RELATIONS PROPERTIES COMPATIBILITY Misha Mikhaylov

E-mail address: [email protected]

ABSTRACT. Thoughts expressed in previous paper [3] were developed. There was shown formally that collection of sets’ properties may not appear chaotically and independently on each other. Presence or absence of one leads to rise or drop of another.

Contents 1. Something about “reflexivity” of tolerances ...... 1 2. Dependence of reversibility on the other relation’s properties ...... 3 3. Equivalences’ negative powering features ...... 6 References ...... 7

1. Something about “reflexivity” of tolerances

One may think that reflexivity and symmetry are necessary and sufficient conditions to appear transitivity. However, often tolerances are determined as reflexive and symmetric rela- tions which transitivity is not required (e.g., according to [1]). Meanwhile, P. M. Cohn [2] mentioned1 that similar relations are intransitive and even they are not reflexive. To under- stand who of them was right there may be applied general method that was developed in my earlier article [3].

First of all, talking about relations, it was meant that they are established at least on the binary Cartesian product – more accurate – Cartesian square. But firstly, it needs to determine lower Cartesian powers2. So, the first Cartesian power is just a non-empty disordered set as it is commonly determined

= = { : } . (1.1) 1 This is disjoint union of disordered퐴 “pairs”퐴 푥 푥 ∈ 퐴 ≠ ∅

= { , } = { }. (1.2)

They are usual one-element sets3 but〈푎 〉written푎 푎 in terms푎 of Cartesian products. As far as all of those “pairs” are exclusively diagonal, there wouldn’t be another Cartesian product but a kind of power. Nullary Cartesian power just coincides with Boolean of empty set

1 But he did not advance an argument to prove it. Perhaps it has based on empirical observation. Who knows? 2 Obviously, another kind of Cartesian product simply doesn’t exist – it will be proved a little bellow. 3 Initial letters of Latin alphabet (e.g. , , ) denote here fixed sets’ elements, the final ones (e.g. , , ) -- any element of sets running all of the meanings. For example, they may be connected by formula { } = { , , … , }. 푎 푏 푐 1 푥 푦 푧 푥 푎 푏 푐 = 2 = { } = { }. (1.3) 0 ∅ That’s why it is not empty. 퐴 � � ∅

Such definitions allow determining relations of lower arity. They differ from binary ones just by randomness. Since the first Cartesian power may be represented by set union, its subset – unary relation – may be represented by set intersection

( = = = ) ( ). (1.4) 1 Here, idempotent set union퐴 plays퐴 ⋃ 퐴the role퐴 ⋃ of퐼 ⊇Cartesian퐴 ⋂ 퐼 “square”,퐼 ↤ 퐴 ⊇and퐼 – relation’s one – its in- tersection with self-subset . Due to commutativity of these operations, such relations are ex- clusively symmetrical. Any “pairs” compiling such relations are exclusively diagonal 퐼 { , } = { } = { : } = .

“Diagonal” of such set is determined푖 푖 by 푖the expression푖 푖 ∈ 퐴 ⋂ 퐼 퐼

id = { : { , } { , } }. (1.5)

Such relations seem to be established퐼 푖 reflexive,푖 푥 ∈ 퐼 implying⋀ 푥 푖 ∈ their퐼 following transitivity. But the left part of transitivity condition (inclusion (3.4.1)4), due to “pair’s” randomness, yet, may be simplified by elements

푥 { : { , } { , } } { : }. (1.6)

But this inclusion has the opposite푥 푥 푖 direction∈ 퐼 ⋀ 푖 푥 that∈ 퐼occurs⊇ 푖 while푖 ∈ 퐼 one deals with transitive rela- tion. To make clear the reason of the phenomenon, there may be noticed that formula (1.5) is not unique definition of this one. There exists another one that is not quite equivalent to the previous definition. Because a set doesn’t consists any elements but just diagonal’s disor- dered “pairs”, it totally coincides with diagonal and there may be written inclusion 퐴 = { } = id id = id = { } = . (1.7) ′ Such definition let conclude퐴 that 푥relation퐴 is⊇ contained퐴 퐼 in diagonal푖 퐼 does be pseudo-reflexive. So, no one could state that such relations are equivalences – due to identity’s definition ambi- guity of such sets. Apparently, such pseudo-equivalences may be called tolerances. Inclusion direction in formula (1.6) doesn’t correspond to conjunction’s one – instead of this followings occur

{ : { , } { , }} = { : { , } { , }} { , } . (1.8) { : { , } { , }} = { : { , } { , }} { , } 푥 푥 푗 ⋀ 푘 푥 푥 푗 푥 ⋀ 푥 푘 ⊈ 푗 푘 � These formulae express푥 intransitivity푥 푗 ⋀ 푘 푥 of tolerances.푥 푗 푥 Among⋀ 푥 푘 sub⊉sets푗 푘 of set obligatorily exists those that equal to null. Choosing it as index set, there may be written for such tolerance 퐴

So, disordered sets are perfectly defined by one of their elements. All equipotential disordered sets are isomor- phic each other. 4 Below in all of strange numerical notations the 1st number will mean source in reference list. E.g., notation (3.4.1) means formula (4.1) from the source [3]. 2

( = ) ( = ) ( = ) ( ) ( ) ( ) . (1.9) 퐵 ⋂ 퐼 ∅ ⋀ 퐶 ⋂ 퐼 ∅ ↮ 퐵 ⋂ 퐶 ∅ � In addition, arity of tolerance퐵 ⋂ 퐼is≠ not∅ so⋀ clear.퐶 ⋂ 퐼 On≠ ∅ the↮ one퐵 hand,⋂ 퐶 ≠ relation∅ arity is determined by multiplicands’ quantity of Cartesian power – in this sense it is the unary relation. But, on the other hand, this is a correspondence between two sets. They are different, in spite of obligato- ry inclusion between them, so, this is binary.

The only relation established on empty set is equivalence anyway – its only improper subset coincides with its diagonal5.

2. Dependence of reversibility on the other relation’s properties

However, stated reasons are rather similar to some empirical statement of transitivity deficiency that, as a matter of fact, doesn’t expose an origin of phenomenon than explains this. To do it there may be noticed following. As it has been shown earlier (at [3]) in relation definition in dependence on its kind of symmetry relation intersection with its reversal one plays the major role, than the relation of its own. There gives rise to doubt the fact of transi- tivity, symmetry and reflexivity keeping while relations go through some activity6. E.g., the definition (3.4.9) contains such intersection but it is the part of diagonal and, anyway, it is pseudo-reflexive still remaining symmetrical. It’s too hard to understand without knowledge of what anti-symmetric relation would be during reversion. But starting from asymmetrical relations there may be shown that their intersection with any other relation is asymmetrical too. According to formula (3.4.21), assuming intersection’s commutativity and associativity, there may be written

( ) ( ) = . (2.1) −1 −1 −1 But it confirms just relation’s휇 ⋂ anti휈 ⋂-reflexivity.휇 ⋂ 휈 Due휇 ⋂ to휇 the⋂ relation휈 ⋂ 휈 = ¬ is ambiguous; its intersection with another relation is empty. Therefore, there may be −written1 휇 휇 ( = ¬ ) [( ) = ]. (2.2) −1 −1 And already it proves the statement휇 about휇 ↦ asymmetry휇 ⋂ 휈 of such∅ intersection. But union with any relation may be not asymmetric – even when relation is empty, there may be written −1 ( ) ( ) = ( 휇 ) ( ). −1 −1 −1 휇 ⋃ 휈 ⋂ 휇 ⋃ 휈 휇 ⋂ 휈 ⋃ 휇 ⋂ 휈

5 Such certainty is something akin to that natural numbers’ series (which are linearly ordered) generation may be obtained by sequential ordering of empty set. I mean – perhaps, it may be a cause of their linear order. As it will be shown below, there is no such certainty of identity definition in the most other cases – in previous article [3] double-sided diagonal was called “almost” identity non-randomly. 6 Terms operations and actions aren’t used here for a while because they have concrete algebraic meaning. But sets differ from algebraic structures just by probability of operation lack. As it widely known (e.g. from [2]) bi- nary operation is ternary relation that established at a set transforming it into algebraic structure (group, module etc.). If it is transitive there might talk about algebraic closure. But this is the matter of another research. 3

Anyway, the 2nd operand in the right part is empty, but there may not say the same about the 1st one. The 1st one is empty when it is satisfied to implication

( = ) [( ) ( ) = ]. (2.3) −1 −1 Notably, sets and are휇 ⋂ disjoint휈 independently∅ ↦ 휇 ⋃ 휈 on⋂ set휇 ⋃ 휈 is empty∅ or not. In other cases it is not asymmetric due− to1 it is not anti-reflexive. −1 휇 휈 휈 There may be done similar discourses on the union and intersection of anti-symmetric relations. But firstly, it needs to be done preliminary notes. It’s easy to show that union of di- agonal or its fragment with anti-symmetric relation is reflexive or pseudo-reflexive with keep- ing of anti-symmetry of a result. Because union and intersection are distributive there may be written formula

( id) ( id) = ( ) id. (2.4) −1 −1 It coincides with one of the훼 ⋃definitions⋂ 훼 ⋃ (3.4.9) confir훼 ⋂ming훼 by⋃ this anti-symmetry of the result. Coincidence with equality (3.4.13) confirms its reflexivity. Apparently, it occurs for union with non-empty diagonal’s fragment too, that may be written as

훼� ( ) id = id −1 id −1 . (2.5) 훼� ⋂ 훼� ⊂ 훼 ⋂ 훼id ⋃ 훼� ⋂ id≠ ∅ � 훼� ⊈ This inclusion coincides with one of the훼 �definitions⊉ (3.4.9) confirming pseudo-reflexivity of such union. Using pseudo-reflexivity of initial and final relations, there may be written equali- ties

( ) ( ) = ( ) ( ) id id = id. (2.6) −1 −1 −1 As it was in asymmetry훼 ⋂ 휉 ⋂case,훼 ⋂ anti휉 -symmetry훼 ⋂ 훼may be⋂ not휉 ⋂ keeping휉 ⊆ at ⋂a union. Reason by analo- gy leads to expression

( ) ( ) = ( ) ( ) = ( ) ( ). (2.7) −1 −1 −1 −1 −1 If relation 훼 ⋃is 휉asymmetric,⋂ 훼 ⋃ 휉 then it훼 ⋂is 훼the case⋃ 휉 described⋂ 휉 by훼 ⋂equality휉 ⋃ (2.3),휉 ⋂ 훼 but if it is anti- symmetric, then it depends on reversion features of pseudo-reflexive relation. 휉 Talking about reversibility in dependence on kind of reflexivity display there may be noticed that, indeed, intersection of pseudo-reflexive anti-symmetric relation with its reversal one is irreversible. In fact, reversing intersection id, it would lead to that the result were reflexive −1 훼� ⋂ 훼� ⊂ ( ) = id. −1 −1 −1 However, it would contradict to훼 �statement⋂ 훼� of intersection훼 ⋂ 훼 ⊃ commutativity. Reversing here is provided by pairs’ elements permutation and in logical terms7 it looks like

7 It means the usage of conjunction and disjunction instead of signs cap ( ) and cup ( ). 4 ∩ ∪ , , , , . (2.8) −1 ( ) −1 〈푥 푗〉 ∈ 훼� ⋀〈푥 푘〉 ∈ 훼� ⇏ 〈푗 푥〉 ∈ 훼� ⋀〈푘 푥〉 ∈ 훼� −1 −1 −1 � Because pseudo-reflexive anti-symmetric훼� ⋂ 훼� relation⊈ 훼� ⋂ 훼is� represented by direct sum with diago- nal’s fragment, then this object is irreversible.

Returning to the matter of relation transitivity in dependence on its kind of symmetry appearance, there may be written for asymmetric relation

= . −1= 휇 ⋂ 휇 ∅ � Also, these expressions may be written as휇 something⋂ 휇 휇 like that

, , ( ) . , , −1 ( = ) 〈푥 푗〉 ∈ 휇 ⋀〈푥 푘〉 ∉ 휇 ⇒ 푗 ≠ 푘 � Elements’ permutation in 2nd 〈operand푥 푗〉 ∈ 휇 of⋀ 〈1푥st 푘of〉 ∈these휇 ⇒ expressions푗 푘 leads to transfer from com- mutative intersection to non-commutative multiplication

, , , id . = −¬1 = 〈푥 푗〉 ∈ 휇 ⋀〈푗 푥〉 ∉ 휇 ⇒ 〈푥 푥〉 ∉ −1 � Changing of intersection by multiplication휇 ∘ 휇 in the휇 ∘ idem휇 -potency∅ condition leads to formulae , , , . (2.9) 〈푥 푗〉 ∈ 휇 ⋀〈푘 푥〉 ∈ 휇 ⇏ 〈푗 푘〉 ∈ 휇 � It points to intransitivity of asymmetric relation휇 ∘ 휇 –⊈ just휇 due to elements’ permutation in pairs of asymmetric relations is impossible, the left part of this is non-commutative.

There may be written something like that for anti-symmetric relations

id . =−1 ∅ ⊆ 훼 ⋂ 훼 ⊆ nd �st Elements’ permutation in 2 pair in the left훼 ⋂ part훼 of훼 the 1 expression (2.8) may be written as

, , .

If such relation is reversible, there may〈푥 푗be〉 ∈ written훼 ⋀〈푘 sequence푥〉 ∈ 훼

, , , , , . −1 −1 −1 And it is already〈푥 푗 〉plain∈ 훼 ⋀to〈 푘see푥 〉that∈ 훼 only⇒ 〈 푘anti푥〉-reflexive∈ 훼 ⋀ 〈anti푥 푗〉-symmetric∈ 훼 ⇒ 〈 푘relation푗〉 ∈ 훼 or its direct sum with diagonal is transitive. Summarizing, there may be obtained criterion for anti-symmetric relation to be transitive

( = ) ( ) . (2.10) ( −1 = id) ( 2 = ) 훼 ⋂ 훼 ∅ ↦ 훼 ⊂ 훼 −1 2 � As a simple example the훼 case⋂ 훼 of proper↦ inclusion훼 훼 may be observed. Transitivity condi- tion here is written as 5

( ) ( ) ( ). (2.11)

Composition of relations is described퐴 ⊂ 퐵 here,⋀ 퐵 ⊂but퐶 it ⇒is not퐴 ⊂their퐶 intersection, which is empty due to their anti-reflexivity.

3. Equivalences’ negative powering features

Natural powering was defined, at least, for relations. But it’s too hard to say definitely about negative powering and its coincidence with their reversing even for relations, in spite of symbol similarity for these activities. General property (3.3.16) is not fulfilled in most cases – due to inequality (3.3.12) occurs

id . (3.1) 푛+1 −1 푛 But there is exclusion from this 훽rule for∘ 훽 multiplicative≠ 퐴 ∘ 훽 idempotent. Only in this case there may talk about such powering and its coincidence with reversing. Nevertheless, regardless of non- commutativity anti-symmetric relations’ multiplication, composition is equivalence – it may be powered in natural degree and general property (3.4.2) is inherent−1 to it. Anyway for 훼 ∘ 훼 both it and any of its own natural power there may be written sequence of transitivity condi- tions

id . (3.2) 푛+1 푛 푛−1 2 Therefore, there may say that⊆ equivalences휀 ⊆ 휀 ⊆ may휀 have⊆ ⋯ proper⊆ 휀 ⊆ subsets퐴 – they are equivalences too. But it could be powered only as integration – “associativity” here is inappropriate, be- cause “degree” is undetermined −푛 훼 [( ) ] [ = ( ) ]. (3.3) −1 푛 −푛 푛 −1 푛 푛 −푛 It doesn’t coincide with훼 reversal≠ 훼relation↦ powering.휀 훼 ∘ Even훼 when≠ 훼 both∘ 훼 multiplicands are equiva- lences it is not always possible due to absence of commutativity

= id = . (3.4) 푛+1 −1 = 푛 −1 = id푛 = 푛 훼−1 ∘ 훼푛+1 훼−1∘ 훼 ∘ 훼 푛 ⊃ 훼 ∘ 푛 훼푛� All the more, it is not possible훼 ∘ 훼 when one훼 of∘ 훼compositions∘ 훼 ∘ 훼is tolerant훼 = = id = . (3.5) 푛+1 −1 = 푛 −1 id푛 = 푛 훼−1 ∘ 훼푛+1 훼−1∘ 훼 ∘ 훼 푛 훼 ∘ 푛 훼푛� Anyway there may훼 be∘ 훼 written inclusions’훼 ∘ 훼 ∘ 훼 sequence⊂ ∘ 훼 훼 id . (3.6) −1 −1 2 It determines partial order relation.휏 ⋂ 휏 But⊆ among⊆ 휏 varieties⋃ 휏 ⊆ of퐴 the last of these inclusions equali- ty is possible for some relation

휏̂ = . (3.7) −1 2 휏 ⋃ 휏 퐴 6

In this case set is appeared to be fully indicated and ordered – this order is linear. Thus, any relation is subset of this relation

. (3.8)

It휌 ⊆may휏̂ be called set’s transitive closure. Asymmetric relation is not observed as a candidate to play the role. There may be shown that this role can play only reflexive anti-symmetric and transitive relations – they are called order relations.

Talking about Cartesian product of its own as binary relation – there may say that it is surely equivalence. Anyway there may be written for this

id =2 ( ) . (3.9) 2퐴 ⊇=2 −1 퐴2 2퐴 2� The same as in formula (3.2.9), defining퐴 ∘ 퐴 diagonal,퐴 here transitivity condition may be inter- preted like idempotent intersection of sets , but as opposed to diagonal this activity is not distributive with “intersection”8 and that’s why2 it is not associative 퐴 ( × ) ( × ) × × . (3.10)

Square of Cartesian square퐴 as a퐴 binary∘ 퐴 relation퐴 ≠ 퐴 surely퐴 ⋂ differs퐴 퐴 from 4th Cartesian power (obvi- ously, the last one is quaternary relation) by using composition forming activity “ ” but not by Cartesian square “×” between these multiplicands. Although, by analogy with intersection, ∘ idem-potency of such composition lets write expression that is similar to formula (3.3.14)

= = . (3.11) 2 2 Therefore, as a matter of fact, diagonal휌 ∘ 퐴 is not퐴 unique∘ 휌 휌identity that may be established on a set. It explains the reason of usage adverb “almost9”. Any equivalence differs from diagonal usu- ally is called non-trivial, and one differs from Cartesian square – is proper.

References 1. Sheela Ramanna Discovering Image Similarities. Tolerance Near Set Approach. in Rough Fuzzy Image Analysis ed. by Sankar K. Pal and James F. Peters 12-2 2. P.M. Cohn Universal Algebra. D.Reidel Publishing Company. 1981. p.66 3. Misha Mikhaylov. An Origin of Transitivity and Other Useful Relation's Properties. http://vixra.org/pdf/1409.0056v2.pdf

8 As any intersection is not surely distributive with relations’ multiplication. But this statement is not supplied by proof here. 9 If there could say so, anyway, in this situation. 7