COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

More documents

Recommendations

Info

254 Takayuki Tamura TABLE 5. Semigroups of Order 2 11 is exactly the same as 1, i.e. lo = 11. 1’ denotes %Z ’ n omitted from Table 5. Table 6 shows all r] such that 8,, - 7. We may pick 13~ from all non-isomorphic semigroups, but must select 7 from all semigroups. Generally the following holds : 8 - 7 implies 8g, r~ 7~ for all permutations v of S (see 9 2.2), (2.12) e - 7 implies +j’ - 8’. (2.13) TABLE 6 e. 1 1’ 2 3 4 rl 1 1’ 2, 3 2, 3 4, 4, From the table we also have e. = 21, 7 = 21, 31, e. = 31, r = 21, 31, e. = 41, ?j = 4, 41. Semigroups and groupoids 255 Table 7 shows all non-isomorphic left general products D of S, 1 S 1 = 2, by a right zero semigroup of order 2. TABLE 7 6 cz. % 1 1 1’ 1’ 2 2 2 3 3 3 4 j 4 As an application of the above results, we have THEOREM~.~. Let Sbeaset, IS/ = 2, andTbearight zerosemigroup of order n. A left general product D of S by T is isomorphic onto either the direct product of a semigroup S of order 2 and a right zero semigroup T of order n DrSXT, lSl=2, jTl=n or the union of the two direct products D = (Sl X TI) U (Sz X T2), where TI and T2 are right zero semigroups, / TII + I T2/ = n and SI is a null semigroup of order 2 and S2 is a semilattice of order 2. II. Left generalproduct of S, ISI = 3, by right zero semigroup. a B Let T = {a, ,4}, cc cc /? . B.u B The method is the same as in case I, and we use the same notation. Let Gs denote the set of all semigroups defined on S, [ S/ = 3. Table 8 shows Ga except the dual forms. Those were copied from [8], [lo]. Table 9 shows all 17 for given B. such that B. IV 7. This table shows, for example, that 2 = 20 = 21, 22 = 23, 24 = 25. In the following family X of ten subsets of Gs, each set satisfies the property: Any two elements of each set are --equivalent, and each set is a maximal set with this property. z {I}, (2, 3, 15>, (4, 52, lb}, 6, 62, 64}, (7, 72, 1 l}, 1 (7, 122}, (2, 8, 14’, 14;}, (2, 9, 18}, (2, 10, 101}, (2, 13, 131 17). Let S’ denote the family obtained from S by replacing (6, 62, 64,) by (6) and (2, 10, 101) by (2, lo} and leaving the remaining sets unchanged.
256 Takayuki Tamura TABLE 8. AI1 Semigroups of Order 3 up to Isomorphism andDual-isomorphism 1 2 3 4 5 6 r-l abc abc abc ana uua aaa r-l 0 0 0 aaa aca aaa -I sac sac act cca El LCUU caa 0 I 1 UUC sac llCZC II aaa act ace 2 2 0 I i cbu bbb abc I ! 0 ccc ccc ccc ” ccc cat ccc Ir L- 1- caa act act 17 cca cca sac 17 sac / sac / ccc L / bat ccc L-l 4 4 4 0 bbc bbc , ccc ’ I i bat I ( / abc, / ccc ~ 7 11 12 13 14 15 16 17 18 r Semigroups and groupoids TABLE 8 (continued) o(:E) l(Z) 4::) 3(E) aaa aaa sac i I/ aaa uba I -I sac 0 aaa abc act L 0 bb61 bbbj bbc’ bba Fq abb i bbb’ bbc! Iab6/ j bbb / ebb 2 I / sac abc ccc i 4 L-l 1 a b c bbc ccc We have the following theorem, in which we do not assume T is finite: THEOREM 2.8. A left general product D of S, [ SI = 3, by a right zero semigroup T is determined by a mapping z of the set T into one of the sets belonging to 2’ in such a way that 8,. = n(a), r. E T. Every left general product D of S, ISI = 3, by T is isomorphic or anti-isomorphic onto one of those thus obtained. Accordingly D is the disjoint union of at most four
Page 1 and 2:
COMPUTATIONAL PROBLEMS IN ABSTRACT
Page 3 and 4:
vi Contents E. KRAUSE and K. WESTON
Page 5 and 6:
X Preface I am indebted to the auth
Page 7 and 8:
4 J. Neubiiser 2.3.5. Added in proo
Page 9 and 10:
8 J. Neubiiser Investigations of gr
Page 11 and 12:
12 J. Neubiiser For 1 es 4 r, ws is
Page 13 and 14:
16 J. Neubiiser Investigations of g
Page 23 and 24:
Some examples using coset enumerati
Page 25 and 26:
40 C. M. Campbell Some examples usi
Page 27 and 28:
44 N. S. Mendelsohn for example, in
Page 29 and 30:
48 H. Jiirgensen Calculation with e
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
60 V. Fe&h and J. Neubiiser 2. The
Page 37 and 38:
64 L. Gerhards and E. Altmann Autom
Page 39 and 40:
68 L. Gerhards and E. Altmann Autom
Page 41 and 42:
72 L. Gerhards and E. Altmann ni E
Page 43 and 44:
76 W. Lindenberg and L. Gerhards Se
Page 45 and 46:
80 W. Lindenberg and L. Gerhards Se
Page 47 and 48:
84 K. Ferber and H. Jiirgensen The
Page 49 and 50:
7 The construction of the character
Page 51 and 52:
92 John McKay defining multiplicati
Page 53 and 54:
96 John McKay Construction of chara
Page 55 and 56:
100 John McKay In the table, c indi
Page 57 and 58:
104 C. Brott and J. Neubiiser irred
Page 59 and 60:
108 C. Brott and J. Neubiiser eleme
Page 61 and 62:
112 J. S. Frame The characters of t
Page 63 and 64:
116 J. S. Frame 3. The decompositio
Page 65 and 66:
Page 67 and 68:
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 77 and 78:
144 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 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: 252 Takayuki Tamura Let U be a subs
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
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
Page 185 and 186:
H. F. Trotter Computations in knot
Page 187 and 188:
364 H. F. Trotter to a. Schubert [1
Page 189 and 190:
368 Shen Lin sequence of squares, t
Page 191 and 192:
372 Harvey Cohn We also consider GZ
Page 193 and 194:
376 Harvey Cohn In the case of Fig.
Page 195 and 196:
380 Harvey Cohn always dictated by
Page 197 and 198:
384 Hans Zassenhaus A real root cal
Page 199 and 200:
388 Hans Zassenhaus the elements A,
Page 201 and 202:
392 Hans Zassenhaus It is clear fro
Page 203 and 204:
396 R. E. Kalman Invariant factors
Page 205 and 206:
400 List of participants DR. J. J.
show all

COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?