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


  1. Computing the sets of totally symmetric and totally conjugate orthogonal partial Latin squares by means of a SAT solver Ra´ ul M. Falc´ on Department of Applied Mathematics I. University of Seville. (Joint work with ´ Oscar J. Falc´ on and Juan N´ u˜ nez). Rota. July 4, 2017. on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  2. CONTENTS 1 Preliminaries. 2 Binary constraints for T SPLS n and T COPLS n . 3 Lie partial quasigroup rings. on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  3. I. Preliminaries. on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  4. Partial Latin squares. Partial Latin square : n × n array where each cell is empty or contains one symbol of [ n ] := { 1 , . . . , n } . 1 each symbol occurs at most once in each row and each column. 2 Weight : Number of non-empty cells. PLS n ; m : Partial Latin squares of order n and weight m . 2 1 P ≡ 1 ∈ PLS 3;4 . 3 Entry set : { ( row , column , symbol ) } . Ent ( P ) = { (1 , 1 , 2) , (1 , 2 , 1) , (2 , 1 , 1) , (3 , 3 , 3) } . LS n = PLS n ; n 2 : Latin squares of order n . Any (P)LS is the multiplication table of a (partial) quasigroup . on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  5. Totally symmetric partial Latin squares. Let π ∈ S 3 , the symmetric group of { 1 , 2 , 3 } . π -conjugate of P ∈ PLS n ; m : P π ∈ PLS n ; m such that E ( P π ) { ( p π (1) , p π (2) , p π (3) ): ( p 1 , p 2 , p 3 ) ∈ E ( P ) } . 1 2 1 1 3 P (12) ≡ P (13) ≡ P ≡ 3 2 3 1 1 1 2 1 2 1 3 1 P (23) ≡ P (123) ≡ P (132) ≡ 2 2 1 2 3 2 3 Totally symmetric ( T SPLS n ; m ): The six conjugates coin- cide. 2 1 1 3 on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  6. Orthogonal partial Latin squares. Q = ( q ij ) ∈ PLS n ; m . P = ( p ij ), Orthogonal : All the ordered pairs ( p ij , q ij ) on non-empty en- tries are distinct. 1 2 1 3 P (13) ≡ P = and are orthogonal . 3 1 1 2 1 2 1 P (12) ≡ P = and are not orthogonal . 3 2 3 1 1 π -orthogonal : Orthogonal to its π -conjugate. π = (12) → self-orthogonal . 1 2 P (23) ≡ 2 3 on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  7. Totally conjugate orthogonal partial Latin squares. Totally conjugate orthogonal ( T COPLS n ; m ): The six con- jugates are distinct and pairwise orthogonal. 3 1 3 P (12) ≡ P (13) ≡ P ≡ 2 3 2 1 3 3 2 3 1 3 1 3 P (23) ≡ P (123) ≡ P (132) ≡ 3 3 3 1 2 3 2 2 1 on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  8. 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´ on): Self-orthogonal PLS ( n ≤ 4). 2015 (Falc´ on and Stones): #PLS ( n ≤ 7). on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  9. Binary constraints for PLS n . X := { x ijk | i , j , k ∈ [ n ] } .  x ijk x i ′ jk = 0 , ∀ i , i ′ , j , k ≤ n such that i � = i ′ ,   x ijk x ij ′ k = 0 , ∀ i , j , j ′ , k ≤ n such that j � = j ′ ,     x ijk x ijk ′ = 0 , ∀ i , j , k , k ′ ≤ n such that k � = k ′ ,     PLS n ≡ � j , k ∈ [ n ] x ijk ≥ 1 , ∀ i ∈ [ n ] , (1) �  i , k ∈ [ n ] x ijk ≥ 1 , ∀ j ∈ [ n ] ,     � i , j ∈ [ n ] x ijk ≥ 1 , ∀ k ∈ [ n ] ,      x ijk ∈ { 0 , 1 } , ∀ i , j , k ≤ n . P = ( p ij ) ∈ PLS n ↔ ( x 111 , . . . , x nnn ) � 1 , if p ij = k , x ijk = 0 , otherwise. on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  10. II. Binary constraints for T SPLS n and T COPLS n . on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  11. Binary constraints. Lemma Let n ≤ m ≤ n 2 . a) m > n ⇒ Every pair of orthogonal conjugates of a TCOPLS n ; m are distinct. b) |T COPLS n ; m | = 0 ⇒ |T COPLS n ; m ′ | = 0 , ∀ m < m ′ ≤ n 2 . 3 1 3 P (12) ≡ P (13) ≡ P ≡ 2 3 2 1 3 3 2 3 1 3 1 3 P (23) ≡ P (123) ≡ P (132) ≡ 3 3 3 1 2 3 2 2 1 on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  12. Binary constraints. x π i 1 i 2 i 3 := x i π (1) i π (2) i π (3) . S 3 := { π 1 = Id , π 2 = (12) , π 3 = (13) , π 4 = (23) , π 5 = (123) , π 6 = (132) } . Proposition a) T SPLS n ≡ (1) and x π s ijk = x ijk , ∀ i , j , k ≤ n and s ∈ { 1 , 2 , 3 } . (2) b) T SPLS n ; m ≡ (1), (2) and � x ijk = m . (3) i , j , k ≤ n c) T COPLS n ≡ (1) and x π s ijp x π s klp x π t ijq x π t klq = 0 , ∀ i , j , k , l , p , q ≤ n ; s , t ≤ 3; s. t. ( i , j ) � = ( k , l ) , s ≤ t . (4) d) T COPLS n ; m ≡ (1), (3) and (4). on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  13. Computation of T SPLS n ; m and T COPLS n ; m . on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  14. Computation of T SPLS n ; m and T COPLS n ; m . Run time (seconds) Run time (seconds) n m T SPLS n ; m T COPLS n ; m 5 5 0 22 10 0 3 6 6 0 8561 12 0 10 15 0 74 10 10 69 Out of memory 50 0 ” 15 15 > 3 hours ” 60 2 ” 20 100 Out of memory ” on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  15. Example: T SPLS 7;12 and T COPLS 7;12 . on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  16. Example: T SPLS 7;12 . on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  17. Example: T SPLS 7;12 . on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  18. Example: T SPLS 7;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 on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  19. Example: T COPLS 7;12 . on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  20. Example: T COPLS 7;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 on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  21. III. Lie partial quasigroup rings. on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

  22. 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 . on, ´ Ra´ ul M. Falc´ Oscar J. Falc´ on and Juan N´ u˜ nez Computing TSPLSs and TCOPLSs by means of a SAT solver

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