COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA. COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
258 Takayuki Tamura distinct (but not necessarily isomorphically distinct) direct products, i.e. D = fi (S(Oi)XTi), m G 4, i=l where T = b . : T,, T, s are right zero semigroups and either Bi = n(Ti) 1=1 (i= 1, . . . . m) or 0: = Z(Ti) (i = 1, . . ., m). III. Right generalproduct of S by right zero semigroup. First let T = {a, /3}, 0: a /I . The equations (2.9’) are tJ UP e *u. a 95 e+ = e+ ++b e., e+ ag e., = e., +b e., 8 *(L (I x e., = e., x0 e., e., a+k e+ = e., ++c, e+ 1 A relation w is defined on Gs as follows: 8 x 7 if and only if Recall that Using these notations, fu7== h7 b+d3== he zy: = zex, zy: = xez. for all a E S. (2.14) Therefore x -, 1ys? is an anti-homomorphism of a semigroup S(0) into the left regular representation of a semigroup S(q). We have obtained all non-isomorphic right general products of S, ISI =z 3, by a right zero semigroup of order 2. The results will be published elsewhere. IV. General product of S bq’ a right zero semigroup T of order 2. aB LetT={a,/3}, a a/3. B aB - - e. 1 - - 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 .- 662, 6, _. _. I 1 Semigroups and groupoids TABLE 9+ 4 - 7 rl 2, 3, 3,, 8, 8’, 9, 9,, 10, lo,, 13, 13,, 14, 14,, 14’, 14;, 15, 15,. 17, 18, 18, 2, 3, 15 4, 52, 16 42, 5, 162 7, 71, 11, 12, 2, 8, 14’. 14; 2, 9, 18 2, 10, 10, 7, 71, 11 7*, 12 2, 13, 131, 17 2, 8’, 14, 14, 2, 3, 15 4, 52, 16 2, 13, 131, 17 2, 9, 18 I t These were computed by P. Dubois, J. Youngs, T. Okamoto, R. Kaneiwa, and A. Ohta under the author’s direction. 259
I The author and R. Dickinson have computed all non-isomorphic general products of S, /Sj < 3, by a right zero semigroup of order 2 using a CDC 6600. The results will be published elsewhere. Acknowledgements. The author is grateful to the following people bfor their assistance in the computation by hand or by machine: Mr. Richard Biggs for Part I, Mr. Robert Dickinson, Professor Morio Sasaki and his students for Part II. The author also wishes to thank the Editor, Mr. John Leech, for his thoughtful elaboration of the manuscript. REFERENCES (2.16) 1. A. H. CLIFFORD and G. B. PRESTON: The Algebraic Theory of Semigroups, vol. 1, Amer. Math. Sot. Survey 7 (Providence, 1961). 2. G. E. FORSYTHE: SWAC computes 126 distinct semigroups of order 4. Proc. Amer. Math. Sot. 6 (1955), 443-445. 3. M. HALL: The Theory of Groups (Macmillan Co., New York, 1959). 4. T. S. MOTZKIN and J. L. SELFRIDGE: Semigroups of order five. (Presented in Amer. Math. Sot. Los Angeles Meeting on November 12,1955). 5. R. J. PLEMMONS: Cayley tables for all semigroups of order == 6. (Distributed by Department of Mathematics, Auburn University, Alabama 1965.) 6. E. SHENKMAN: Group Theory (Van Nostrand, Princeton, 1965). 7. T. TAMURA: On the system of semigroup operations defined in a set. J. Gakugei, Tokushima Univ. 2 (1952), 1-18. 8. T. TAMURA: Some remarks on semigroups and all types of semigroups of order 2,3. J. Gakugei, Tokushima Univ. 3 (1953), l-11. 9. T. TAMURA, Notes on finite semigroups and determination of semigroups of order 4. J. Gakugei, Tokushima Univ. 5 (1954), 17-27. 10. T. TAMURA et al.: All semigroups of order at most 5. J. Gakugei, Tokushima Univ. 6 (1955), 19-39. 11. T. TAMURA: Commutative nonpotent archimedean semigroup with cancellation law 1. J. Gakugei, Tokushima Univ. 8 (1957), 5-11. 12. T. Tm: Distributive multiplications to semigroup operations. J. Gakugei, Tokushima Univ. 8 (1957), 91-101. 13. T. TAMURA et al.: Semigroups of order < 10 whose greatest c-homomorphic images are groups. J. Gakugei, Tokushima Univ. 10 (1959), 43-64. 14. T. TAMURA: Semigroups of order 5, 6, 7, 8 whose greatest c-homomorphic images are unipotent semigroups with groups. J. Gakugei, Tokushima Univ. 11 (1960), 53-66. Semigroups and groupoids 261 15. T. TAMURA: Note on finite semigroups which satisfy certain grouplike condition. Proc. Jap. Acad. 36 (1960), 62-64. 16. T. TAMURA: Some special groupoids. Math. Jap. 8 (1963), 23-31. 17. T. TAMURA and R. DICKINSON: Semigroups connected with equivalence and congruence relations. Proc. Jap. Acad. 42 (1966), 688-692. 18. R. YOSHIDA: I-compositions of semigroups I. Memoirs of the Research Inst. of Sci. and Eng., Ritsumeikan Univ. 14 (1965), 1-12. 19. R. YOSHIDA: Z-compositions of semigroups II. Memoirs of the Research Inst. of Sci. and Eng., Ritsumeikan Univ. 15 (1966), l-5. 20. T. TAMURA: Note on automorphism group of groupoids. Proc. Jap. Acad. 43 (1967), 843-846.
- Page 83 and 84: 156 Marshall Hall Jr. Simple groups
- 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 and 118: 224 Robert J. Plemmons such class [
- Page 119 and 120: 228 Robert J. Plemmons ! TOTALS Sem
- 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: 256 Takayuki Tamura TABLE 8. AI1 Se
- Page 137 and 138: 264 Donald E. Knuth and Peter B. Be
- Page 139 and 140: 268 Donald E. Knuth and Peter B. Be
- Page 141 and 142: 272 Donald E. Knuth and Peter B. Be
- Page 143 and 144: 276 Donald E. Knuth and Peter B. Be
- Page 145 and 146: 280 Donald E. Knuth and Peter B. Be
- Page 147 and 148: 284 Donald E. Knuth and Peter B. Be
- Page 149 and 150: 288 Donald E. Knuth and Peter B. Be
- Page 151 and 152: 292 Donald E. Knuth and Peter B. Be
- Page 153 and 154: 296 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
- Page 169 and 170: 328 W. D. Maurer mod p; for a facto
- Page 171 and 172: 332 J. H. Conway Enumeration of kno
- Page 173 and 174: J. H. Conway Enumeration of knots a
- Page 175 and 176: 340 J. H. Conway -thus our symmetry
- Page 177 and 178: 344 J. H. Conway 2 string links to
- Page 179 and 180: 348 J. H. Conway 3 and 4 string lin
- Page 181 and 182: 352 Non-alternating J. H. Conway L
- Page 183 and 184: 356 J. H. Conway Alternating 11 cro
I<br />
The author and R. Dickinson have computed all non-isomorphic general<br />
products of S, /Sj < 3, by a right zero semigroup of order 2 using a CDC<br />
6600. The results will be published elsewhere.<br />
Acknowledgements. The author is grateful to the following people bfor<br />
their assistance in the computation by hand or by machine:<br />
Mr. Richard Biggs for Part I,<br />
Mr. Robert Dickinson, Professor Morio Sasaki and his students for<br />
Part II.<br />
The author also wishes to thank the Editor, Mr. John Leech, for his<br />
thoughtful elaboration of the manuscript.<br />
REFERENCES<br />
(2.16)<br />
1. A. H. CLIFFORD and G. B. PRESTON: The Algebraic Theory of Semigroups, vol. 1,<br />
Amer. Math. Sot. Survey 7 (Providence, 1961).<br />
2. G. E. FORSYTHE: SWAC computes 126 distinct semigroups of order 4. Proc. Amer.<br />
Math. Sot. 6 (1955), 443-445.<br />
3. M. HALL: The Theory of Groups (Macmillan Co., New York, 1959).<br />
4. T. S. MOTZK<strong>IN</strong> and J. L. SELFRIDGE: Semigroups of order five. (Presented in Amer.<br />
Math. Sot. Los Angeles Meeting on November 12,1955).<br />
5. R. J. PLEMMONS: Cayley tables for all semigroups of order == 6. (Distributed by<br />
Department of Mathematics, Auburn University, Alabama 1965.)<br />
6. E. SHENKMAN: Group Theory (Van Nostrand, Princeton, 1965).<br />
7. T. TAMURA: On the system of semigroup operations defined in a set. J. Gakugei,<br />
Tokushima Univ. 2 (1952), 1-18.<br />
8. T. TAMURA: Some remarks on semigroups and all types of semigroups of order 2,3.<br />
J. Gakugei, Tokushima Univ. 3 (1953), l-11.<br />
9. T. TAMURA, Notes on finite semigroups and determination of semigroups of order 4.<br />
J. Gakugei, Tokushima Univ. 5 (1954), 17-27.<br />
10. T. TAMURA et al.: All semigroups of order at most 5. J. Gakugei, Tokushima Univ.<br />
6 (1955), 19-39.<br />
11. T. TAMURA: Commutative nonpotent archimedean semigroup with cancellation<br />
law 1. J. Gakugei, Tokushima Univ. 8 (1957), 5-11.<br />
12. T. Tm: Distributive multiplications to semigroup operations. J. Gakugei,<br />
Tokushima Univ. 8 (1957), 91-101.<br />
13. T. TAMURA et al.: Semigroups of order < 10 whose greatest c-homomorphic<br />
images are groups. J. Gakugei, Tokushima Univ. 10 (1959), 43-64.<br />
14. T. TAMURA: Semigroups of order 5, 6, 7, 8 whose greatest c-homomorphic images<br />
are unipotent semigroups with groups. J. Gakugei, Tokushima Univ. 11 (1960),<br />
53-66.<br />
Semigroups and groupoids 261<br />
15. T. TAMURA: Note on finite semigroups which satisfy certain grouplike condition.<br />
Proc. Jap. Acad. 36 (1960), 62-64.<br />
16. T. TAMURA: Some special groupoids. Math. Jap. 8 (1963), 23-31.<br />
17. T. TAMURA and R. DICK<strong>IN</strong>SON: Semigroups connected with equivalence and congruence<br />
relations. Proc. Jap. Acad. 42 (1966), 688-692.<br />
18. R. YOSHIDA: I-compositions of semigroups I. Memoirs of the Research Inst. of Sci.<br />
and Eng., Ritsumeikan Univ. 14 (1965), 1-12.<br />
19. R. YOSHIDA: Z-compositions of semigroups II. Memoirs of the Research Inst. of Sci.<br />
and Eng., Ritsumeikan Univ. 15 (1966), l-5.<br />
20. T. TAMURA: Note on automorphism group of groupoids. Proc. Jap. Acad. 43 (1967),<br />
843-846.
Change language