Classification of Maps by Their Membership on Maximal Clones that Contain Minimum and Complement

Rade Doroslovački, Jovanka Pantović

Faculty of Engineering, University of Novi Sad Trg D. Obradovića 6, 21000 Novi Sad, Yugoslavia

Gradimir Vojvodić

Institute of Mathematics, University of Novi Sad Trg D. Obradovića 4, 21000 Novi Sad, Yugoslavia

Abstract In this paper, five-valued logic functions are classified according to their membership in the

maximal clones which contain min(x, y) and \overline{x} = 4 − x.

1 Notation and Preliminaries

Denote by N the set {1, 2,…} of positive integers. For k, n ∈ N, E_{k} = {0, 1,…, k − 1}, denote by P_{k}^{(n)} the set of all maps E_{k}^{n} → E_{k}, and P_{k} = ⋃_{n∈N} P_{k}^{(n)}. We say that f is an i-th projection of arity

n (1 ≤ i ≤ n) if f ∈ P_{k}^{(n)} and f satisfies the identity f(x_{1},…, x_{n}) ≈ x_{i}. We say that f ∈ P_{k}^{(n)} is essential if it depends on at least two variables and it takes all values from E_{k}. Let π_{i}^{n} denote the i-th projection of arity n, and let Π_{k} denote the set of all projections over E_{k}. For n, m ≥ 1, f ∈ P_{k}^{(n)} and g_{1},…, g_{n} ∈ P_{k}^{(m)}, the superposition of f and g_{1},…, g_{n}, denoted by f(g_{1},…, g_{n}), is defined by f(g_{1},…, g_{n})(a_{1},…, a_{m}) = f(g_{1}(a_{1},…, a_{m}),…, g_{n}(a_{1},…, a_{m})) for all (a_{1},…, a_{m}) ∈ E_{k}^{m}. A set F ⊆ P_{k} is a clone of operations on E_{k} (or clone for short) if Π_{k} ⊆ F and F is closed with respect to superposition. For F ⊆ P_{k}, ⟨F ⟩_{CL} stands for the clone generated by F. We say that clone F is maximal if there is no clone G such that F ⊂ G ⊂ P_{k}. F ⊆ P_{k} is complete if ⟨F ⟩_{CL} = P_{k}.

Let ϱ ⊆ E_{k}^{h} be an h-ary relation and f ∈ P_{k}^{(n)}. We say that f preserves ϱ if for all h-tuples (a_{11},…, a_{1h}),…, (a_{n1},…, a_{nh}) from ϱ we have (f(a_{11},…, a_{n1}),…, f(a_{1h},…, a_{nh})) ∈ ϱ. Pol ϱ is the set of all f ∈ P_{k} which preserve ϱ. For F ⊆ P_{k}, Inv F denotes the set of all the relations preserved by each f ∈ F.

Let C be a clone on E_{k} and F ⊆ P_{k}. F is a complete relative to C (or C-complete) if ⟨F ∪C⟩_{CL} = P_{k}. A relative complete set F in P_{k} is called a relative base of P_{k} with respect to C if no proper subset

of F is relative complete in P_{k} with respect to C.

The functions from P_{k} may be classified into nonempty equivalence classes by their membership in the relative maximal clones. Then we can discuss the completeness properties in P_{k} in terms of these classes instead of individual functions: if a set is complete then replacing a function in the set by any function in the corresponding equivalence class yields another complete set.

Let M_{1}, ..., M_{m} be relative maximal clones. Define the map ϕ : P_{k} → {0, 1}^{m} by setting ϕ(f) := a_{1}...a_{m} where a_{i} = 0 if f ∈ M_{i} and a_{i} = 1 if f ∉ M_{i}. We call ϕ(f) the characteristic vector of f. We put fϱg if f, g ∈ P_{k} have the same characteristic vector, i.e. if ϕ(f) = ϕ(g). It means that for all j ∈ {1, 2, ..., n}, either {f, g} ⊂ M_{j} or {f, g} ∩ M_{j} = ∅. Clearly, ϱ is an equivalence relation on P_{k} and so it partitions P_{k} into pairwise disjoint nonempty sets called (equivalence) classes.

To a_{1}, ..., a_{m} ∈ {0, 1}^{m} we associate A = {i : a_{i} = 1} and if A_{1}, ..., A_{l} are the subsets of {1, ..., m} corresponding to the characteristic vectors, the relative completeness problem is reduced to the listing of subsets of {A_{1}, ..., A_{l}} covering {1, ..., m}.

All maximal clones in P_{k} containing the functions min and complement and all maximal clones in P_{k}, 2 < k < 9 containing two special unary operations are determined in [3].

2 Results and discussion

Using the statements of paper [3] it follows: Theorem 2.1 There are 11 relative maximal clones in P_{5} :

Q_{1} = P ol\binom{012340134}{012341043} = P ol(E_{5}^{2} − P_{02} − P_{03} − P_{04} − P_{12} − P_{13} − P_{14} − P_{23} − P_{24})

Q_{2} = P ol\binom{01234121323}{01234213132} = P ol(E_{5}^{2} − P_{01} − P_{02} − P_{03} − P_{04} − P_{14} − P_{24} − P_{34})

Q_{3} = P ol(0134) Q_{4} = P ol(04) Q_{5} = P ol(123) Q_{6} = P ol(13) Q_{7} = P ol(024) Q_{8} = P ol(2)

Q_{9} = P ol\binom{01234010203121314232434}{01234102030213141324243} = P ol(E_{5}^{2} − P_{04})

Q_{10} = P ol\binom{0123401021213232434}{0123410202131324243} = P ol(E_{5}^{2} − P_{04} − P_{03} − P_{14})

Q_{11} = P ol\binom{01234010212232434}{01234102021324243} = P ol(E_{5}^{2} − P_{04} − P_{03} − P_{13} − P_{14})

From theorems 1-14 [4] it follows respectively: Theorem 2.2

1. Q_{5}Q_{7} ⊂ Q_{8} ∧ Q_{3}Q_{5} ⊂ Q_{6} ∧ Q_{3}Q_{7} ⊂ Q_{4}.

2. \overline{Q}_{9}Q_{10} ⊂ Q_{8} ∧ \overline{Q}_{9}Q_{11} ⊂ Q_{8} ∧ \overline{Q}_{10}Q_{11} ⊂ Q_{8}.

3. Q_{2}Q_{6} ⊂ Q_{5} ∧ Q_{2}Q_{8} ⊂ Q_{5} ∧ Q_{1}Q_{4} ⊂ Q_{3} ∧ Q_{1}Q_{6} ⊂ Q_{3}.

4. Q_{1}Q_{7}Q_{9} ⊂ Q_{10} ∧ Q_{1}Q_{7}Q_{10} ⊂ Q_{11}.

5. Q_{7}\overline{Q}_{8}\overline{Q}_{10} ⊂ \overline{Q}_{11} ∧ Q_{7}\overline{Q}_{8}\overline{Q}_{11} ⊂ \overline{Q}_{10} ∧ Q_{7}\overline{Q}_{8}\overline{Q}_{9} ⊂ \overline{Q}_{10} ∧ Q_{7}\overline{Q}_{8}\overline{Q}_{9} ⊂ \overline{Q}_{11}.

6. Q_{2}\overline{Q}_{5}\overline{Q}_{11} ⊂ \overline{Q}_{10} ∧ Q_{2}\overline{Q}_{5}\overline{Q}_{9} ⊂ \overline{Q}_{10} ∧ Q_{2}\overline{Q}_{5}\overline{Q}_{9} ⊂ \overline{Q}_{11}.

7. Q_{4}Q_{9}\overline{Q}_{10} ⊂ \overline{Q}_{11}

8. Q_{3}Q_{10} ⊂ Q_{9} ∧ Q_{6}Q_{10} ⊂ Q_{9} ∧ Q_{4}Q_{10} ⊂ Q_{11}.

9. Q_{1}Q_{2} ⊂ Q_{9}.

10. Q_{2}Q_{10} ⊂ Q_{9} ∧ Q_{2}Q_{11} ⊂ Q_{9}.

11. Q_{2}Q_{11} ⊂ Q_{10}.

12. Q_{1}Q_{2}\overline{Q}_{5} ⊂ Q_{9} ∧ Q_{1}Q_{2}\overline{Q}_{5} ⊂ Q_{10} ∧ Q_{1}Q_{2}\overline{Q}_{5} ⊂ Q_{11}.

13. Q_{1}\overline{Q}_{5}\overline{Q}_{9} ⊂ \overline{Q}_{10} ∧ Q_{1}\overline{Q}_{5}\overline{Q}_{11} ⊂ \overline{Q}_{10}.

14. Q_{1}Q_{2}\overline{Q}_{5} ⊂ Q_{3}.

Remarks It is important to notice that theorem 2.2. is useful for classification of functions in P_{5} (according to their membership in all maximal clones), which is still an open problem.

Theorem 2.3 There are 391 classes of functions in P_{5} with respect to the clone generated by min(x, y) and \overline{x} = 4 − x.

Proof In P_{5} we have 2048 possible classes of functions. According to the previous theorem we have found 1657 empty classes of functions. Representatives are efectively constructed for other 391 classes of functions. They are given in the following tables.

class of funct. repre.
1 00000000000  #72
2 00000010000 #113
3 00000010011 #114
4 00000011000  #62
5 00000011011  #71
6 00001101000  #1
7 00001111000 #115
8 00010010000  #73
9 00010010001 #116
10 00010010011 #117
11 00010011000  #63
12 00010011001 #108
13 00010011011  #80
14 00011111000  #2
15 00110100000  #79
16 00110110000 #110
17 00110110001 #118
18 00110110011 #119
19 00110111000  #69
20 00110111001 #111
21 00110111011  #84
22 01000000000 #120
23 01000000110 #121
24 01000000111 #122
25 01000010000 #123
26 01000010011 #124
27 01000010110 #124
28 01000010111 #126
29 01000011000 #127
30 01000011011 #128
31 01000011111 #129
32 01001000000 #130
33 01001000110 #131
34 01001000111 #132
35 01001001000  #53
36 01001001111  #61
37 01001010000 #133
38 01001010011 #134
39 01001010110 #135
40 01001010111 #136
41 01001011000 #137
42 01001011011 #138
43 01001011111 #139
44 01001100000  #32
45 01001100110  #45
46 01001100111 #140
47 01001101000  #6
48 01001101111  #19
49 01001110000 #141
50 01001110011 #142
51 01001110110 #143
52 01001110111 #144
53 01001111000  #23
54 01001111011  #49
55 01001111111  #30
56 01010010000 #145
57 01010010001 #146
58 01010010010 #147
59 01010010011 #148
60 01010010110 #149
61 01010010111 #150
62 01010011000 #151
63 01010011001 #152
64 01010011011 #153
65 01010011111 #154
66 01011010000 #155
67 01011010010 #156

class of funct. repre.
68 01011010011 #157
69 01011010110 #158
70 01011010111 #159
71 01011011000 #54
72 01011011011 #60
73 01011011111 #85
74 01011110000 #33
75 01011110010 #44
76 01011110011 #160
77 01011110110 #48
78 01011110111 #161
79 01011111000  #7
80 01011111011 #18
81 01011111111 #22
82 01110100000 #162
83 01110100100 #163
84 01110100110 #164
85 01110100111 #165
86 01110110000 #166
87 01110110001 #167
88 01110110010 #168
89 01110110011 #169
90 01110110100 #170
91 01110110101 #171
92 01110110110 #172
93 01110110111 #173
94 01110111000 #174
95 01110111001 #175
96 01110111011 #176
97 01110111111 #177
98 01111100000 #40
99 01111100110 #178
100 01111100111 #179
101 01111101000 #13
102 01111101111 #52
103 01111110000 #102
104 01111110010 #180
105 01111110011 #181
106 01111110110 #182
107 01111110111 #183
108 01111111000 #27
109 01111111011 #51
110 01111111111 #103
111 10000000000 #184
112 10000000011 #76
113 10000000111 #185
114 10000010000 #186
115 10000010011 #187
116 10000010111 #188
117 10000011000 #189
118 10000011011 #66
119 10000011111 #190
120 10001101000 #191
121 10001101011 #192
122 10001101111  #5
123 10001111000 #193
124 10001111011 #194
125 10001111111 #195
126 10010010000 #196
127 10010010001 #75
128 10010010011 #95
129 10010010111 #197
130 10010011000 #198
131 10010011001 #65
132 10010011011 #82
133 10010011111 #199
134 10011111000 #200

class of funct. repre.
135 10011111011  #4
136 10011111111 #99
137 10100000000 #201
138 10100000011 #202
139 10100000111 #203
140 10100010000 #204
141 10100010011 #205
142 10100010111 #206
143 10100011000 #207
144 10100011011 #208
145 10100011111 #209
146 10100100000 #77
147 10100100011 #210
148 10100100111 #211
149 10100110000 #212
150 10100110011 #213
151 10100110111 #214
152 10100111000 #67
153 10100111011 #70
154 10100111111 #215
155 10101101000 #216
156 10101101011 #217
157 10101101111 #218
158 10101111000 #219
159 10101111011 #220
160 10101111111 #221
161 10110000000 #74
162 10110000001 #222
163 10110000011 #96
164 10110000111 #223
165 10110010000 #109
166 10110010001 #224
167 10110010011 #225
168 10110010111 #226
169 10110011000 #64
170 10110011001 #107
171 10110011011 #81
172 10110011111 #227
173 10110100000 #91
174 10110100001 #228
175 10110100011 #98
176 10110100111 #229
177 10110110000 #78
178 10110110001 #230
179 10110110011 #97
180 10110110111 #231
181 10110111000 #68
182 10110111001 #106
183 10110111011 #83
184 10110111111 #232
185 10111101000  #3
186 10111101011 #233
187 10111101111 #100
188 10111111000 #101
189 10111111011 #112
190 10111111111 #234
191 11000000000 #235
192 11000000011 #236
193 11000000110 #237
194 11000000111 #238
195 11000010000 #239
196 11000010011 #240
197 11000010110 #241
198 11000010111 #242
199 11000011000 #243
200 11000011011 #244
201 11000011111 #245

class of funct. repre.
202 11001000000 #246
203 11001000011 #247
204 11001000110 #248
205 11001000111 #249
206 11001001000 #250
207 11001001011  #58
208 11001001111  #57
209 11001010000 #251
210 11001010011 #252
211 11001010110 #253
212 11001010111 #254
213 11001011000 #255
214 11001011011 #256
215 11001011111 #257
216 11001100000 #258
217 11001100011  #42
218 11001100110 #259
219 11001100111  #36
220 11001101000 #260
221 11001101011  #16
222 11001101111  #10
223 11001110000 #261
224 11001110011 #262
225 11001110110 #263
226 11001110111 #264
227 11001111000 #265
228 11001111011  #29
229 11001111111  #24
230 11010010000 #266
231 11010010001 #267
232 11010010010 #268
233 11010010011 #269
234 11010010110 #270
235 11010010111 #271
236 11010011000 #272
237 11010011001 #273
238 11010011011 #274
239 11010011111 #275
240 11011010000 #276
241 11011010001 #277
242 11011010010 #278
243 11011010011 #279
244 11011010110 #280
245 11011010111 #281
246 11011011000 #282
247 11011011001 #283
248 11011011011  #56
249 11011011111  #87
250 11011110000 #284
251 11011110001  #37
252 11011110010 #285
253 11011110011  #35
254 11011110110 #286
255 11011110111  #46
256 11011111000 #287
257 11011111001  #25
258 11011111011  #9
259 11011111111  #20
260 11100000000 #288
261 11100000011 #289
262 11100000110 #290
263 11100000111 #291
264 11100010000 #292
265 11100010011 #293
266 11100010110 #294
267 11100010111 #295
268 11100011000 #296

24

class of funct. repre.
269 11100011011 #297
270 11100011111 #298
271 11100100000 #299
272 11100100011 #300
273 11100100100 #301
274 11100100110 #302
275 11100100111 #303
276 11100110000 #304
277 11100110011 #305
278 11100110100 #306
279 11100110110 #307
280 11100110111 #308
281 11100111000 #309
282 11100111011 #310
283 11100111111 #311
284 11101000000 #312
285 11101000011 #313
286 11101000110 #314
287 11101000111 #315
288 11101001000 #316
289 11101001011 #317
290 11101001111 #318
291 11101010000 #319
292 11101010011 #320
293 11101010110 #321
294 11101010111 #322
295 11101011000 #323
296 11101011011 #324
297 11101011111 #325
298 11101100000 #38
299 11101100011 #326

class of funct. repre.
300 11101100100 #41
301 11101100110 #327
302 11101100111 #90
303 11101101000 #11
304 11101101011 #88
305 11101101111 #15
306 11101110000 #328
307 11101110011 #329
308 11101110100 #330
309 11101110110 #331
310 11101110111 #332
311 11101111000 #26
312 11101111011 #50
313 11101111111 #28
314 11110000000 #333
315 11110000001 #334
316 11110000010 #335
317 11110000011 #336
318 11110000110 #337
319 11110000111 #338
320 11110010000 #339
321 11110010001 #340
322 11110010010 #341
323 11110010011 #342
324 11110010110 #343
325 11110010111 #344
326 11110011000 #345
327 11110011001 #346
328 11110011011 #347
329 11110011111 #348
330 11110100000 #349

class of funct. repre.
331 11110100001 #350
332 11110100010 #351
333 11110100011 #352
334 11110100100 #353
335 11110100101 #354
336 11110100110 #355
337 11110100111 #356
338 11110110000 #357
339 11110110001 #358
340 11110110010 #359
341 11110110011 #360
342 11110110100 #361
343 11110110101 #362
344 11110110110 #363
345 11110110111 #364
346 11110111000 #365
347 11110111001 #366
348 11110111011 #367
349 11110111111 #368
350 11111000000 #369
351 11111000001 #370
352 11111000010 #371
353 11111000011 #372
354 11111000110 #373
355 11111000111 #374
356 11111001000 #55
357 11111001011 #59
358 11111001111 #86
359 11111010000 #375
360 11111010001 #376
361 11111010010 #377

class of funct. repre.
362 11111010011 #378
363 11111010110 #379
364 11111010111 #380
365 11111011000 #104
366 11111011001 #381
367 11111011011 #105
368 11111011111 #382
369 11111100000  #34
370 11111100001 #383
371 11111100010  #43
372 11111100011 #384
373 11111100100  #93
374 11111100101 #385
375 11111100110  #47
376 11111100111 #386
377 11111101000  #8
378 11111101011  #17
379 11111101111  #21
380 11111110000  #39
381 11111110001 #387
382 11111110010 #388
383 11111110011  #89
384 11111110100  #94
385 11111110101 #389
386 11111110110 #390
387 11111110111  #92
388 11111111000  #12
389 11111111001 #391
390 11111111011  #14
391 11111111111  #31

UNARY REPRESENTATIVES:

# 01234 1 00000 2 00001 3 00002 4 00003 5 00004 6 00010 7 00011 8 00012 9 00013 10 00014 11 00020 12 00021 13 00022 14 00023 15 00024 16 00030 17 00032 18 00033 19 00034 20 00041 21 00042 22 00043 23 00100 24 00104 25 00113 26 00120 27 00122 28 00124 29 00130 30 00134 31 00142 32 00200 33 00201 34 00202 35 00203 36 00204 37 00213 38 00220 39 00221 40 00222 41 00224 42 00230 43 00232 44 00233 45 00234 46 00241 47 00242 48 00243 49 00300 50 00320 51 00322 52 00422 53 01010 54 01011 55 01012 56 01013 57 01014 58 01030 59 01032 60 01033 61 01034 62 01110 63 01111 64 01112 65 01113 66 01114 67 01120 68 01121 69 01122 70 01124 71 01134 72 01210 73 01211 74 01212 75 01213 76 01214 77 01220 78 01221 79 01222 80 01311 81 01312 82 01313 83 01321 84 01322 85 01411 86 01412 87 01413 88 02030 89 02203 90 02204 91 02222 92 02241 93 02242 94 02243 95 03211 96 03212 97 03221 98 03222 99 04441 100 04442 101 10002 102 10222 103 10422 104 11012 105 11032 106 11123 107 11132 108 11133 109 11212 110 11222 111 11322 112 14442

BINARY REPRESENTATIVES : f(00)f(01)f(02)f(03)f(04) | f(10)f(11)f(12)f(13)f(14) | f(20)f(21)f(22)f(23)f(24) | f(30)f(31)f(32)f(33)f(34) | f(40)f(41)f(42)f(43)f(44)

113 43334 43334 43234 43334 44444 114 01334 11333 22233 33333 43334 115 01110 00000 00000 00000 00000 116 11333 11333 33233 33333 33333 117 10001 11111 33211 11111 11111 118 22133 22133 22222 22222 22222 119 10001 11111 33222 22222 22222 120 44244 43234 22222 43234 44244 121 00200 01210 22222 01234 00244 122 44444 43333 43234 43133 44044 123 01100 11111 11211 11111 01110 124 44444 43334 43234 43134 44144 125 01110 11111 22222 43334 44444 126 44444 43334 43233 43134 44044 127 43334 33334 33333 33334 43334 128 44444 43334 43334 43134 44144 129 44444 43334 43333 43134 44044 130 44444 43434 44244 43434 44444 131 00000 11111 22222 33433 44444 132 44444 43334 43244 43134 44044 133 44444 43434 44243 43434 44444 134 44444 43334 43244 43134 44144 135 00000 11111 22222 33433 43334 136 44344 43334 43244 43134 44044 137 44444 43434 44344 43434 44444 138 44444 43334 43344 43134 44144 139 44444 43334 43344 43134 44044 140 44444 43344 43244 43134 44044 141 44444 44444 44243 44444 44444 142 00100 00100 33200 00000 00000 143 00000 11111 22222 34433 43334 144 44344 43344 43244 43134 44044 145 44233 43233 22222 43234 44244 146 11311 11311 11111 11111 11011 147 11111 01111 22222 33333 33333 148 44233 43233 43211 43334 44444

25

149 01111 11111 22222 43333 34444 150 11111 11110 11234 11111 11111 151 11111 11111 01111 11111 11111 152 11311 11311 11111 11111 11011 153 33333 33333 33311 33334 33333 154 33333 33333 33310 33334 33333 155 44443 43434 44244 43434 44444 156 11111 11011 22222 33333 33333 157 33333 33333 33211 33434 33333 158 11110 11111 22222 43434 44444 159 33333 33333 33200 33334 33333 160 33333 33343 33211 33333 33333 161 33333 33343 33201 33333 33333 162 22222 22222 22222 43222 44222 163 01222 01222 22222 22233 22244 164 22210 22211 22222 01234 00244 165 22222 22222 22210 22234 22244 166 22243 22234 22222 22222 22222 167 22022 22122 11222 33222 33222 168 22211 22211 22222 22234 22244 169 22222 22222 22211 22234 22244 170 11222 01222 22222 22233 22243 171 11133 01133 22222 22233 22243 172 01222 11222 22222 43222 33222 173 22222 22222 33210 22334 22444 174 22011 22111 11111 11111 11111 175 22133 22133 22322 22333 22344 176 22222 22222 22322 22133 22144 177 22222 22222 22333 22133 22044 178 22222 22222 00244 22222 22222 179 22222 22222 01244 22422 22422 180 22222 22222 11244 22222 22222 181 22222 22222 11244 22322 22322 182 01122 10122 22222 43333 33333 183 22222 22222 10243 22322 22322 184 01210 11210 22220 11210 01210 185 44444 43330 43230 43330 44444 186 00000 11211 11211 11111 01110 187 41114 11111 11211 11111 41114 188 40004 11111 11211 11111 41114 189 01110 11111 11112 11111 01110 190 40004 11111 11111 11111 41114 191 01210 00000 00000 00000 00000 192 41214 44444 44444 44444 44444 193 01210 10000 10000 10000 00000 194 41114 44444 44444 44444 44444 195 40004 44444 44444 44444 43334 196 11211 01211 02222 01211 11211 197 34443 33333 33233 33333 30004 198 33333 33233 32323 33233 33233 199 33333 33333 23333 33333 04443 200 11111 10001 20001 10001 11111 201 44444 23233 22222 23233 43234 202 41214 43233 42222 43233 42224 203 40004 43233 42222 43233 42224 204 02220 21212 22222 21212 02120 205 41114 23334 23234 23334 44444 206 40004 23333 23233 23333 43334 207 02220 21212 22122 21212 02220 208 41114 23333 23333 23333 43334 209 40004 23333 23333 23333 43334 210 44444 42224 42224 42124 44444 211 40004 42224 42224 42124 44444 212 02120 22222 22222 22222 02220 213 41114 22223 22223 22223 42224 214 40004 22223 22223 22223 42224 215 40004 22223 22323 22223 42224 216 00000 20000 20000 20000 00000 217 42224 14444 24444 14444 44444 218 42224 04444 04444 04444 44444 219 01110 20000 20000 20000 00000 220 41114 24444 24444 24444 44444 221 02220 34444 34444 34444 04440 222 22222 11111 21232 21112 22222 223 22222 23334 23234 23334 20002 224 11121 11113 11211 11111 11111 225 12221 23234 22224 23234 23332 226 23333 33333 33233 33333 30004 227 23333 33333 33333 33333 30004 228 22222 22221 22232 22222 22222 229 22222 22224 22224 22324 20002 230 21111 12111 11213 11111 11111 231 02222 22220 12220 22220 24442 232 04442 22332 13332 23332 22222 233 44444 14444 24444 14444 24444 234 04443 44444 44444 44444 24440 235 43234 33233 22224 33233 43244 236 03030 33233 02220 33233 03030 237 00200 11110 22222 43333 43234 238 40000 01110 01212 01110 00000 239 00100 01110 21210 01110 00200 240 03330 33233 02223 33233 03330 241 01110 11211 22222 43233 44444 242 40200 01110 11214 01110 00000 243 00100 01110 21110 01110 00000 244 03330 33233 02323 33233 03330 245 40200 01110 01110 01114 00000 246 00000 01010 02200 01010 00000 247 03030 33033 03230 33333 03030 248 43434 43433 22222 11011 00200 249 00000 01210 00200 01014 00000 250 00000 01010 02000 01010 00000 251 00100 01010 00202 01010 00000 252 03330 33033 03233 33333 03330 253 43334 43433 22222 11011 00200 254 40444 41414 44242 41414 44344 255 00200 01010 00100 01010 00100 256 03330 33033 03233 33333 03330 257 00000 01010 00104 01010 00000 258 00000 00000 02200 00000 00000 259 00200 00000 22222 44444 44444 260 00200 00000 20002 00000 00200 261 00100 00000 00202 00000 00000 262 03330 33033 03233 30333 03330 263 01110 11211 22222 44444 44444 264 00200 00200 42223 00000 00200 265 01210 10111 01111 11111 01110 266 11111 11211 01211 11111 11111 267 11111 11111 01213 11111 11111 268 11211 11111 22222 43333 43333 269 33333 03333 33233 33333 33333 270 01211 11111 22222 43333 33333 271 11411 11111 21210 11111 11111 272 11111 11111 01111 11111 11211 273 11111 11111 01113 11111 11111 274 33333 03333 33333 33333 33333 275 11411 11111 21110 11111 11111 276 10000 01010 02200 01010 00000 277 11111 11111 10213 11111 11111 278 11111 11211 22222 43434 44444 279 03333 33033 03233 33333 33333 280 01111 11011 22222 43233 33333 281 31211 11210 22224 11011 11211 282 11111 11111 20111 11111 11111

26

283 11111 11111 20113 11111 11111 284 11111 10211 01211 11111 11111 285 33233 34333 22222 11111 11111 286 11211 10111 22222 44444 44444 287 10000 00200 22100 00000 00000 288 00220 21212 22222 21212 02220 289 02030 33333 03230 33333 03030 290 02220 11212 22222 43232 42224 291 02420 01112 21212 23112 02220 292 00100 21110 01210 01110 00000 293 02330 33333 03233 33333 03330 294 02120 11212 22222 43232 42224 295 02020 21114 11211 21111 02120 296 00000 21110 01110 01110 00000 297 02330 33333 03333 33333 03330 298 02010 11114 11111 11111 01110 299 02220 22222 22220 22222 02220 300 02230 32332 03230 33333 03230 301 42200 22222 22222 22222 02220 302 02220 12222 22222 42222 42224 303 02020 22224 02220 22222 02020 304 01110 22000 01211 11111 01110 305 02330 32333 03233 33333 03330 306 02120 22222 22222 42222 42224 307 02020 22112 31213 21112 02420 308 00000 22110 01110 01110 00000 309 02330 32333 03333 33333 03330 310 02020 22224 22323 22222 02420 311 00000 21010 00200 01010 00000 312 00000 21010 00200 01010 00000 313 02030 33333 03230 33033 03030 314 02220 21012 22222 23332 42424 315 00220 21014 20222 21212 02220 316 00000 21010 00000 01010 00000 317 02030 33333 03030 33333 03030 318 02420 21112 21412 23112 02220 319 00100 21010 00200 01010 00000 320 02330 33033 33233 33333 03330 321 02110 01011 22222 33333 44444 322 02020 21212 42223 21412 02220 323 01110 21011 01111 11111 01110 324 02330 33033 33333 33333 03330 325 02020 21212 42322 21412 02220 326 00030 23323 03200 33333 03030 327 02220 12222 22222 44222 42224 328 02120 20222 22222 22222 02220 329 03330 32023 33233 32323 03330 330 02220 00222 22222 22222 42324 331 02120 22022 22222 22322 42424 332 02020 22422 22223 22222 02220 333 20222 21212 22222 21212 22222 334 22222 01213 22222 21212 22222 335 02222 21210 22222 33232 22222 336 20230 33333 23230 33333 23232 337 22222 01212 22222 33233 42222 338 23030 33332 03230 33334 03030 339 20221 21212 22222 21212 22222 340 21111 01111 11211 11111 11311 341 22222 01212 22222 23232 22322 342 33333 23330 33233 33333 33333 343 22222 01212 22222 23232 42322 344 23330 33332 33230 33334 03330 345 23333 43333 23333 33333 33333 346 22211 43211 22313 23311 11111 347 33333 23330 33333 33333 33333 348 20222 43232 32122 23232 22222 349 20222 22222 22222 22222 22222 350 22222 42213 22222 22222 22222 351 22222 23222 22222 02222 22222 352 20202 02333 23232 33333 23232 353 42202 22222 22222 22222 22222 354 02242 22222 22224 12322 23222 355 42222 42222 22222 02222 22222 356 20202 22233 02224 22224 02224 357 20221 22222 22222 22222 22222 358 22222 42222 22213 22222 22222 359 22221 23222 22222 02222 22222 360 33333 02323 33233 32323 33333 361 42201 22222 22222 22222 22222 362 02242 22222 22224 22222 13222 363 42221 42222 22222 02222 22222 364 20202 22233 02223 22224 02223 365 22222 42222 22322 22222 22222 366 22222 21222 22322 22222 22422 367 33333 02323 33333 32323 33333 368 10211 01311 43131 11321 11121 369 22222 20222 22123 22222 22222 370 22222 21212 20212 21213 22222 371 22222 21112 22222 23432 22222 372 23202 33033 23202 33333 23232 373 22222 21012 22222 23232 22422 374 20222 41012 22222 21212 22222 375 11111 11011 11211 11111 21111 376 22222 21211 20223 21212 22222 377 22122 21012 22222 23332 22222 378 33332 33033 33233 33333 33333 379 22122 21012 22222 23232 22422 380 21111 01111 34211 11411 11111 381 11111 21111 10113 11111 11111 382 12011 11111 11104 11114 11111 383 22222 20222 22212 22223 22222 384 20000 02023 00200 02020 00000 385 22222 20223 22212 22222 24222 386 20202 22243 02224 22224 02224 387 22222 20221 22223 22222 22222 388 22122 20222 22222 23222 22222 389 22222 20221 22223 22222 24222 390 22122 20222 22222 24222 22222 391 22222 20222 22123 22222 22222

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References

[1] D. Simovici, I. Stojmenović, R. Tošić, Functional Completeness and Weak Completeness in Set Logic, Proceedings of 23rd International Symposium on Multiple-Valued Logic, May 25–27, Boston, 1993, pp. 251–256

[2] I. G. Rosenberg, Completeness Properties of Multiple-Valued Logic Algebra, in D. C. Rhine (ed.): Computer Science and Multiple-Valued Logic: Theory and Application, North-Holland, 1977, pp. 144–186

[3] R. Tošić, G. Vojvodić, D. Mašulović, R. Doroslovački, J. Rosić, Two Examples of Relative Completnes Multi.Val. Logic, 1996, Vol.2 pp.67-78

[4] R.Doroslovački, J. Pantović,G. Vojvodić, Note on Intersections of Maximal Clones (in print).