≡ 1 3 3 2 P(132) ≡ 3 3 2 1

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Chronology.

1944 (Bruck): TSLS. 1957 (Stein): Quasigroups orthogonal to a conjugate. 1974 (Brayton et al.): Self-orthogonal LS (n = 2, 3, 6). 1976 (Lindner and Cruse): Embedding TCOPLS into TCOLS. 1978 (Phelps): (132)-, (123)-, (23)- and (13)- orthogonal LS. 1979 (Bailey): Isomorphism classes of TSLS (n ≤ 10). 1979 (Lindner et al.): Idempotent TCOLS (prime power n ≥ 8). 1981 (Bennet): Idempotent TCOLS (n > 5594). 1985 (Bennet): Idempotent TCOLS (n > 5074). 2004 (Bennet and Zhang): Latin squares for which each one of their con- jugates is orthogonal to its transpose (prime power n ∈ {2, 3, 5}). 2004 (Kaski and ¨ Osterg˚ ard): Isomorphism classes of TSLS (n ≤ 19). 2011 (Hulpke et al.): #LS (n ≤ 11). 2012 (Belyavskaya and Popovich): TCOLS (n ≥ 11). 2015 (Falc´

• n): Self-orthogonal PLS (n ≤ 4).

2015 (Falc´

• n and Stones): #PLS (n ≤ 7).

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Binary constraints for PLSn.

X := {xijk | i, j, k ∈ [n]}. PLSn ≡                     

xijkxi′jk = 0, ∀ i, i′, j, k ≤ n such that i = i′, xijkxij′k = 0, ∀ i, j, j′, k ≤ n such that j = j′, xijkxijk′ = 0, ∀ i, j, k, k′ ≤ n such that k = k′,

• j,k∈[n] xijk ≥ 1, ∀ i ∈ [n],
• i,k∈[n] xijk ≥ 1, ∀ j ∈ [n],
• i,j∈[n] xijk ≥ 1, ∀ k ∈ [n],

xijk ∈ {0, 1}, ∀ i, j, k ≤ n.

(1) P = (pij) ∈ PLSn ↔ (x111, . . . , xnnn) xijk =

• 1, if pij = k,

0, otherwise.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

• II. Binary constraints for

T SPLSn and T COPLSn.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Binary constraints.

Lemma Let n ≤ m ≤ n2. a) m > n ⇒ Every pair of orthogonal conjugates of a TCOPLSn;m are distinct. b) |T COPLSn;m| = 0 ⇒ |T COPLSn;m′| = 0, ∀m < m′ ≤ n2.

P ≡ 3 2 1 3 P(12) ≡ 1 3 3 2 P(13) ≡ 3 2 3 1 P(23) ≡ 3 3 1 2 P(123) ≡ 1 3 3 2 P(132) ≡ 3 3 2 1

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Binary constraints.

xπ

i1i2i3 := xiπ(1)iπ(2)iπ(3). S3 := {π1 = Id, π2 = (12), π3 = (13), π4 = (23), π5 = (123), π6 = (132)}.

Proposition a) T SPLSn ≡ (1) and xπs

ijk = xijk, ∀ i, j, k ≤ n and s ∈ {1, 2, 3}.

(2) b) T SPLSn;m ≡ (1), (2) and

• i,j,k≤n

xijk = m. (3) c) T COPLSn ≡ (1) and

xπs

ijp xπs klpxπt ijq xπt klq = 0, ∀i, j, k, l, p, q ≤ n; s, t ≤ 3; s. t. (i, j) = (k, l), s ≤ t.

(4)

d) T COPLSn;m ≡ (1), (3) and (4).

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Computation of T SPLSn;m and T COPLSn;m.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Computation of T SPLSn;m and T COPLSn;m.

Run time (seconds) Run time (seconds) n m T SPLSn;m T COPLSn;m 5 5 22 10 3 6 6 8561 12 10 15 74 10 10 69 Out of memory 50 ” 15 15 > 3 hours ” 60 2 ” 20 100 Out of memory ”

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T SPLS7;12 and T COPLS7;12.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T SPLS7;12.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T SPLS7;12.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T SPLS7;12.

0000000 0000000 0000000 0000000 0000000 0000000 0000001 0000000 0000000 0000000 0000000 0000000 0000001 0000010 0000000 0000000 0010000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0001000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000100 0000000 0000000 0000000 0000001 0000000 0000000 0000000 0000000 0100000 0000001 0000010 0000000 0000000 0000000 0100000 1000000

• 7

7 6 3 4 5 7 2 7 6 2 1

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T COPLS7;12.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T COPLS7;12.

0000000 0000000 0000000 0000000 0000000 0000000 0000001 0000000 0000000 0000000 0000000 0000000 0000000 0000010 0000000 0000000 0000000 0000000 0000000 0000000 0000100 0000000 0000000 0000000 0000000 0000000 0000000 0001000 0000000 0000000 0000000 0000000 0000000 0000000 0100000 0000000 0000000 0000000 0000000 0000000 0000000 0010000 0000010 0000100 0000001 0100000 1000000 0001000 0000000

• 7

6 5 4 2 3 6 5 7 2 1 4

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

• III. Lie partial quasigroup rings.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Lie algebras.

Sophus Lie, 1842–1899. Carl Gustav Jacob Jacobi, 1804–1851.

Lie algebra: Anti-commutative algebra A that holds the Ja- cobi identity J(a, b, c) := (ab)c + (bc)a + (ca)b = 0, ∀a, b, c ∈ A.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Conjugate-extension of a PLS.

P = (pij) ∈ PLSn;m → P′ = (p′

ij), P′′ = (p′′ ij) ∈ PLSn;m.

p′

ij :=

• pij + n, if pij ∈ [n],

0, otherwise. and p′′

ij :=

• pij + 2n, if pij ∈ [n],

0, otherwise. Conjugate-extension: P :≡ P′′ P′(23) P′′(12) P(132) P′(123) P(13)

P = 1 2 3 1 → P = 7 8 4 5 9 5 7 6 7 1 8 9 1 2 7 3 4 6 1 3 5 1 5 2

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Conjugate-extension of a PLS.

P = (pij) ∈ PLSn;m → P′ = (p′

ij), P′′ = (p′′ ij) ∈ PLSn;m.

p′

ij :=

• pij + n, if pij ∈ [n],

0, otherwise. and p′′

ij :=

• pij + 2n, if pij ∈ [n],

0, otherwise. Conjugate-extension: P :≡ P′′ P′(23) P′′(12) P(132) P′(123) P(13)

P = 1 2 3 1 → P = 7 8 4 5 9 5 7 6 7 1 8 9 1 2 7 3 4 6 1 3 5 1 5 2

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Conjugate-extension of a PLS.

Lemma If P ∈ PLSn;m, then P ∈ T SPLS3n;6m.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Conjugate-extension of a PLS.

Lemma If P ∈ PLSn;m, then P ∈ T SPLS3n;6m. What if P ∈ T SPLSn;m?

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Conjugate-extension of a PLS.

Lemma If P ∈ PLSn;m, then P ∈ T SPLS3n;6m. What if P ∈ T SPLSn;m? Conjugate-extension: P :≡ P′′ P′ P′′ P P′ P

P = 3 1 2 1 3 → P = 9 7 6 4 8 5 7 9 4 6 9 7 3 1 8 2 7 9 1 3 6 4 3 1 5 2 4 6 1 3

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Quasigroup rings.

Richard Hubert Bruck 1914-1991

Quasigroup ring related to a quasigroup ([n], ·): Algebra of basis {ei | i ∈ [n]} over a base field K such that eiej = ei·j, ∀i, j ∈ [n]. ↓ Partial quasigroup ring Partial quasigroup ([n], ·).

P = 1 2 3 1 →          e1e1 = e1, e1e2 = e2, e2e2 = e3, e3e3 = e1.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Lie partial quasigroup rings.

Theorem Let P ∈ TSPLSn be the multiplication table of a quasigroup ([n], ·) satisfying the left invertive law (a · b) · c = (c · b) · a, for all a, b, c ∈ [n]. Then, every partial quasigroup ring related to the conjugate- extension P over a base finite field of characteristic two is a 3n- dimensional Lie algebra.

xijkxklsxljt(xtis − 1) = 0, ∀i, j, k, l, s, t ≤ n.  

k≤n

xijk − 1    

k≤n

xljk   xljt  

k≤n

xtik   = 0, ∀i, j, l, t ≤ n. xijk  

s≤n

xkls − 1    

s≤n

xljs   xljt  

s≤n

xtis   = 0, ∀i, j, k, l, t ≤ n.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T SPLS3;5 and T SPLS6;12.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T SPLS3;5.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Example: T SPLS6;12.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Lie partial quasigroup rings.

Examples of TSPLSs satisfying the left invertive law:

3 1 2 1 3 2 1 1 2 4 3 3 4 6 5 5 6

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Open problem and further works.

Problem: Let n be a positive integer and let π ∈ S3. Which is the maximum positive integer m for which a partial Latin square P ∈ T COPLS(n; m) exists? Further work:Computation of TSPLSs and TCOPLSs for higher

• rders.

Further work: Distribution of TSPLSs and TCOPLSs into iso- morphism, isotopism and main classes. Further work: Distribution of Lie partial quasigroup rings into isomorphism classes.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

References

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gendum, Utilitas Math. 16 (1979) 302.

• G. B. Belyavskaya, T. V. Popovich, Totally conjugate-orthogonal quasigroups and complete graphs, J.
• Math. Sci. 185 (2012) 184–191.
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• F. E. Bennett, On conjugate orthogonal idempotent Latin squares, Ars Combin. 19 (1985) 37–49.
• F. E. Bennett, H. Zhang, Latin squares with self-orthogonal conjugates, Discrete Math.

284 (2004) 45–55.

• R. K. Brayton, D. Coppersmith, A. J. Hoffman, Self-orthogonal Latin squares of all orders n = 2, 3 or 6,
• Bull. Amer. Math. Soc. 80 (1974) 116–118.
• R. H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc. 55 (1944) 19–52.
• O. J. Falc´
• n, R. M. Falc´
• n, J. N´

u˜ nez, A. Pacheco, M. T. Villar, Computation of isotopisms of algebras over finite fields by means of graph invariants, J. Comput. Appl. Math., 318 (2017), 307–315.

• R. M. Falc´
• n, R. J. Stones, Classifying partial Latin rectangles, Electron. Notes Discrete Math. 49

(2015) 765–771. Hulpke, P. Kaski, P. R. J. ¨ Osterg˚ ard, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197–1219.

• P. Kaski, P. R. J.¨

Osterg˚ ard, The Steiner triple systems of order 19, Math. Comp. 73 (2004) 2075–2092.

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Theory 3 (1979) 325–338.

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• S. K. Stein, On the foundations of quasigroups, Trans. Amer. Math. Soc. 85 (1957) 228–256.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

Many thanks!!

Computing the sets of totally symmetric and totally conjugate orthogonal partial Latin squares by means of a SAT solver

Ra´ ul M. Falc´

• n

Department of Applied Mathematics I. University of Seville. (Joint work with ´ Oscar J. Falc´

• n and Juan N´

u˜ nez).

• Rota. July 4, 2017.

Ra´ ul M. Falc´

• n, ´

Oscar J. Falc´

• n and Juan N´

u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver