Algebras of finitary

V P Tsvetov1

1Samara National Research University, Moskovskoe Shosse 34А, Samara, Russia, 443086

e-mail: [email protected]

Abstract. of finitary relations naturally generalize the of binary relations with the left composition. In this paper, we consider some properties of such algebras. It is well known that we can study the as finitary relations. In this way the results can be applied to graph and theory, automatons and artificial intelligence.

1. Introduction It is obvious that graphs and binary relations are closely related. We often use the facts of the binary relations theory in graph theory to solve some algorithmic problems. In the same way, we can consider hypergraphs as finitary relations. This could be a good idea for IT and AI, especially for pattern recognition and machine learning [1-13]. By now it has become common to use universal algebras [14] in various applications [15]. Algebraic methods can also be efficiently applied in graph theory. For example, the shortest path problem can be solved by transitive closure algorithm for binary [16]. In this way, and following by [17], we are going to study hypergraphs as elements of algebraic structures. At first, we define a (n-uniform) hypergraph as a finitary relation on finite U , in other words, as a of U n . In case of n = 2 this leads to graph as a . Boolean algebras n 2UU× ,(∪∩ , ,, ∅ ,UU × ) and 2Un ,(∪∩ , ,, ∅ ,U ) are well known to us. It is less trivial to define the inverse and the left composition for finitary relations. We have to start from inverse operation, left and right compositions for binary relations: −1 R= {()() uu21,|, uu 12∈ R}, (1)

R1 R 2={()()() uu 12,| ∃ u 0 uu 10 , ∈∧ R 1 uu 02 , ∈ R 2}, (2) (3) R1◦R2 = R2◦R1 = {(u1,u2) |Ǝ u0(u0,u2)ЄR1^(u1,u0) Є R2} Note that are isomorphic monoids, where I is identity relation on U . By the way, we can define operations −1 RRRR11 2= 1 2, (4) (5) −1 R12 R 2= RR 1 2, (6)

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(7) −−11 RRRR13 2= 1 2, (8) (9) This makes it possible to set the following pairs of isomorphic magmas. UU××UU 2,,()11II 2,,() are isomorphic magmas with left identity elements.

UU××UU 2,,()22II 2,,()e ar isomorphic magmas with right identity elements.

UU××UU 2,()33 2,() are isomorphic magmas without identity elements.

− ∞ It is easy to see that in the symmetric case RR= 1 all of monogenic monoids RIn ,,() , {}n=0

∞ ∞ ∞ RIn ,,() , RIn ,,() , RIn ,,() ( i ∈1..3) are equal. {}n=0 {}n=0 i {}n=0 i

∞ ∞ The monogenic monoid RIn ,,() and distributive RIn ,() ,∪∅ ,, {}n=0 {}n=0 are useful to treat all-pairs shortest path problem [16]. We are going to define and study hypergraph operations similar to (1)-(9).

2. Algebras of finitary relations n Let us consider the underlying set of finitary relations 2U , and define the following unary and binary operations for ij≠ (ij) (ji) R= R = {( uuuu11,..,jin ,.., ,..,)( | uuuu ,.., i ,.., jn ,.., )∈ R}, (10)

RR1ij 2={( uuuu 1,..,i ,.., jn ,..,)( |∃ uuuuu01 ,.., 0 ,..,jn ,.., )∈∧ R1( uuuu 1,..,i ,.., 0 ,.., n)∈ R2}. (11) Obviously, the operation (10) is an involution. (ij) ()RR(ij) = . (12) Moreover,

RRR1ij 22= ji R 1. (13) It is easy to prove that operation (11) is associative. Actually,

(uuuu1,..,i ,.., j ,.., n )()(∈ R1ij R 23 ij R⇔∃ uuuuu 010,.., ,..,j ,.., n )∈ R11 ∧( uuuu,..,i ,.., 0 ,.., n )∈ R 23ij R ⇔

⇔∃uuuuu010( ,.., ,..,jn ,.., )∈ R1∧( ∃ uuuuu 0100′′( ,.., ,.., ,.., n)(∈ R 2 ∧ uuuu 1,..,i ,.., 0′ ,.., n)∈ R 3) ⇔ ′ ′′ ⇔∃uuuuuu0010( ∃ ( ,.., ,..,jn ,.., )∈ Ruuuu1 ∧( 100,.., ,.., ,.., n)∈ R 2) ∧( uuuu 1,..,i ,.., 0 ,.., n)∈ R 3 ⇔

⇔∃uuuuu01′′( ,.., 0 ,..,j ,.., n )∈ RR1ij 2 ∧( uuuu 1,..,i ,.., 0′ ,.., n )∈ R3 ⇔

⇔∈(uuuu1,..,i ,.., j ,.., n )() RR12ij ij R 3. Then we set U n Iij ={( uuuuk1,..,i ,..,j ,.., n ) |∈ 1.. nuUuu∧∈∧k j = i } ∈2 . (14) It is easy to see

(uuuu1,..,i ,.., j ,.., n)∈ I ij ij R ⇔∃ uuuuu01( ,.., 0 ,..,j ,.., n)∈ I ij ∧( uuuu1,..,i ,.., 0 ,.., n )∈ R ⇔

⇔∃uuuuu01( ,..,i ,.., 0 ,.., n)∈ Ruu∧ j =0 ⇔( uuuu 1,..,i ,.., jn ,.., )∈ R, and similarly

(uuuu1,..,i ,.., j ,.., n)∈ RI ij ij ⇔∃ uuuuu01( ,.., 0 ,..,j ,.., n )∈ Ruuuu∧( 1,..,i ,.., 0 ,.., n)∈ I ij ⇔

⇔∃uuuuu01( ,.., 0 ,..,jn ,.., )∈ Ruu∧ i =0 ⇔( uuuu 1,..,i ,.., jn ,.., )∈ R. Thus,

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Iij ij RR=ij I ij = R. (15) Hence we have just proved the U n Lemma 1. 2,()ij ,I ij is a monoid. Note that (ij) (uuuu1,..,i ,.., j ,.., n )()(∈ RR12ij ⇔ uuuu 1,..,j ,.., i ,.., n ) ∈⇔ RR12ij

⇔∃uuuuu01( ,.., 0 ,..,in ,.., )∈ R1∧( uuuu 1,..,j ,.., 0 ,.., n)∈ R2 ⇔ (ij) (ij) ⇔∃uuuuu01( ,..,i ,.., 0 ,.., n)∈ R 1∧( uuuu 1,.., 0 ,..,jn ,.., )∈ R2 ⇔ (ij) (ij) (ij) (ij) ⇔∈⇔∈(uuuu1,..,i ,.., j ,.., n ) R1ji R2( uuuu 1,..,i ,.., j ,.., n ) R21ij R. In that way (ij) (ij) (ij) (ij) (ij) ()RR12ij = R 1 ji R 2 = R 2ij R 1. (16)

(ij) U n Hence the bijective fR() := R is an isomorphism of monoids 2,()ij ,I ij and

U n 2,() ji ,I ji . Moreover, (ij) (uuu1,..,ik ,.., ,.., uu jn ,.., )∈() R12 ik R ⇔( uuu1,..,jkin ,.., ,.., uu ,.., )∈⇔ R12 ik R

⇔∃uuuu01( ,.., 0 ,..,kin ,.., uu ,.., )∈ R1∧( uuuuu 1,..,j ,.., 0 ,.., in ,.., )∈ R2 ⇔ (ij) (ij) ⇔∃uuuu01( ,..,ik ,.., ,.., uu0 ,.., n)∈ R1∧( uuuuu 1,..,i ,.., 0 ,.., jn ,.., )∈ R2 ⇔ (ij) (ij) (ij) (ij) ⇔∈⇔∈(uuuu1,..,i ,.., jn ,.., ) R1 jk R2( uuuu 1,..,i ,.., jn ,.., ) R21 kj R. From which we obtain (ij) (ij) (ij) (ij) (ij) ()RR12ik = R 1 jk R 2 = R 2kj R 1. (17) Hence we have proved the U n U n Lemma 2. Monoids 2,()ik ,I ik and 2,() jk ,I jk are isomorphic, as well as monoids

U n U n 2,()ij ,I ij and 2,() ji ,I ji .

U n Let us set an algebraic structure 2,()ij , ik ,,II ij ik and then we can write the following logical consequences:

(uuuuu1,..,i ,.., j ,.., k ,.., n )∈ R1ij() R 2 ik R 3⇔∃ uuuuuu 01( ,.., 0 ,..,j ,.., k ,.., n )∈ R1 ∧

∧(uuuuu1,..,i ,.., 0 ,.., kn ,.., )∈ R2 ik R 3⇔∃ uuuuuuu 0 ∃ 01′ ( ,.., 0 ,..,jkn ,.., ,.., )∈ R1 ∧

∧(uuuuu100,..,′′ ,.., ,..,kn ,.., )(∈∧ R2 uuuuu 1,..,i ,.., 00 ,.., ,.., n)∈⇔ R 3

∃∃uuuuuuu0010′′( ,.., ,..,jkn ,.., ,.., )∈ Ruuuuu1∧( 100,.., ,.., ,..,kn ,.., )∈ R2 ∧

∧(uuuuu1,..,in ,.., 00 ,..,′ ,.., )∈⇒ R 3 ′′ ∃uuuuuuu0010( ∃( ,.., ,..,jkn ,.., ,.., )∈∧ Ruuuuu1( 100,.., ,.., ,..,kn ,.., )∈ R2) ∧

∧( ∃uuuuuu01( ,..,i ,.., 0 ,.., 0′ ,.., n )∈ R3) ⇔∃ uuuuuu 01′′( ,.., 0 ,..,j ,.., k ,.., n )∈ RR1ij 2 ∧ ′′ ∧∃( uuuuuu01( ,..,i ,.., 0 ,.., 0 ,.., n)∈ R3 ∧( uuuuu 1,.., 0 ,..,j ,.., 0 ,.., nR)∈ 1 ) ⇔

⇔∃uuuuuu01′′( ,.., 0 ,..,j ,.., k ,.., n )∈ RR1ij 2 ∧( uuuuu 1,..,i ,.., j ,..,0′ ,..,n )∈ R3ij 1 R ⇔

⇔∈(uuuu1,..,i ,.., j ,.., k ,.., u n )()() RR12ij ik R 3 ij1 R . This means that the following Lemma is true.

V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 121 Data Science V P Tsvetov

U n Lemma 3. In an ordered algebra 2 ,(ij , ik ,⊆ ,II ij , ik ,0 R ,1 R ) , the pseudo distributive law holds

R1ij() R 2 ik R 3⊆ ()() RR 12 ij ik R 3 ij1 R . (18) n According to [17], we use the notation 1:R = U and 0:R = ∅ .

Then look at composition (ij) (ij) (uuuu1,..,i ,.., jn ,.., )∈ RR ij ⇔∃ uuuuu01( ,..,0 ,..,jn ,.., )∈ R∧( uuuu1,..,i ,.., 0 ,.., n)∈ R ⇔ (19) ⇔∃uuuuu010( ,.., ,..,jn ,.., )∈ R∧( uuuu10,.., ,..,i ,..,n)∈ R . Definition 1. The finitary relation R is called a function from i-th to j-th if

∀u11,..., uuuuuuuui ,.., jjn ,′ ,..,( ,.., i ,.., jn ,.., )(∈∧ R uuuu 1,..,i ,..,′′ jn ,.., )∈→= R uu jj. (20) We can obtain from (19) - (20) the following set inclusion (ij) (ij) (uuuu11,..,i ,.., j ,.., n )∈ Rij R ⇒= uui j ⇔( uuuu,..,i ,.., j ,.., n)∈ I ij ⇔ Rij R ⊆ Iij . (21) Definition 2. The finitary relation R is called a surjection from i-th argument if

∀u1,..., uuii−+ 1 , 1 ,.., uuuuuuu jn ,.., ∃∈01( ,.., 0 ,..,jn ,.., ) R. (22) From (21) - (22) we can get the reverse set inclusion (ij) Iij⊆ RR ij . (23) Thus, in the case of R is a surjective function from i-th to j-th argument we have the (ij) RRij = Iij . (24) Similarly, in the case of R is a surjective function from j-th to i-th argument we have the equality (ij) Rij RI= ij . (25)

Let us denote the set of surjective functions from both (i-th to j-th and j-th to i-th) as Fij .

It is easy that Fij is closed by ij , and hence we have proved the

U n Lemma 4. FIij,,() ij ij is a subgroup of the monoid 2,()ij ,I ij . As well as binary relations, finitary relations have the following properties [17]

R1ij () RR 23∪=()() RR 12 ij ∪ RR 13ij , (26)

()RR23∪=ij R 1()() RR 21ij ∪ RR 31 ij , (27)

R1ij () RR 23∩⊆()() RR 12 ij ∩ RR 13ij , (28)

()RR23∩⊆ij R 1()() RR 21ij ∩ RR 31 ij , (29)

U n (ij) and so we can set an algebraic structures FIij,,() ij ij , 2,(∪∩ ,,,ij ik ,,,0,1, ⊆ R RII ij , ik ) that have properties (12)-(18), (24)-(29).

3. Conclusion and examples We have defined algebraic structures of finitary relations as a common case of well-known algebraic n structures of binary relations. We have considered the algebraic structures on an underlying set 2U U n and sometimes called a finitary relation R ∈2 by a (n-uniform) hypergraph. The operation ij can be called the “straightening the edges” or “deleting shared intermediate vertices”. Let us take an example. U 3 Example 1 (algebraic). Let us set U= {} uuuu0123,,, , 2,()23 ,I 23 , and

R= {()() uuu103,, , uuu 120 ,, }. Now we can get

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()()()()uuu000,,,,,,,,,,,, uuu 011 uuu 022 uuu 033  ()()()()uuu10,,,,,,,,,,,, 0 uuu 111 uuu 12 2 uuu 133 I23 = , ()()()()uuu200,,,,,,,,,,,, uuu 211 uuu 222 uuu 233  ()()()()uuu300,,,,,,,,,,, uuu 311 uuu 322 uuu 333

R23 R= {}() uuu1,, 2 3 ,

RRR23 23 = ∅ .

Despite its simplicity, operation 23 has some interesting applications. In examples 2, 3 we are going to denote 3-tuple ()uuuiii,, as a word uuuiii. 123 123

Example 2 (feature selection). Let R= { uuu100,,,,, uuu 110 uuu 120 uuu 121 uuu 123 uuu 130} be a set of (23) words, and Rf = {} uuu103 , Rf = {} uuu130 are filters. First, apply the filter R f

Rf 23 R= { uuu 103,,, uuu 113 uuu 123 uuu 133}. (23) Then apply the filter R f (23) Rff23 R 23 R= { uuu 100,,, uuu 110 uuu 120 uuu 130} .

Example 3 (crossover). Let R= { uuu100,,, uuu 110 uuu 121 uuu 132} be a population. Let us define the evolution operator Ε=∪()R RR23 R and start a first step of evolution

Ε=(){R uuu100,,,,, uuu 110 uuu 120 uuu 121 uuu 131 uuu 132}.

In example 4 we are going to denote 3-tuple ()uuuiii,, as an implies ui→→() uu ii. 123 1 23

Example 4 (AI). Let Ru=→→{ 111112()() uuuuu, →→} be a base set of AI premises. Let us

∞ k ()23 kk+1 1 define the semantic closure of R as []R=() RR ∪ , where R= RR23 and RR= . By k=1 definition we have

[]Ruuuuuuuuuuuu=→→{ 1()()()() 1 11,,, →→ 1 21 →→ 2 11 →→ 2 2}.

Note that ((uuuuuu1→→() 03) ∧→→( 1() 20)) →→→( uuu 1() 23) is tautology, so the rule uuuuuuuuu1→→()()() 031201, →→ђ →→ 23 preserves truth.

We also note that ((()uu10→→∧ u 3) (() uu 12 →→ u 0)) →(() uu 12 →→ u 3) is tautology, too.

n U n It makes perfect sense to use an χR :U  {} 0,1 for R ∈2 , that is defined as

∈1,()uu1 ,.., n R χRn()uu1,.., =  . 0,()uu1 ,.., n ∉ R

In the case of Uuu= {}1,.., m , we can use this function to define a join-vertices logical n array ψ R :(){} 1..m false, true for (n-uniform) hypergraph. Let f:1.. mU be a total bijection

n and R ∈2U . We define true, χ fk−−11()(),.., fk= 1 RR  Rn( 1 ) ψψkk,.., = ()kk1,.., n=  . 1 n χ −−11=  false, Rn( f()() k1 ,.., f k ) 0 n Let us denote {}false, true as D and a set of logical array defined above as D()1..m .

V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 123 Data Science V P Tsvetov

We also can set a logical algebra that generalized adjacency matrices algebra. In this way we define n ()1..m a ∗ij on D m ψψ12∗= ψ 1 ∧ ψ2 . kk1,.., n ijkk 1,.., n ∨ kkskkk111,..,i−+ , , i ,.., jn ,.., kkk1 ,..,i ,.., j− 1 , skk , j + 1 ,.., n s=1 n U n ()1..m By our construction semigroups 2,()ij and D ,()∗ij are isomorphic.

∞ n More interesting is the case of algebraic structures on an underlying set 2U and operations  n=1 U n U m from 2 to 2 . For example, let us define the operations “gluing edges” g and “replacing chains”

r .

RR12112g={( uuuu,.., mn−− ,′′ ,..,)()() |∃ uuuu 0110102 ,.., m , ∈∧ Ruuu,′′ ,.., n ∈ R 2}, ′′ ′ ′′ RR12111r={( uuu,..,ijn−+ , ,.., u) |∃ uijuuuu 0∃∃ ( 110 ,..,i− , ,.., m)∈ R 1101 ∧( uuuu,.., ,jn+ ,.., )∈ R 2}.

For the finitary relation R= {()() uuu103,, , uuu 120 ,, } from Example 1 we can get (13) R= {()() uuu301,, , uuu 021 ,, }, (13) Rg R= {()() uuuu1001,,, , uuuu 1221 ,,, },

Rr R= {()()()()() u1,, uu 03 ,, uu 10 ,, uu 12 ,, uu 13 ,,()() uu 20 ,,, uuu 123}. It is clear that even in the case of finite set U we would never make a finite representation for such algebraic structures. But in particular cases, maybe we can. This case is of interest.

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[11] Ghoshdastidar D, Ambedkar D 2015 A Provable Generalized Tensor Spectral Method for Uniform Hypergraph Partitioning Proceedings of the 32nd International Conference on Machine Learning, ICML 400-409 [12] Ghoshdastidar D, Ambedkar D 2017 Consistency of spectral hypergraph partitioning under planted partition model Ann. Statist. 45(1) 289-315 [13] Chien I, Lin C and Wang I 2018 Community Detection in Hypergraphs: Optimal Statistical Limit and Efficient Algorithms Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, in PMLR 84 871-879 [14] Mal’tsev A I 1973 Algebraic systems (Springer) p 319 [15] Chernov V M 2018 Ternary number systems in finite fields Computer Optics 42(4) 704-711 DOI: 10.18287/2412-6179-2018-42-4-704-711 [16] Tsvetov V P 2014 On a syntactic algorithm on graphs Proceedings of the International Conference Advanced Information Technologies and Scientific Computing (Samara, Russia) 235-238 [17] Tsvetov V P 2018 Dual ordered structures of binary relations Proceedings of the International Conference Information Technology and Nanotechnology. Session Data Science (Samara, Russia) 2635-2644

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