= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = COMBINATORIAL ANALYSIS PROGRAMS = = = = = = = = = = = = (c) Copyright 2017 by J.E. Glover, Ph.D. = = = = All Rights Reserved = = = = = = = = = = = = = = = = ( ABSTRACTS ) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = ======================================================================== ABSTRACTS OF COMBINATORIAL ANALYSIS ROUTINES ======================================================================== * * * TRANSVSL * * * DISJOINT TRANSVERSALS (HAVING FULL CARDINALITY) FOR A QUASIGROUP OR LATIN SQUARE THIS ROUTINE READS A FINITE SEQUENCE, , OF LATIN SQUARES OR QUASIGROUPS OF ORDER R AND LISTS, FOR EACH QUASIGROUP, L, IN THE SEQUENCE, A SET OF R DISJOINT TRANSVERSALS, EACH OF SIZE R, REFERRED TO AS A "TRANSVERSAL SYSTEM", PROVIDED THAT SUCH TRANSVERSALS EXIST FOR THE GIVEN LATIN SQUARE, L. EACH QUASIGROUP/LATIN SQUARE IS PRE-TESTED TO DETERMINE WHETHER OR NOT IT IS A VALID QUASIGROUP. THE SET OF DISJOINT TRANSVERSALS OF SIZE R, (THE SIZE OF EACH QUASIGROUP), IS LISTED AS A COLUMN VECTOR. EACH QUASIGROUP/ LATIN SQUARE IS PRINTED AS AN RXR ARRAY. THE ROUTINE ALSO DETERMINES AND PRINTS THE TRANSVERSAL CONFIGURATIONS FOR EACH QUASIGROUP/LATIN SQUARE. A 'TRANSVERSAL CONFIGURATION' FOR AN RXR LATIN SQUARE IS AN RXR ARRAY IN WHICH TRANSVERSAL T(I) CONTAINS THE SYMBOL I, FOR EACH I = 1,2,3 ,..., R. A MENDELSOHN TRIPLE SYSTEM IS DETERMINED AND PRINTED FOR EACH QUASIGROUP WHICH IS BOTH IDEMPOTENT AND SEMI-SYMMETRIC. (SUCH A QUASIGROUP IS REFERRED TO AS A MENDELSOHN QUASIGROUP). ======================================================================== * * * QUASAR * * * THIS ROUTINE CONSTRUCTS A COMPLETE SET OF (R-1) MUTUALLY ORTHOGONAL LATIN SQUARES OF SIZE RXR (WITH ELEMENTS 1 , 2 ,..., R) FROM ADDITION AND MULTIPLICATION TABLES OF A GIVEN FINITE FIELD OF PRIME POWER ORDER. GIVEN SUCH TABLES OF A FINITE FIELD OF ORDER R, THEY ARE READ AND USED TO CALCULATE THE ENTRIES OF A SEQUENCE OF ORTHOGONAL QUASIGROUPS/LATIN SQUARES , , OF SIZE RXR, HAVING WHAT IS KNOWN AS AN ORTHOGONAL MATE (I.E., ADMITTING R DISJOINT TRANSVERSALS, REFERRED TO AS A "TRANSVERSAL SYSTEM".) A VIRTUAL OR PRINTED BLOCK IS, OPTIONALLY, GENERATED FOR THE SEQUENCE OF (R-1) SUCH LATIN SQUARES PRODUCED FOR FURTHER ANALYSIS, E.G., DETERMINING ENTRIES IN COMMON, ETC. THE ROUTINE ALSO DETERMINES WHETHER OR NOT EACH SUCH LATIN SQUARE GENERATED IS A VALID LATIN SQUARE AND DETERMINES AN ACTUAL "TRANSVERSAL SYSTEM" FOR EACH OF THE (R-1) LATIN SQUARES, CONSTRUCTING THE TRANSVERSAL CONFIGURATION FOR EACH. A TRANSVERSAL CONFIGURATION FOR AN RXR LATIN SQUARE IS AN RXR MATRIX DETERMINED BY PLACING AN I IN TRANSVERSAL T(I), FOR EACH I = 1, 2, 3 ,..., R. ======================================================================== * * * NEXLAT * * * THIS ROUTINE READS THE RXR ARRAY, L1, VALIDATES IT AS A LATIN SQUARE/QUASIGROUP, PRINTS IT, DETERMINES AND PRINTS A "TRANSVERSAL SYSTEM" FOR L1, IF THERE EXISTS ONE, AND GENERATES AND PRINTS THE NEXT LATIN SQUARE/QUASIGROUP, L2, FROM L1. SUBSEQUENTLY, L2 IS, IN TURN, PROCESSED IN THE SAME WAY AS L1. AS A RESULT, A FINITE SEQUENCE, , IS GENERATED AND PROCESSED, AS ABOVE. MOREOVER, THE SEQUENCE, , IS GENERATED LEXICOGRAPHICALLY. EACH LATIN SQUARE/QUASIGROUP IS, OPTIONALLY, PRINTED AS OUTPUT FOR FURTHER ANALYSIS, E.G., DETERMINING ENTRIES IN COMMON. SUBROUTINE 'NEXTLS' IS USED TO GENERATE THE LATIN SQUARES, IN SEQUENCE, LEXICOGRAPHICALLY. ======================================================================== * * * LEXICOG2 * * * THIS ROUTINE READS AN MTS, M, OF ORDER Q, HAVING R=Q*(Q-1)/3 CYCLIC TRIPLES. THEREAFTER, IT ARRANGES AND WRITES THE TRIPLES OF M IN LEXICOGRAPHIC ORDER AND DISPLAYS THE Q PERMUTATIONS CORRESPONDING TO THE T-VECTOR CYCLES FOR M. THE CYCLE LENGTHS ARE CALCULATED, CLASSIFIED, AND PRINTED, MAKING IT EASY TO INTERPRET THE 'T-VECTOR' REPRESENTATION FOR M. THIS WILL BE A VECTOR, THE ENTRIES OF WHICH ARE CARDINALITIES OF THE PARTITIONS OF THE INTEGER (Q-1) OF A GIVEN TYPE. THUS, THE K-TH ENTRY OF THE T-VECTOR FOR M WILL BE THE CARDINALITY OF TYPE K IN THE T-VECTOR CYCLE REPRESENTATION FOR M. SUBROUTINE LEXMTS IS CALLED TO INSURE THE LEXICOGRAPHIC ORDERING OF M. SUBROUTINE TESTMT IS CALLED TO VALIDATE M AS AN AUTHENTIC MTS. SUBROUTINES READMT AND PRTMTS HANDLE INPUT/OUTPUT OF M. SUBROUTINE TVECT DISPLAYS THE PERMUTATIONS CORRESPONDING TO THE T-VECTOR CYCLES AND, MOREOVER, CALCULATES AND PRINTS THE IDEMPOTENT, SEMI-SYMMETRIC QUASIGROUP CORRESPONDING TO M, REFERRED TO AS THE MENDELSOHN QUASIGROUP FOR M. ======================================================================== * * * LEXICOG1 * * * THIS ROUTINE READS AN MTS, M, OF ORDER Q, HAVING R=Q*(Q-1)/3 TRIPLES. THEREAFTER, IT ARRANGES AND WRITES THE TRIPLES OF M IN LEXICOGRAPHIC ORDER. SUBROUTINE LEXMTS IS CALLED TO DETERMINE THE LEXICOGRAPHIC ORDERING OF M. MOREOVER, WHAT WILL BE REFERRED TO AS 'T-VECTOR CYCLES' FOR M ARE CALCULATED AND PRINTED. Q SUCH CYCLES FOR M ARE REPRESENTED AND PRINTED AS PERMUTATIONS IN CYCLIC NOTATION. SUCH T-VECTOR CYCLES WILL BE USED TO DISTINGUISH BETWEEN NON-ISOMORPHIC PAIRS OF MTS(Q). ======================================================================== * * * LEXAR1 * * * (ALMOST RESOLVABILITY) THIS ROUTINE TESTS THE MENDELSOHN TRIPLE SYSTEM, M, FOR ALMOST RESOLVABILITY. IT ALSO PRODUCES AT LEAST ONE (OR MORE, IF DESIRED) RESOLUTION OF M INTO PARALLEL CLASSES. IN EFFECT, THE ROUTINE READS AN MTS, M, OF ORDER Q, HAVING R=Q*(Q-1)/3 TRIPLES. THEREAFTER, IT ARRANGES AND WRITES THE TRIPLES OF M IN LEXICOGRAPHIC ORDER. SUBROUTINE LEXMTS IS CALLED TO DETERMINE THE LEXICOGRAPHIC ORDERING OF M. WHEREUPON, SUBROUTINE ALMRES IS CALLED TO TEST THE ALMOST RESOLVABILITY OF M. (NOTE THE NECESSARY CONDITION FOR ALMOST RESOLVABILITY: THE ORDER OF THE MTS, M, MUST BE CONGRUENT TO 1 MOD 3.) IN THE FINAL ANALYSIS, THIS ROUTINE ALSO CALCULATES AND PRINTS THE R-QUASIGROUP (OR RESOLUTION QUASIGROUP) FROM THE DETERMINED RESOLUTION OF M INTO Q PARALLEL CLASSES. ======================================================================== * * * LEXAR2 * * * (ALMOST RESOLVABILITY) THIS ROUTINE TESTS THE MENDELSOHN TRIPLE SYSTEM, M, FOR ALMOST RESOLVABILITY. IT ALSO PRODUCES AT LEAST ONE (OR MORE, IF DESIRED) RESOLUTION OF M INTO PARALLEL CLASSES. IN EFFECT, THIS ROUTINE READS AN MTS, M, OF ORDER Q, HAVING R = Q*(Q-1)/3 TRIPLES. THEREAFTER, IT ARRANGES AND WRITES THE TRIPLES OF M IN LEXICOGRAPHIC ORDER. SUBROUTINE LEXMTS IS CALLED TO DETERMINE THE LEXICOGRAPHIC ORDERING OF M. WHEREUPON, SUBROUTINE ALMRES IS CALLED TO TEST THE ALMOST RESOLVABILITY OF M. (NOTE THE NECESSARY ALMOST RESOLVABILITY CONDITION: THE ORDER OF THE MTS, M, MUST BE CONGRUENT TO 1 MOD 3.) IN THE FINAL ANALYSIS, THE ROUTINE ALSO CALCULATES AND PRINTS THE R-QUASIGROUP (RESOLUTION QUASIGROUP), FOR THE MTS, M, FROM THE DETERMINED RESOLUTION OF M INTO Q PARALLEL CLASSES. MOREOVER, THE MENDELSOHN QUASIGROUP, AN IDEMPOTENT, SEMI-SYMMETRIC QUASIGROUP OF ORDER Q, CORRESPONDING TO THE MTS, M, IS DETERMINED AND PRINTED. ======================================================================== * * * LEXAR3 * * * (ALMOST RESOLVABILITY) THIS ROUTINE TESTS THE MENDELSOHN TRIPLE SYSTEM, M, FOR ALMOST RESOLVABILITY. IT ALSO PRODUCES AT LEAST ONE (OR MORE, IF DESIRED) RESOLUTION OF M INTO PARALLEL CLASSES. IN EFFECT, THIS ROUTINE READS AN MTS, M, OF ORDER Q, HAVING R = Q*(Q-1)/3 TRIPLES. THEREAFTER, IT ARRANGES AND WRITES THE TRIPLES OF M IN LEXICOGRAPHIC ORDER. SUBROUTINE LEXMTS IS CALLED TO DETERMINE THE LEXICOGRAPHIC ORDERING OF M. WHEREUPON, SUBROUTINE ALMRES IS CALLED TO TEST THE ALMOST RESOLVABILITY OF M. NOTE THE ALMOST RESOLVABILITY REQUIREMENT: THE ORDER OF THE MTS, M, MUST BE CONGRUENT TO 1 MOD 3. IN THE FINAL ANALYSIS, THIS ROUTINE ALSO CALCULATES AND PRINTS THE R-QUASIGROUP (RESOLUTION QUASIGROUP), FOR THE MTS, M, FROM THE DETERMINED RESOLUTION OF M INTO Q PARALLEL CLASSES IT ALSO DETERMINES AND PRINTS THE MENDELSOHN QUASIGROUP, AN IDEMPOTENT, SEMI-SYMMETRIC QUASIGROUP OF ORDER Q CORRESPONDING TO THE MTS, M. * IN THIS ROUTINE, THE PROBABILISTIC ASPECT IS UTILIZED IN THE SEARCH FOR TRIPLES TO DETERMINE THE RESOLUTION OF THE MTS INTO THE SET OF PARALLEL CLASSES.*** ALSO, PARTIAL RESOLUTIONS INTO PARALLEL CLASSES ARE CALCULATED AND PRINTED IN CASES WHERE THE SEARCH LEADS TO A SPECIFIED NUMBER OF DEAD ENDS.*** NOTE THAT THIS ROUTINE HAS CAPABILITIES EQUIVALENT TO THOSE OF LEXRES, WHICH TESTS FOR RESOLVABILITY OF AN MTS, M, HAVING ORDER Q CONGRUENT TO 0 (MOD 3). ======================================================================== * * * LEXRES * * * (RESOLVABILITY) THIS ROUTINE TESTS THE MENDELSOHN TRIPLE SYSTEM SYSTEM, M, FOR RESOLVABILITY. IT ALSO PRODUCES AT LEAST ONE (OR MORE, IF DESIRED) RESOLUTION OF M INTO PARALLEL CLASSES. IN EFFECT, THE ROUTINE READS AN MTS, M, OF ORDER Q, HAVING R = Q*(Q-1)/3 TRIPLES. THEREAFTER, IT ARRANGES AND WRITES THE TRIPLES OF M IN LEXICOGRAPHIC ORDER. SUBROUTINE LEXMTS IS CALLED TO DETERMINE THE LEXICOGRAPHIC ORDERING OF M, WHEREUPON, SUBROUTINE RESOLV IS CALLED TO TEST THE RESOLVABILITY OF M. (NOTE THE RESOLVABILITY REQUIREMENT: A NECESSARY CONDITION IS THAT THE ORDER, Q, OF THE MTS, M, MUST BE CONGRUENT TO 0 (MOD3).) IN THE FINAL ANALYSIS, THE ROUTINE ALSO CALCULATES AND PRINTS THE MENDELSOHN QUASIGROUP, AN IDEMPOTENT, SEMI-SYMMETRIC QUASIGROUP OF ORDER Q ASSOCIATED WITH THE MTS, M. THE RESOLUTION QUASIGROUP (OR R-QUASIGROUP) OF ORDER Q FOR THE MTS, M, IS CALCULATED AND PRINTED FROM THE DETERMINED RESOLUTION OF M INTO (Q-1) PARALLEL CLASSES. (IN THIS ROUTINE, THE PROBABILISTIC ASPECT IS UTILIZED IN THE SEARCH FOR TRIPLES TO DETERMINE THE RESOLUTION OF THE MTS INTO THE SET OF PARALLEL CLASSES.*** ALSO, PARTIAL RESOLUTIONS INTO PARALLEL CLASSES ARE CALCULATED AND PRINTED IN CASES WHERE THE SEARCH LEADS TO A SPECIFIED NUMBER OF DEAD ENDS). ======================================================================== * * * TESTARM2 * * * THIS ROUTINE READS A FINITE SEQUENCE OF SETS OF TRIPLES, , DEFINED ON Q ELEMENTS AND DETERMINES WHETHER OR NOT EACH M IN IS A VALID MENDELSOHN TRIPLE SYSTEM AND WHETHER OR NOT EACH M IS RESOLVABLE OR ALMOST RESOLVABLE. IN SEQUENCE, THE ROUTINE IS DESIGNED TO READ AND TEST THE SET OF TRIPLES, M: IF M IS NOT A VALID MTS, THEN THE ROUTINE INDICATES THIS AND PRINTS A LIST OF ORDERED PAIRS WHICH ARE REPEATED AND/OR MISSING IN THE GIVEN SET OF TRIPLES, M. IN THE EVENT THAT M IS A VALID TRIPLE SYSTEM AND IS RESOLVABLE OR ALMOST RESOLVABLE, A SET OF PARALLEL CLASSES (OR, OPTIONALLY, MORE IF MORE THAN ONE EXISTS) IS PRINTED. THIS PROCESS IS REPEATED FOR EACH M IN THE SEQUENCE, . THE INHERENT FEATURE OR VALUE OF TESTARM2 IS THAT THE ROUTINE ALLOWS ONE TO OBTAIN A COMPLETE MENDELSOHN TRIPLE SYSTEM FROM A PARTIAL TRIPLE SYSTEM. ======================================================================== * * * MTSCON1 * * * (THE 4V CONSTRUCTION) THIS ROUTINE READS THE ARMTS(4), M0, WHICH IS DEFINED BY THE CYCLIC TRIPLES: (1,2,3), (1,3,4), (1,4,2), (2,4,3), AND A FINITE SEQUENCE, <(Q,T)>, OF ARMTS(V), AS WELL AS A FINITE SEQUENCE, <(Q,0)>, OF QUASIGROUPS HAVING ORTHOGONAL MATES, AND USES THEM TO CONSTRUCT A FINITE SEQUENCE, , OF NEW MENDELSOHN TRIPLE SYSTEMS OF ORDER 4V BY WAY OF THE 4V CONSTRUCTION. IF EACH INITIAL MTS, (Q,T), IS ALMOST RESOLVABLE (OR RESOLVABLE) AND THE LATIN SQUARE CORRESPONDING TO EACH QUASIGROUP, (Q,0), HAS AN ORTHOGONAL MATE, THEN EACH MTS, S, IN THE CONSTRUCTED SEQUENCE, , WILL BE ALMOST RESOLVABLE (OR RESOLVABLE). ======================================================================== * * * MTSCON2 * * * THE V+1 ---> 4V+1 CONSTRUCTION (THE V ---> 4V-3 CONSTRUCTION) THIS ROUTINE READS THE ARMTS(4), M0, WHICH IS DEFINED BY THE CYCLIC TRIPLES: (1,2,3), (1,3,4),(1,4,2),(2,4,3), AND A FINITE SEQUENCE, , OF ARMTS(V), EACH DEFINED ON A POINT AT INFINITY UNION A (V-1)-SET, AND A FINITE SEQUENCE OF QUASIGROUPS, , (OF ORDER V-1), CONSTRUCTING A FINITE SEQUENCE OF NEW MENDELSOHN TRIPLE SYSTEMS, , (OF ORDER (4V-3)), BY WAY OF THE (V--->4V-3) CONSTRUCTION. IF EACH INITIAL MTS, M, IS ALMOST RESOLVABLE (OR RESOLVABLE) AND THE LATIN SQUARE CORRESPONDING TO EACH INITIAL QUASIGROUP, Lk, HAS AN ORTHOGONAL MATE, THEN EACH NEWLY CONSTRUCTED MTS, MSTAR, WILL BE ALMOST RESOLVABLE (OR RESOLVABLE). **NOTE: THIS ROUTINE BECOMES AN ANALYSIS BY WAY OF THE (V+1--->4V+1) CONSTRUCTION IF EACH INITIAL ARMTS, Mk, IS OF ORDER V+1 AND EACH INITIAL QUASIGROUP, LK, IS OF ORDER V.** ======================================================================== * * * MTSCON3 * * * (THE 3V+1 CONSTRUCTION) THIS ROUTINE READS A FINITE SEQUENCE OF IDEMPOTENT QUASIGROUPS/ LATIN SQUARES, , OF ORDER V, (IN GROUPS OF THREE), DEFINED ON THE ELEMENTS 1,2,3,...,V, AND USES THE QUASIGROUPS TO CONSTRUCT A FINITE SEQUENCE OF MENDELSOHN TRIPLE SYSTEMS, , (OF ORDER 3V+1). ONE MSTAR IN THE SEQUENCE, , IS CONSTRUCTED FROM EACH SET OF THREE GIVEN QUASIGROUPS (L1,L2,L3: DEFINED ON SYMBOLS 1,2,...,V) IN THE SEQUENCE, . FURTHER RESTRICTIONS CAN BE MADE ON THE SEQUENCE , IN AN ATTEMPT TO OBTAIN AN ALMOST RESOLVABLE SEQUENCE, . FOR EXAMPLE, MIGHT BE CHOSEN TO EACH HAVE AN ORTHOGONAL MATE, I.E., TO EACH HAVE V DISJOINT TRANSVERSALS OF SIZE V. ======================================================================== * * * MTSCON4 * * * CONSTRUCTION C* (VERSION I) THIS ROUTINE CONSTRUCTS A FINITE SEQ., , OF MENDELSOHN TRIPLE SYS. OF ORDER Q=16 FROM A FINITE SEQUENCE, , OF ORDER 4 QUASIGROUPS (IN MULTIPLES OF FOUR: L1,L2,L3,L4) DEFINED ON THE ELEMENTS 1,2,3,4 AND FROM THE ALMOST RESOLVABLE MENDELSOHN TRIPLE SYSTEM Q1, OF ORDER 4 WHERE: Q1= <(1,2,3),(1,3,4),(1,4,2),(2,4,3)>. SECTION I: Q1 AND FOUR COPIES OF THE SET L CONSISTING OF THE ELEMENTS 1,2,3, AND 4 ARE USED TO DETERMINE 64 CYCLIC TRIPLES. SECTION II: Q1 AND THE FOUR QUASIGROUPS (L1,L2,L3,L4) ARE USED TO CONSTRUCT 16 CYCLIC TRIPLES. SECTIONS I AND II ARE COMBINED TO CONSTRUCT AN MTS OF ORDER 16. THIS PROCESS IS REPEATED FOR EACH SEQUENCE OF FOUR GIVEN QUASIGROUPS: L1,L2,L3,L4. ======================================================================== * * * MTSCON5 * * * CONSTRUCTION C* (VERSION II) ROUTINE CONSTRUCTS A FINITE SEQUENCE, , OF MENDELSOHN TRIPLE SYSTEMS OF ORDER Q=16 FROM A FINITE SEQUENCE, , OF QUASIGROUPS OF ORDER 4 (IN MULTIPLES OF FOUR: L=L1,L2,L3,L4) DEFINED ON THE ELEMENTS 1,2,3,4 AND FROM THE ALMOST RESOLVABLE MENDELSOHN TRIPLE SYSTEMS, Q1, Q2, BOTH OF ORDER 4, WHERE: Q1=<(1,2,3),(1,3,4),(1,4,2),(2,4,3)> AND Q2=<(1,2,4),(1,3,2),(1,4,3),(2,3,4)>. SECTION I: Q2 AND FOUR COPIES OF THE SET L (CONSISTING OF THE ELEMENTS 1,2,3, AND 4) ARE USED TO CONSTRUCT 64 CYCLIC TRIPLES. SECTION II: Q1 AND THE FOUR GIVEN QUASIGROUPS (L1,L2,L3,L4) ARE USED TO CONSTRUCT 16 CYCLIC TRIPLES. SECTIONS I AND II ARE COMBINED TO CONSTRUCT AN MTS OF ORDER 16. THIS PROCESS IS REPEATED FOR EACH SEQUENCE OF FOUR GIVEN QUASIGROUPS: L1,L2,L3,L4. ======================================================================== * * * MTSCON6 * * * CONSTRUCTION C* (VERSION III) THIS ROUTINE CONSTRUCTS A FINITE SEQUENCE, , OF MENDELSOHN TRIPLE SYSTEMS OF ORDER Q=16 FROM A FINITE SEQUENCE, , OF QUASI-GPS OF ORDER 4 (IN MULTIPLES OF FOUR: L1,L2,L3,L4) ON THE ELEMENTS 1,2,3,4 AND FROM TWO ALMOST RESOLVABLE MENDELSOHN TRIPLE SYSTEMS OF ORDER 4, Q1 AND Q2, WHERE: Q1=<1,2,3),(1,3,4),(1,4,2),(2,4,3)>, AND Q2=<(1,2,4),(1,3,2),(1,4,3),(2,3,4)>. SECTION I: Q2 AND FOUR COPIES OF THE SET L (DEFINED ON THE ELEMENTS 1,2,3,4) ARE USED TO CONSTRUCT A SET OF 64 CYCLIC TRIPLES. SECTION II: Q2 AND THE FOUR GIVEN QUASIGROUPS (L1,L2,L3,L4) ARE USED TO CONSTRUCT A SET OF 16 CYCLIC TRIPLES. SECTIONS I AND II : COMBINED TO CONSTRUCT AN MTS OF ORDER 16. THIS PROCESS IS REPEATED FOR EACH GIVEN SEQUENCE OF FOUR QUASIGROUPS: L1,L2,L3,L4. ======================================================================== * * * MTSCON7 * * * CONSTRUCTION C* (VERSION IV) THIS ROUTINE CONSTRUCTS A FINITE SEQUENCE, , OF MENDELSOHN TRIPLE SYSTEMS OF ORDER Q=16 FROM A FIN. SEQUENCE, , OF QUASIGROUPS (IN MULTIPLES OF FOUR: L1,L2,L3,L4) OF ORDER 4 (DEFINED ON THE ELEMENTS 1,2,3, AND 4) AND FROM TWO ALMOST RESOLVABLE MENDELSOHN TRIPLE SYSTEMS, Q1 AND Q2, WHERE: Q1=<(1,2,3),(1,3,4),(1,4,2),(2,4,3)> AND Q2=<(1,2,4),(1,3,2),(1,4,3),(2,3,4)>.S OF ORDER 4, Q1 AND Q2, (ON THE ELEMENTS 1,2,3, AND 4). SECTION I: Q1 AND FOUR COPIES OF THE SET L (DEFINED ON SYMBOLS 1,2,3, AND 4) ARE USED TO CONSTRUCT A SET OF 64 CYCLIC TRIPLES. SECTION II: Q2 AND THE FOUR GIVEN QUASIGROUPS (L1,L2,L3,L4) ARE USED TO CONSTRUCT A SET OF 16 CYCLIC TRIPLES. SECTIONS I AND II ARE COMBINED TO CONSTRUCT AN MTS OF ORDER 16. THIS PROCESS IS REPEATED FOR EACH GIVEN SEQUENCE OF FOUR QUASIGROUPS: L1,L2,L3,L4. ======================================================================== * * * MTSCON8 * * * (THE 3V CONSTRUCTION) L1, L2, AND L3 ARE IDEMPOTENT QUASIGROUPS/LATIN SQUARES OF ORDER Q (=V) TO BE USED IN THE 3V CONSTRUCTION. (EACH IDEMPOTENT QUASIGROUP IS DEFINED ON THE ELEMENTS 1,2,3,...,Q). THIS ROUTINE READS THE ABOVE QUASIGROUPS AND USES THEM TO CONSTRUCT A MENDELSOHN TRIPLE SYSTEM OF ORDER 3Q (=3V). FURTHER RESTRICTIONS ON L1, L2, AND L3 CAN BE MADE IN AN ATTEMPT TO OBTAIN A RESOLVABLE MTS OF ORDER 3Q (=3V), AT THE OPTION OF THE USER. L1, L2, AND L3 CAN ALL BE CHOSEN, E.G., TO HAVE DISJOINT TRANSVERSALS. THE ROUTINE IS FURTHER DESIGNED TO READ A SEQUENCE OF QUASIGROUPS IN MULTIPLES OF THREE (L1, L2, L3), AS ABOVE, AND TO PRODUCE A SEQUENCE OF MENDELSOHN TRIPLE SYSTEMS, EACH OF ORDER 3Q (=3V). ======================================================================== * * * GENMTS# * * * (MTS GENERATOR) THIS ROUTINE GENERATES A SEQUENCE OF MENDELSOHN TRIPLE SYSTEMS, , OF ORDER Q, REFERRED TO AS A SEQUENCE OF MTS(Q), VIA AN INITIAL MTS(Q), ARRANGED LEXICOGRAPHICALLY UP TO THE ABRIDGEMENT MTS(Q) WITH TRIPLES: (1,2,3), (1,3,5), (1,4,6), (1,5,7), (1,6,8), (1,7,9), ..., ETC., TO INCLUDE PAIRS OF MTS(Q), SIMULTANEOUSLY ISOMORPHIC TO ALL PAIRS, OF MTS(Q) HAVING AT LEAST ONE TRIPLE IN COMMON. THIS COVERS THE SPECTRUM FOR SUCH PAIRS. PAIRS OF MTS(Q) WITHOUT COMMON TRIPLES MUST BE GENERATED OTHERWISE. THE ROUTINE NULLIFIES ABRIDGEMENT (LIMIT ON SYSTEMS) FOR A COMPLETE SEGMENT OF (NSTOP - NPREV) MTS(Q), STARTING AFTER A GIVEN MTS(Q). ========================================================================