Simple Laws about Nonprominent Properties of Binary Relations

06/13/2018 ∙ by Jochen Burghardt, et al. ∙ 0

We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like "not irreflexive, but asymmetric". We considered the non-trivial laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties.

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1 Introduction

In order to flesh out encyclopedic articles111 at https://en.wikipedia.org about less common properties (like e.g. anti-transitivity) of binary relations, we implemented a simple C program to iterate over all relations on a small finite set and to check each of them for given properties. We implemented checks for the properties given in Def. 1 below. Figure 1 shows the C source code to check a relation R for transitivity, where card is the universe size and elemT is the type encoding a universe element.

This way, we could, in a first stage, (attempt to) falsify intuitively found hypotheses about laws involving such properties, and search for illustrative counter-examples to known, or intuitively guessed, non-laws. For example, Fig. 2 shows the source code to search for right Euclidean non-transitive relations over a -element universe, where printRel prints its argument relation in a human-readable form. For a universe of elements, for loops are nested. In Sect. 6.1 we describe an improved way to iterate over all relations.

Relations on a set of up to elements could be dealt with in reasonable time on a 2.3 GHz CPU. Figure 3 gives an overview, where all times are wall clock times in seconds, and “trqt” indicates the task of validating that each transitive binary relation is also quasi-transitive. Note the considerable amount of compile time,222 We used gcc version 7.3.0 with the highest optimization level. presumably caused by excessive use of inlining, deeply nested loops, and abuse of array elements as loop variables.

In a second stage, we aimed at supporting the generation of law hypotheses, rather than their validation.

We used a 5-element universe set, and checked each binary relation for each of the properties.333 For this run, we hadn’t provided checks for left and right quasi-reflexivity (Def. 1.4+5), but only for the conjunction of both, viz. quasi-reflexivity (Def. 1.6). As additional properties, we provided a check for the empty relation () and for the universal relation (). The latter were encoded by bits of a 64-bit word. After that, we applied a poor-man’s Quine-McCluskey algorithm444 See Quine (1952) and McCluskey Jr. (1956) for the original algorithm. (denoted “QMc” in Fig. 3) to obtain a short description of property combinations that didn’t occur at all. For example, an output line “~Irrefl ASym” indicated that the program didn’t find any relation that was asymmetric but not irreflexive, i.e. that each asymmetric relation on a 5-element set is irreflexive. Section 3 shows the complete output on a 5-element universe.

bool isTrans(const bool R[card][card]) {
  elemT x,y,z;
  for (x=0; x<card; ++x)
    for (y=0; y<card; ++y)
      if (R[x][y])
        for (z=0; z<card; ++z)
          if (R[y][z] && ! R[x][z])
            return false;
  return true;
}
Figure 1: Source code for transitivity check
void check03(void) {
  bool R[card][card];
  for (R[0][0]=false; R[0][0]<=true; ++R[0][0])
    for (R[0][1]=false; R[0][1]<=true; ++R[0][1])
      for (R[1][0]=false; R[1][0]<=true; ++R[1][0])
        for (R[1][1]=false; R[1][1]<=true; ++R[1][1])
          if (isRgEucl(R) && ! isTrans(R))
            printRel(R);
}
Figure 2: Source code to search for right Euclidean non-transitive relations
Universe card 2 3 4 5 6
Relation count 16 512 6.55e04 3.35e07 6.87e10
10 140 6.17e03 9.07e05 4.60e08
Compile time 7.123 14.254 20.868 27.923 41.965
Run time trqt 0.007 0.007 0.009 0.132 50.386
Figure 3: Timing vs. universe cardinality

We took each printed law as a suggestion to be proven for all binary relations (on arbitrary sets). Many of the considered laws were trivial, in particular those involving co-reflexivity, as this property applies only to a relatively small number of relations (32 on a 5-element set).

A couple of laws appeared to be interesting, and we could prove them fairly easily by hand for the general case555 We needed to require a minimum cardinality of the universe set in some lemmas, e.g. Lem. 51 and 42. . For those laws involving less usual properties (like anti-transitivity, quasi-transitivity, Euclideanness) there is good chance that they haven’t been stated in the literature before. However, while they may contribute to the completeness of an encyclopedia, it is not clear whether they may serve any other purpose.

Disregarding the particular area of binary relations, the method of computing law suggestions by the Quine-McCluskey algorithm might be used as a source of fresh exercises whose solutions are unlikely to be found on web pages.

Some of the laws, e.g. Lem. 40, appeared surprising, but turned out during the proof to be vacuously true. The proof attempt to some laws gave rise to the assertion of other lemmas that weren’t directly obtained from the computed output: Lemma 4 was needed for the proof of Lem. 19, and Lem. 52 was needed for Lem. 42.

Our Quine-McCluskey approach restricts law suggestions to formulas of the form , where the quantification is over all binary relations, and is one of the considered properties or a negation thereof.

Figure 4: Encoding scheme for relations for a Burghardt (2002) approach
Figure 5: Tree grammar sketch for Burghardt (2002) approach

For an approach to compute more general forms of law suggestions, see Burghardt (2002); however, due to its run-time complexity this approach is feasible only for even smaller universe sets. In order to handle all relations on a 3-element set, a regular tree grammar of 512 nonterminals, one for each relation, plus 2 nonterminals, one for each truth value, would be needed. Using the encoding scheme from Fig. 4, the original grammar would consist of rules as sketched666 For sake of simplicity, only one unary and one binary operation on relations is considered, viz. symmetric closure and union . Only two properties of relations are considered, viz. reflexivity and symmetry . It should be obvious how to incorporate more operators and predicates on relations. By additionally providing a sort for sets, operations like , , restriction, etc. could be considered also. in Fig. 5. However, this grammar grows very large, and its -fold product would be needed if all laws in variables were to be computed.

The rest of this paper is organized as follows. In Sect. 2, we formally define each considered property, and introduce some other notions. In Sect. 3, we show the annotated output for a run of our algorithm on a 5-element set, also indicating which law suggestions gave rise to which lemmas. The latter are stated and proven in Sect. 4, which is the main part of this paper. In addition, we state the proofs of some laws that weren’t of the form admitted by our approach; some of them were, however, obtained using the assistance of the counter-example search in our C program. In Sect. 5, we discuss those computed law suggestions that lead to single examples, rather than to general laws. In Sect. 6, we comment on some program details.

This paper is a follow-up version of https://arxiv.org/abs/1806.05036v1. Compared to the previous version, we considered more properties (see Def. 1), including being the empty and being the universal relation, to avoid circumscriptions like “IrreflCoReflASym” in favor of “EmptyASym”; in the new setting, we found a total of law suggestions, and proved or disproved all of them. I am thankful to all people who have helped with their comments and corrections.

2 Definitions

Definition 1

(Binary relation properties) Let be a set. A (homogeneous) binary relation on is a subset of . The relation is called

  1. reflexive (“Refl”, “rf”)   if   ;

  2. irreflexive (“Irrefl”, “ir”)   if   ;

  3. co-reflexive (“CoRefl”, “cr”)   if   ;

  4. left quasi-reflexive (“lq”)   if   ;

  5. right quasi-reflexive (“rq”)   if   ;

  6. quasi-reflexive (“QuasiRefl”)   if   it is both left and right quasi-reflexive;

  7. symmetric (“Sym”, “sy”)   if   ;

  8. asymmetric (“ASym”, “as”)   if   ;

  9. anti-symmetric (“AntiSym”, “an”)   if   ;

  10. semi-connex (“SemiConnex”, “sc”) if   ;

  11. connex (“Connex”, “co”) if   ;

  12. transitive (“Trans”, “tr”)   if   ;

  13. anti-transitive (“AntiTrans”, “at”)   if   ;

  14. quasi-transitive (“QuasiTrans”, “qt”)   if   ;

  15. right Euclidean (“RgEucl”, “re”)   if   ;

  16. left Euclidean (“LfEucl”, “le”)   if   ;

  17. semi-order property 1 (“SemiOrd1”, “s1”)   if   ;

  18. semi-order property 2 (“SemiOrd2”, “s2”)   if   .

  19. right serial (“RgSerial”, “rs”)   if  

  20. left serial (“LfSerial”, “ls”)   if  

  21. dense (“Dense”, “de”)   if   .

  22. incomparability-transitive (“IncTrans”, “it”)   if   .

  23. left unique (“LfUnique”, “lu”)   if   .

  24. right unique (“RgUnique”, “ru”)   if   .

The capitalized abbreviations in parentheses are used by our algorithm; the two-letter codes are used in tables and pictures when space is scarce.

The “left” and “right” properties are dual to each other. All other properties are self-dual. For example, a relation is left unique iff its converse, , is right unique; a relation is dense iff its converse is dense.

We say that are incomparable w.r.t. , if holds. ∎

Definition 2

(Kinds of binary relations) A binary relation on a set is called

  1. an equivalence   if   it is reflexive, symmetric, and transitive;

  2. a partial equivalence   if   it is symmetric and transitive;

  3. a tolerance relation   if   it is reflexive and symmetric;

  4. idempotent   if   it is dense and transitive;

  5. trichotomous   if   it is irreflexive, asymmetric, and semi-connex;

  6. a non-strict partial order   if   it is reflexive, anti-symmetric, and transitive;

  7. a strict partial order   if   it is irreflexive, asymmetric, and transitive;

  8. a semi-order   if   it is asymmetric and satisfies semi-order properties 1 and 2;

  9. a preorder   if   it is reflexive and transitive;

  10. a weak ordering   if   it is irreflexive, asymmetric, transitive, and incomparability-transitive;

  11. a partial function   if   it is right unique;

  12. a total function   if   it is right unique and right serial;

  13. an injective function   if   it is left unique, right unique, and right serial;

  14. a surjective function   if   it is right unique and and left and right serial;

  15. a bijective function   if   it is left and right unique and left and right serial. ∎

Definition 3

(Operations on relations)

  1. For a relation on a set and a subset , we write for the restriction of to . Formally, is the relation on defined by for each .

  2. For an equivalence relation on a set , we write for the equivalence class of w.r.t. . Formally, .

  3. For a relation on a set and , we write for the set of elements is related to, and for the set of elements that are related to . Formally, and . ∎

3 Reported law suggestions

In this section, we show the complete output produced by our Quine-McCluskey algorithm run.

In the Fig. 6 to 13, we list the computed prime implicants for missing relation property combinations on a 5-element universe set. We took each prime implicant as a suggested law about all binary relations. These suggestions are grouped by the number of their literals (“level”).

In the leftmost column, we provide a consecutive law number for referencing. In the middle column, the law is given in textual representation, “P” denoting the negation of P, and juxtaposition used for conjunction. The property names correspond to those used by the C program; they should be understandable without further explanation, but can also be looked up via Fig. 42, if necessary. In the rightmost column, we annotated a reference to the lemma (in Sect. 4) where the law has been formally proven or to the example (in Sect. 5) where it is discussed.

For example, line 039, in level 2 (Fig. 6 left), reports that no relation was found to be asymmetric (property 1.8) and non-irreflexive (negation of property 1.2); we show the formal proof that every asymmetric relation is irreflexive in Lem. 13.1.777 A warning about possible confusion appears advisable here: In the setting of the Quine-McCluskey algorithm, a prime implicant is a conjunction of negated and/or unnegated variables. However, its corresponding law suggestion is its complement, and hence a disjunction, as should be clear from the example. Where possible, we used the term “literal” in favor of “conjunct” or “disjunct”.

Laws that could be derived from others by purely propositional reasoning and without referring to the property definitions in Def. 1 are considered redundant; they are marked with a star “”.888 We marked all redundancies we became aware of; we don’t claim that no undetected ones exist. For example, law 044 (“no relation is asymmetric and reflexive”) is marked since it follows immediately from 046 (“no relation is irreflexive and reflexive”) and 039.

No laws were reported for level 1 and level 9 and beyond.   A text version of these tables is available in the ancillary file reportedLaws.txt at arxiv.org.

In Fig. 14 to 17, we summarize the found laws. We omitted suggestions that couldn’t be manually verified as laws, and suggestions marked as redundant.

Figure 14 and 15 shows the left and right half of an implication table, respectively. Every field lists all law numbers that can possibly be used to derive the column property from the row property.

For example, law 129 appears in line “tr” (transitive) and column “as” (asymmetric) in Fig. 14 because that law (well-known, and proven in Lem. 12.2) allows one to infer a relation’s asymmetry from its transitivity, provided that it is also known to be irreflexive.

Fields belonging to the table’s diagonal are marked by “X”. Law numbers are colored by number of literals, deeply-colored and pale-colored numbers indicating few and many literals, respectively.

Similarly, the table consisting of Fig. 16 and 17 shows below and above its diagonal laws about required disjunctions and impossible conjunctions, respectively.

For example, law 223 appears below the diagonal in line “co” (connex) and column “em” (empty) of Fig. 16, since the law (proven in Lem. 29) requires every relation to be connex or empty, provided it is quasi-reflexive and incomparability-transitive.

Law 145 appears above the diagonal in line “le” (left Euclidean) and column “lu” (left unique), since the law (proven in Lem. 45) ensures that no relation can be left Euclidean and left unique, provided it isn’t anti-symmetric.

Figure 18 shows all proper implications (black) and incompatibilities (red) from level 2, except for the empty and the universal relation. Vertex labels use the abbreviations from Fig. 14, edge labels refer to law numbers in Fig. 6.

001 Empty Univ 74.12
002 Empty CoRefl 74.1
003 Univ CoRefl 75.10
004 Empty LfEucl 74.2
005 Univ LfEucl 75.1
006 CoRefl LfEucl 8.1
007 Empty RgEucl 74.2
008 Univ RgEucl 75.1
009 CoRefl RgEucl 8.2
010 Empty LfUnique 74.3
011 Univ LfUnique 75.12
012 CoRefl LfUnique 8.3
013 Empty RgUnique 74.3
014 Univ RgUnique 75.12
015 CoRefl RgUnique 8.4
016 Empty Sym 74.4
017 Univ Sym 75.2
018 CoRefl Sym 8.5
019 Empty AntiTrans 74.5
020 Univ AntiTrans 75.9
021 Empty ASym 74.6
022 Univ ASym 75.9
023 Empty Connex 74.14
024 Univ Connex 75.3
025 CoRefl Connex 8.8
026 LfUnique Connex 51
027 RgUnique Connex 51
028 AntiTrans Connex 51
029 ASym Connex 10
030 Empty Trans 74.7
031 Univ Trans 75.4
032 CoRefl Trans 8.7
033 Empty SemiOrd1 74.9
034 Univ SemiOrd1 75.5
035 Connex SemiOrd1 66
036 Empty Irrefl 74.6
037 Univ Irrefl 75.9
038 AntiTrans Irrefl 22
039 ASym Irrefl 13.1
040 Connex Irrefl 10
041 Empty Refl 74.13
042 Univ Refl 75.6
043 AntiTrans Refl 10
044 ASym Refl 10
045 Connex Refl 50
046 Irrefl Refl 10
047 Empty QuasiRefl 74.1
048 Univ QuasiRefl 75.6
049 CoRefl QuasiRefl 8.9
050 Connex QuasiRefl 50
051 Refl QuasiRefl 9
052 Empty AntiSym 74.6
053 Univ AntiSym 75.11
054 CoRefl AntiSym 8.6
055 ASym AntiSym 13.2
056 Empty SemiConnex 74.14
057 Univ SemiConnex 75.3
058 CoRefl SemiConnex 8.8
059 LfUnique SemiConnex 51
060 RgUnique SemiConnex 51
061 AntiTrans SemiConnex 51
062 Connex SemiConnex 50
063 Empty IncTrans 74.10
064 Univ IncTrans 75.7
065 Connex IncTrans 25
066 SemiConnex IncTrans 25
067 Empty SemiOrd2 74.9
068 Univ SemiOrd2 75.5
069 Connex SemiOrd2 66
070 SemiConnex SemiOrd2 25
071 IncTrans SemiOrd2 34
072 Empty QuasiTrans 74.8
073 Univ QuasiTrans 75.4
074 CoRefl QuasiTrans 8.7
075 LfEucl QuasiTrans 40
076 RgEucl QuasiTrans 40
077 Sym QuasiTrans 18
078 Trans QuasiTrans 18
079 Empty Dense 74.11
080 Univ Dense 75.6
081 CoRefl Dense 8.10
082 LfEucl Dense 48.5
083 RgEucl Dense 48.6
084 Connex Dense 48.8
085 Refl Dense 48.1
086 QuasiRefl Dense 48.3
087 Empty LfSerial 74.13
088 Univ LfSerial 75.6
089 Connex LfSerial 54
090 Refl LfSerial 54
091 Empty RgSerial 74.13
092 Univ RgSerial 75.6
093 Connex RgSerial 54
094 Refl RgSerial 54
Figure 6: Reported laws for level 2
095 CoRefl RgEucl LfUnique 7.3
096 CoRefl LfEucl RgUnique 7.4
097 LfEucl RgEucl Sym 39
098 LfEucl RgEucl Sym 15.2
099 LfEucl RgEucl Sym 15.2
100 LfUnique RgUnique Sym 15.4
101 LfUnique RgUnique Sym 15.4
102 Empty CoRefl AntiTrans 11
103 Empty LfEucl AntiTrans 11
104 Empty RgEucl AntiTrans 11
105 Empty CoRefl ASym 11
106 Empty LfEucl ASym 11
107 Empty RgEucl ASym 11
108 Empty Sym ASym 16
109 LfUnique AntiTrans ASym 23
110 RgUnique AntiTrans ASym 23
111 Univ LfEucl Connex 53.2
112 Univ RgEucl Connex 53.3
113 Univ Sym Connex 53.1
114 LfEucl RgEucl Trans 39
115 LfEucl LfUnique Trans 44
116 RgEucl RgUnique Trans 44
117 LfEucl Sym Trans 36
118 AntiTrans ASym Trans 12.5
119 AntiTrans ASym SemiOrd1 12.6
120 LfUnique Trans SemiOrd1 62.2
121 RgUnique Trans SemiOrd1 62.3
122 AntiTrans Trans SemiOrd1 62.5
123 ASym Trans SemiOrd1 62.4
124 Empty CoRefl Irrefl 11.1
125 Empty LfEucl Irrefl 11.4
126 Empty RgEucl Irrefl 11.5
127 LfUnique AntiTrans Irrefl 23
128 RgUnique AntiTrans Irrefl 23
129 ASym Trans Irrefl 12.2
130 ASym SemiOrd1 Irrefl 12.3
131 LfEucl RgEucl Refl 37
132 LfEucl RgEucl Refl 37
133 CoRefl LfUnique Refl 7.5
134 CoRefl RgUnique Refl 7.6
135 CoRefl SemiOrd1 Refl 5.4
136 Connex SemiOrd1 Refl 66
137 LfEucl RgEucl QuasiRefl 39
138 LfEucl RgEucl QuasiRefl 37
139 LfEucl RgEucl QuasiRefl 37
140 CoRefl LfUnique QuasiRefl 7.1
141 CoRefl RgUnique QuasiRefl 7.2
142 Empty AntiTrans QuasiRefl 11
143 Empty ASym QuasiRefl 11
144 Empty Irrefl QuasiRefl 11.2
145 LfEucl LfUnique AntiSym 45
146 LfEucl LfUnique AntiSym 45
147 RgEucl RgUnique AntiSym 45
148 RgEucl RgUnique AntiSym 45
149 CoRefl Sym AntiSym 7.7
150 AntiTrans ASym AntiSym 12.4
151 LfUnique Trans AntiSym 58
152 RgUnique Trans AntiSym 58
153 ASym Irrefl AntiSym 12.1
154 LfEucl Trans SemiConnex 41
155 RgEucl Trans SemiConnex 41
156 LfEucl SemiOrd1 SemiConnex 61.3
157 RgEucl SemiOrd1 SemiConnex 61.4
158 Trans SemiOrd1 SemiConnex 61.2
159 Connex Refl SemiConnex 50
160 Connex QuasiRefl SemiConnex 50
161 Empty CoRefl IncTrans 6.2
162 LfEucl Trans IncTrans 28
163 RgEucl Trans IncTrans 28
164 LfEucl SemiOrd1 IncTrans 28
165 RgEucl SemiOrd1 IncTrans 28
166 Trans SemiOrd1 IncTrans 61.1
167 Connex Refl IncTrans 27
168 SemiOrd1 QuasiRefl IncTrans 61.6
169 Empty CoRefl SemiOrd2 6.1
170 LfEucl Trans SemiOrd2 68
171 RgEucl Trans SemiOrd2 68
172 AntiTrans Trans SemiOrd2 24
173 LfEucl SemiOrd1 SemiOrd2 69
174 RgEucl SemiOrd1 SemiOrd2 69
175 Connex Refl SemiOrd2 66
176 SemiOrd1 QuasiRefl SemiOrd2 73
177 LfEucl IncTrans SemiOrd2 35.5
178 RgEucl IncTrans SemiOrd2 35.6
179 Sym IncTrans SemiOrd2 35.3
180 QuasiRefl IncTrans SemiOrd2 35.1
181 ASym Trans QuasiTrans 19
182 Trans AntiSym QuasiTrans 19
183 LfEucl LfUnique Dense 47
184 RgEucl RgUnique Dense 47
185 Empty AntiTrans Dense 49
186 Empty ASym Dense 76
187 Sym SemiOrd1 Dense 48.7
188 Sym SemiConnex Dense 48.9
Figure 7: Reported laws for level 3 (a)
189 LfEucl RgEucl LfSerial 38
190 LfUnique RgUnique LfSerial 59
191 AntiTrans Trans LfSerial 24
192 ASym Trans LfSerial 14
193 CoRefl SemiOrd1 LfSerial 5.4
194 RgUnique SemiOrd1 LfSerial 65.2
195 CoRefl Refl LfSerial 55
196 RgEucl Refl LfSerial 55
197 Refl QuasiRefl LfSerial 55
198 LfEucl SemiConnex LfSerial 56
199 Sym SemiConnex LfSerial 56
200 RgUnique IncTrans LfSerial 33
201 LfEucl RgEucl RgSerial 38
202 LfUnique RgUnique RgSerial 59
203 AntiTrans Trans RgSerial 24
204 ASym Trans RgSerial 14
205 CoRefl SemiOrd1 RgSerial 5.4
206 LfUnique SemiOrd1 RgSerial 65.1
207 CoRefl Refl RgSerial 55
208 LfEucl Refl RgSerial 55
209 Refl QuasiRefl RgSerial 55
210 RgEucl SemiConnex RgSerial 56
211 Sym SemiConnex RgSerial 56
212 LfUnique IncTrans RgSerial 33
213 LfUnique LfSerial RgSerial 59
214 RgUnique LfSerial RgSerial 59
215 Sym LfSerial RgSerial 15.3
216 Sym LfSerial RgSerial 15.3
Figure 8: Reported laws for level 3 (b)
217 LfEucl LfUnique AntiTrans SemiOrd1 47
218 RgEucl RgUnique AntiTrans SemiOrd1 47
219 LfEucl Sym SemiOrd1 QuasiRefl 63.1
220 Empty LfUnique RgUnique IncTrans 32.4
221 LfEucl LfUnique AntiTrans IncTrans 47.5
222 RgEucl RgUnique AntiTrans IncTrans 47
223 Empty Connex QuasiRefl IncTrans 29
224 LfEucl LfUnique AntiTrans SemiOrd2 47.4
225 RgEucl RgUnique AntiTrans SemiOrd2 47
226 LfUnique RgUnique ASym SemiOrd2 72
227 Trans SemiOrd1 AntiSym SemiOrd2 77
228 LfUnique ASym IncTrans QuasiTrans 32.2
229 RgUnique ASym IncTrans QuasiTrans 32.2
230 LfUnique ASym SemiOrd2 QuasiTrans 71.3
231 RgUnique ASym SemiOrd2 QuasiTrans 71
232 Sym AntiTrans IncTrans Dense 30
233 Trans SemiOrd1 SemiOrd2 Dense 61.5
234 Trans IncTrans SemiOrd2 Dense 35.4
235 LfUnique Sym AntiTrans LfSerial 60
236 LfEucl LfUnique Trans LfSerial 47.2
237 Empty LfEucl IncTrans LfSerial 28
238 Empty Sym IncTrans LfSerial 31
239 LfUnique ASym IncTrans LfSerial 32.3
240 LfUnique ASym IncTrans LfSerial 32.5
241 LfUnique SemiOrd1 IncTrans LfSerial 64.1
242 LfUnique ASym SemiOrd2 LfSerial 71.4
243 RgUnique ASym SemiOrd2 LfSerial 78
244 RgUnique Sym QuasiTrans LfSerial 21.1
245 RgEucl RgUnique Trans RgSerial 47
246 Empty RgEucl IncTrans RgSerial 28
247 RgUnique ASym IncTrans RgSerial 32
248 RgUnique ASym IncTrans RgSerial 32
249 RgUnique SemiOrd1 IncTrans RgSerial 64.2
250 RgUnique ASym SemiOrd2 RgSerial 71
251 ASym SemiOrd2 LfSerial RgSerial 78
Figure 9: Reported laws for level 4
252 LfUnique RgUnique AntiTrans AntiSym QuasiTrans 79
253 LfEucl Trans SemiOrd1 IncTrans LfSerial 26
254 LfUnique RgUnique Trans SemiOrd2 LfSerial 70
255 LfUnique AntiTrans AntiSym QuasiTrans LfSerial 79
256 RgEucl Trans SemiOrd1 IncTrans RgSerial 26
257 RgUnique AntiTrans AntiSym QuasiTrans RgSerial 79
258 Trans SemiOrd1 SemiOrd2 LfSerial RgSerial 80
Figure 10: Reported laws for level 5
259 Trans SemiOrd1 AntiSym IncTrans Dense LfSerial 81
260 Trans SemiOrd1 AntiSym IncTrans Dense RgSerial 81
261 Trans SemiOrd1 AntiSym IncTrans LfSerial RgSerial 82
262 ASym Trans SemiOrd1 SemiOrd2 LfSerial RgSerial 83
263 Trans AntiSym IncTrans SemiOrd2 LfSerial RgSerial 84
264 Trans AntiSym IncTrans SemiOrd2 LfSerial RgSerial 85
265 AntiTrans IncTrans SemiOrd2 QuasiTrans LfSerial RgSerial 35.7
266 Trans SemiOrd1 AntiSym Dense LfSerial RgSerial 86
267 Trans SemiOrd1 AntiSym Dense LfSerial RgSerial 87
268 Trans AntiSym SemiConnex Dense LfSerial RgSerial 88
269 SemiOrd1 SemiConnex QuasiTrans Dense LfSerial RgSerial 89
270 Irrefl SemiConnex QuasiTrans Dense LfSerial RgSerial 90
Figure 11: Reported laws for level 6
271 Trans AntiSym SemiConnex IncTrans Dense
LfSerial RgSerial 91
272 SemiOrd1 IncTrans SemiOrd2 QuasiTrans Dense
LfSerial RgSerial 92
273 SemiOrd1 IncTrans SemiOrd2 QuasiTrans Dense
LfSerial RgSerial 92
Figure 12: Reported laws for level 7
274 Trans SemiOrd1 IncTrans SemiOrd2 QuasiTrans Dense
LfSerial RgSerial 93
Figure 13: Reported laws for level 8
em un cr le re lu ru sy at as co tr
Empty em
X
002
004
007
010
013
016
019
021
030
Univ un
X
005
008
017
024
031
CoRefl cr
124,161
169
X
006
009
012
015
018
032
LfEucl le
125,237
096
X
098
131
138
201
146
097
114,115
154,162
170
RgEucl re
126,246
095
099
132
139
189
X
148
097
114,116
155,163
171
LfUnique lu
220
095
140
183
217
224
236
X
100
202
127,217
224
226,230
242
115,120
254
RgUnique ru
220
096
141
184
218
225
245
101
190
X
244
128,218
225
226,231
250
116,121
254
Sym sy
108,238
113
149
099
117
219
098
101
100
X
232
AntiTrans at
185
X
ASym as
108
X
Connex co
113
X
Trans tr
117
236
245
129
X
SemiOrd1 s1
217
219
218
217,218
136
120,121
Irrefl ir
124,125
126,144
127,128
129,153
Refl rf
132
131
136
167
175
QuasiRefl qr
144
223
140
141
139
219
138
160
223
AntiSym an
149
146
148
153
182
SemiConnex sc
160
154,155
IncTrans it
161,220
223,237
238,246
232
167
223
162,163
SemiOrd2 s2
169
224
225
224,225
226,230
231,242
250
175
170,171
254
QuasiTrans qt