COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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226 Robert J. PIemmons of A. Now suppose that is the set of all distinct diagonals with the property that Di < Dia for each u E Phi. Let denote the set of all non-equivalent groupoids with diagonal Di. Then LDi is the set of all non-equivalent groupoids of order N. Thus to construct these groupoids we need only construct each diagonal Di, lowest in the ordering, determine the subgroup of PN leaving Di fixed, and then construct all non-equivalent groupoids with diagonal Di by using this subgroup in the equivalence checks. This procedure reduces the work factor considerably. In addition to this, terms can often be defined ahead in the table to ensure that the binary operation in the partial table has the desired properties. This algorithm has been coded into FORTRAN and the program has been run, in one form or another, on several computers, including those at Auburn University, the University of Tennessee and the National Security Agency. Programs resulting from the algorithm have been applied to the construction of various types of systems, such as groupoids, semigroups and loops; with the construction of the Cayley tables for all semigroups of order N 4 6 being one of the more noteworthy results. A monograph listing these tables, along with other information, can be obtained from the Department of Mathematics at Auburn University, Auburn, Alabama [8]. 3. Some analysis results. These finite semigroups, constructed by use of the algorithm described in Q 2, have been classified according to several properties, such as being regular, inverse or subdirectly irreducible. Such classification is accomplished by adding appropriate subroutines to the construction program. A table giving the number of (regular, inverse) semigroups of order N < 6 is given at the end of the paper (p. 228). Also included is the number of semigroups containing k idempotents for k = 1, . . ., N. Perhaps the most interesting use of the algorithm has been in the construction of specific finite systems of order N that satisfy certain identities or have certain properties, or else proving that no such systems exist for that order. For example, one such application has solved the problem of finding the smallest order semigroup whose system of identities has no finite basis [5], [6], [7]. THEOREM. There is a semigroup of order 5 whose system of identities has no finite basis, and, moreover, the system of identities for each semigroup of order N < 5 has a finite basis. Non-equivalent finite semigroups 227 Other analysis results were obtained by constructing all the congruence relations on each semigroup of order N G 5 and on selected semigroups of order 6. The algorithm to construct these relations first determines the equivalences on the set and then tests for compatibility with the binary operations of the non-equivalent semigroups of that order. The resulting examples are useful in the study of semigroup decompositions and in the study of homomorphisms, since the consideration of homomorphisms can be limited to the consideration of congruences. These computations have suggested the next result. THEOREM. The following four conditions concerning a semigroup S of order N w 2 are equivalent. (A) Each reflexive relation on S is left compatible. (B) Each equivalence relation on S is a left congruence. (C) For each x, y and z in S either xy = xz or xy = y and xz = z. (D) S = A ~.j B, where An B = 0 and where, for some idempotent function ffrom A to A, the binary operation for S is given by If(x) if xEA. xy = I y if xc B. This theorem, together with its dual, shows that each equivalence relation on a semigroup S is also a congruence relation if and only if S is a [left, right] zero semigroup. The examination of these examples has also led to the following conjecture : If a finite semigroup of order N > 3 has exactly one proper congruence relation, then it is a group or a simple group with zero. In conclusion we mention that the construction of all non-equivalent semigroups of order 7 would be rather difficult, both from the standpoint of running time on any particular computer and from the standpoint of output volume. Although there is no known rule giving the number of non-equivalent finite semigroups as a function of the order, a good estimate for the number of order 7 is around 200,000. However, the construction and analysis of special types for N Z= 7 is sometimes feasible and could be useful in the formulation and testing of conjectures. REFERENCES 1. A. H. CLIFFORD and G. P. PRESTON: The Algebraic Theory of Semigroups, Amer. Math. SOL, Math. Surveys, Vol. I (Providence, 1961). 2. A. H. CLIFFORD and G. P. PRESTON: The Algebraic Theory of Semigroups, Amer. Math. Sot., Math. Surveys, Vol. II (Providence, 1967). 3. G. E. FORSYTHE: SWAC computes 126 distinct semigroups of order 4. Proc. Amer. Math. Sot. 6 (1955), 443-445.
228 Robert J. Plemmons ! TOTALS Semigroups of order 2 / 3 I 1 Total no. non-equivalent I : I 4 I i 18 4/ 5 ! 126 1 1160 NO. commutative 13 1 12 58 ?25 No. regular No. inverse No. with 1 idempotent No. with 2 idempotents No. with 3 idempotents No. with 4 idempotents No. with 5 idempotents I No. with 6 idempotents ‘--.--I - 1 206 ~- 52 132 ___ 216 351 326 135 , 6 35,973 - - 4. M. HALL JR. and D. E. KNUTH: Combinatorial analysis and computers. Amer. Math. Monthly, 72 (1965), 21-28. 5. R. C. LYNDON: Identities in finite algebras. Proc. Amer. Math. Sot. 5 (1954) 8-9. 6. V. L. MURSKLI: The existence in three valued logic. of a closed class with finite’basis not having a finite complete system of identities. Soviet Math. Dok!. 6 (1965), 1020- 1024. 7. P. PERKINS: Bases for Equational Theories of Semigroups, Ph.D. Dissertation, University of California, Berkeley, 1966. 8. R. J. PLEMMONS: Cayley Tablesfor all Semigroups of Order N < 6, Monograph, Auburn University, Auburn, Alabama, 1966. 9. B. M. SCHEIN: Homomorphisms and subdirect decompositions of semigroups. Pacific.J. of Math. 17(1966). 529-547. 10. T. TAMURA: Some remarks on semigroups and all types of orders 2,3. J. Gakugei, Takashima University, 3 (1953), l-11. 11. K. 7. TETSUYA, T. HASHIMOTO, T. AKAZAWA, R. SHIBOTA, T. INUI, T. TAMURA: All semigroups of order at most 5. J. Gakugei, Tukushima University, 6 (1955), 19-39. 2143 1352 208 3107 1780 3093 4157 2961 875 Some contributions of computation to semigroups and groupoids’ TAKAYUKI TAMURA REVIEWING the contribution of computers to the theory of semigroups, we note that G. E. Forsythe computed all semigroups of order 4 [2] in 1954, and T. S. Motzkin and J. L. Selfridge obtained all semigroups of order 5 [4] in 1955. For the ten years from 1955 through 1965, nobody treated the computation of all semigroups of order 6. However, R. Plemmons did all semigroups of order 6 by IBM 7040 in 1965 [5]. On the other hand the author and his students obtained the semigroups of order 3 in 1953 [8], of order 4 in 1954 [9] and of order 5 in 1955 [1o0] by hand, independently of those mentioned above. Beside these, certain special types of semigroups and groupoids of order 3, which are distributive to given semigroups of order 3, were computed by hand [12], [13], [14]. In 1965 we obtained the number of non-isomorphic, non-anti-isomorphic groupoids of order ~4 which have a given permutation group as the automorphism group (@ 1.4, 1.5). Although the result was presented at the meeting of the American Mathematical Society at Reno, 1965, it has not been published. Afterwards R. Plemmons checked the total number by computing machine and wrote to the author that our number was correct; the author wishes to thank Dr. Plemmons. Recently R. Dickinson analyzed the behavior of some operations on the binary relations by machine [17]. In this paper we announce the result concerning the automorphism groups and the total number of groupoids and additionally we introduce the significance of a new concept “general product”, which uses a machine to get a suggestion for an important problem on the extension of semigroups, and further we show the result in a special case which easily computed without using a machine. The detailed proof of some theorems will be omitted because of pressure on space in these Proceedings, and the complete proof will be published elsewhere. 7 This research was supported by NSF GP-5988 and GP-7608. 229
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COMPUTATIONAL PROBLEMS IN ABSTRACT
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vi Contents E. KRAUSE and K. WESTON
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X Preface I am indebted to the auth
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4 J. Neubiiser 2.3.5. Added in proo
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8 J. Neubiiser Investigations of gr
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12 J. Neubiiser For 1 es 4 r, ws is
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16 J. Neubiiser Investigations of g
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Some examples using coset enumerati
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40 C. M. Campbell Some examples usi
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44 N. S. Mendelsohn for example, in
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48 H. Jiirgensen Calculation with e
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60 V. Fe&h and J. Neubiiser 2. The
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64 L. Gerhards and E. Altmann Autom
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68 L. Gerhards and E. Altmann Autom
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72 L. Gerhards and E. Altmann ni E
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76 W. Lindenberg and L. Gerhards Se
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80 W. Lindenberg and L. Gerhards Se
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84 K. Ferber and H. Jiirgensen The
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7 The construction of the character
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92 John McKay defining multiplicati
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96 John McKay Construction of chara
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100 John McKay In the table, c indi
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104 C. Brott and J. Neubiiser irred
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108 C. Brott and J. Neubiiser eleme
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112 J. S. Frame The characters of t
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116 J. S. Frame 3. The decompositio
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120 J. S. Frame The characters of t
Page 67 and 68: 124 J. S. Frame The characters of t
Page 69 and 70: TABLE 5. The Z, character block for
Page 71 and 72: 132 R. Biilow and J. Neubiiser Deri
Page 73 and 74: A search for simple groups of order
Page 75 and 76: 140 Marshall Hall Jr. Simple groups
Page 79 and 80: 148 Marshall Hall Jr. otherwise no
Page 81 and 82: 152 Marshall Hall Jr. easily found
Page 85 and 86: 160 Marshall Hail Jr. This leads to
Page 87 and 88: 164 Marshall Hall Jr. b= (OO)(Ol, 0
Page 89 and 90: 168 Marshall Hall Jr. 11. R. BRAUER
Page 91 and 92: 172 Charles C. Sims THEOREM 2.3. Th
Page 93 and 94: 176 Charles C. Sims has a set of im
Page 95 and 96: 180 Degrc - No. Order t N (-63 Gene
Page 97 and 98: An algorithm related to the restric
Page 99 and 100: A module-theoretic computation rela
Page 101 and 102: 192 A. L. Tritter expressed in the
Page 103 and 104: 196 A. L. Tritter a question is mos
Page 105 and 106: 200 John J. Cannon stack. The Cayle
Page 107 and 108: , I The computation of irreducible
Page 109 and 110: 208 P. G. Ruud and R. Keown product
Page 111 and 112: 212 P. G. Ruud and R. Keown TX wher
Page 113 and 114: 216 P. G. Ruud and R. Keown 4. L. G
Page 115 and 116: 220 N. S. Mendelsohn where it is kn
Page 117: 224 Robert J. Plemmons such class [
Page 121 and 122: 232 Takayuki Tamura 8” = 0B if an
Page 123 and 124: 236 Takayuki Tamura Case ,Q = {e).
Page 125 and 126: 240 Takayuki Tamura The calculation
Page 127 and 128: 244 Takayuki Tamura We calculate46
Page 129 and 130: 248 Takayuki Tamura Immediately we
Page 131 and 132: 252 Takayuki Tamura Let U be a subs
Page 133 and 134: 256 Takayuki Tamura TABLE 8. AI1 Se
Page 135 and 136: I The author and R. Dickinson have
Page 137 and 138: 264 Donald E. Knuth and Peter B. Be
Page 155 and 156: 300 Lowell J. Paige Non-associative
Page 157 and 158: 304 Lowell J. Paige We may lineariz
Page 159 and 160: 308 T(n) U(n) 0) [VI R-4 C. M. Glen
Page 161 and 162: 312 C. M. Glennie where in each cas
Page 163 and 164: 316 A. D. Keedwell of the elements
Page 165 and 166: A projective coqfiguration J. W. P.
Page 167 and 168: The uses of computers in Galois the
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328 W. D. Maurer mod p; for a facto
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332 J. H. Conway Enumeration of kno
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J. H. Conway Enumeration of knots a
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340 J. H. Conway -thus our symmetry
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344 J. H. Conway 2 string links to
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348 J. H. Conway 3 and 4 string lin
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352 Non-alternating J. H. Conway L
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356 J. H. Conway Alternating 11 cro
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H. F. Trotter Computations in knot
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364 H. F. Trotter to a. Schubert [1
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368 Shen Lin sequence of squares, t
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372 Harvey Cohn We also consider GZ
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376 Harvey Cohn In the case of Fig.
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380 Harvey Cohn always dictated by
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384 Hans Zassenhaus A real root cal
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388 Hans Zassenhaus the elements A,
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392 Hans Zassenhaus It is clear fro
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396 R. E. Kalman Invariant factors
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400 List of participants DR. J. J.
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