Algebras of finitary relations
V P Tsvetov1
1
Samara National Research University, Moskovskoe Shosse 34А, Samara, Russia, 443086
e-mail: tsf-su@mail.ru
Abstract. Algebras of finitary relations naturally generalize the algebra 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 hypergraphs as finitary relations. In this way the results can be
applied to graph and hypergraph 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 relation [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 set U , in other words, as
a subset of U n . In case of n = 2 this leads to graph as a binary relation. Boolean algebras
(
2U ×U , ∪, ∩, , ∅,U × U ) and 2 , ( ∪, ∩, , ∅,U ) are well known to us.
Un n
It is less trivial to define the inverse operation and the left composition for finitary relations. We
have to start from inverse operation, left and right compositions for binary relations:
= R −1 {( u2 , u1 ) | ( u1 , u2 ) ∈ R} , (1)
R1=
R2 {( u , u ) | ∃u ( u , u ) ∈ R ∧ ( u , u ) ∈ R } ,
1 2 0 1 0 1 0 2 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
R1 1 R2 = R1−1 R2 , (4)
(5)
R1 2 R2 = R1 R2−1 , (6)
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(7)
−1 −1
R1 3 R2 = R R , 1 2 (8)
(9)
This makes it possible to set the following pairs of isomorphic magmas.
2 , ( 1 , I ) 2U ×U , ( 1 , I )
U ×U
are isomorphic magmas with left identity elements.
2U ×U , ( 2 , I ) 2U ×U , ( 2 , I ) are isomorphic magmas with right identity elements.
2U ×U , ( 3 ) 2U ×U , ( 3 ) are isomorphic magmas without identity elements.
It is easy to see that in the symmetric case R = R −1 all of monogenic monoids {R } , ( , I ) ,
n ∞
n =0
{R } , ( , I ) , {R } , ( , I ) , {R } , ( , I ) ( i ∈1..3 ) are equal.
n ∞
n =0
n ∞
n =0 i
n ∞
n =0 i
The monogenic monoid {R } , ( , I ) and distributive algebraic structure {R } , ( , ∪, I , ∅ )
n ∞
n =0
n ∞
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 i ≠ j
R=
(ij)
R=
(ji)
{( u ,.., u ,.., u ,.., u ) | ( u ,.., u ,.., u ,.., u ) ∈ R} ,
1 j i n 1 i j n (10)
=R1 ij R2 {( u ,.., u ,.., u ,.., u ) | ∃u ( u ,.., u ,.., u ,.., u ) ∈ R ∧ ( u ,.., u ,.., u ,.., u ) ∈ R } . (11)
1 i j n 0 1 0 j n 1 1 i 0 n 2
Obviously, the operation (10) is an involution.
(R ) = R .
(ij) (ij)
(12)
Moreover,
R1 ij R2 = R2 ji R1 . (13)
It is easy to prove that operation (11) is associative. Actually,
( u1 ,.., ui ,.., u j ,.., un ) ∈ R1 ij ( R2 ij R3 ) ⇔ ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ R1 ∧ ( u1 ,.., ui ,.., u0 ,.., un ) ∈ R2 ij R3 ⇔
⇔ ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ R1 ∧ ( ∃u0′ ( u1 ,.., u0′ ,.., u0 ,.., un ) ∈ R2 ∧ ( u1 ,.., ui ,.., u0′ ,.., un ) ∈ R3 ) ⇔
( )
⇔ ∃u0′ ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ R1 ∧ ( u1 ,.., u0′ ,.., u0 ,.., un ) ∈ R2 ∧ ( u1 ,.., ui ,.., u0′ ,.., un ) ∈ R3 ⇔
⇔ ∃u0′ ( u1 ,.., u0′ ,.., u j ,.., un ) ∈ R1 ij R2 ∧ ( u1 ,.., ui ,.., u0′ ,.., un ) ∈ R3 ⇔
⇔ ( u1 ,.., ui ,.., u j ,.., un ) ∈ ( R1 ij R2 ) ij R3 .
Then we set
I ij = {( u ,.., u ,.., u ,.., u ) | k ∈1..n ∧ u ∈U ∧ u = u }∈ 2 .
1 i j n k j i
Un
(14)
It is easy to see
( u1 ,.., ui ,.., u j ,.., un ) ∈ I ij ij R ⇔ ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ I ij ∧ ( u1 ,.., ui ,.., u0 ,.., un ) ∈ R ⇔
⇔ ∃u0 ( u1 ,.., ui ,.., u0 ,.., un ) ∈ R ∧ u =
j u0 ⇔ ( u1 ,.., ui ,.., u j ,.., un ) ∈ R ,
and similarly
( u1 ,.., ui ,.., u j ,.., un ) ∈ R ij I ij ⇔ ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ R ∧ ( u1 ,.., ui ,.., u0 ,.., un ) ∈ I ij ⇔
⇔ ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ R ∧ u=
i u0 ⇔ ( u1 ,.., ui ,.., u j ,.., un ) ∈ R .
Thus,
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I=
ij ij R R=
ij I ij R . (15)
Hence we have just proved the
Lemma 1. 2U , ( ij , I ij ) is a monoid.
n
Note that
( u ,.., u ,.., u ,.., u ) ∈ ( R R ) ⇔ ( u ,.., u ,.., u ,.., u ) ∈ R R ⇔
(ij)
1 i j n 1 ij 2 1 j i n 1 ij 2
⇔ ∃u ( u ,.., u ,.., u ,.., u ) ∈ R ∧ ( u ,.., u ,.., u ,.., u ) ∈ R ⇔
0 1 0 i n 1 1 j 0 n 2
⇔ ∃u ( u ,.., u ,.., u ,.., u ) ∈ R ∧ ( u ,.., u ,.., u ,.., u ) ∈ R ⇔
0 1 i 0 n
(ij)
1 1 0 j n
(ij)
2
⇔ ( u ,.., u ,.., u ,.., u ) ∈ R R ⇔ ( u ,.., u ,.., u ,.., u ) ∈ R R .
1 i j n
(ij)
1 ji
(ij)
2 1 i j n
(ij)
2 ij
(ij)
1
In that way
( R=
R )
(ij)
R=
R
1 R R .
ij 2
(ij)
1 ji
(ij)
2
(ij)
2 ij
(ij)
1 (16)
Hence the bijective function f ( R ) := R (ij) is an isomorphism of monoids 2U , ( ij , I ij )
n
and
2U , ( ji , I ji ) .
n
Moreover,
( u ,.., u ,.., u ,.., u ,.., u ) ∈ ( R R ) ⇔ ( u ,.., u ,.., u ,.., u ,.., u ) ∈ R R ⇔
1 i k j n 1 ik 2
(ij)
1 j k i n 1 ik 2
⇔ ∃u ( u ,.., u ,.., u ,.., u ,.., u ) ∈ R ∧ ( u ,.., u ,.., u ,.., u ,.., u ) ∈ R ⇔
0 1 0 k i n 1 1 j 0 i n 2
⇔ ∃u ( u ,.., u ,.., u ,.., u ,.., u ) ∈ R ∧ ( u ,.., u ,.., u ,.., u ,.., u ) ∈ R ⇔
0 1 i k 0 n
(ij)
1 1 i 0 j n
(ij)
2
⇔ ( u ,.., u ,.., u ,.., u ) ∈ R R ⇔ ( u ,.., u ,.., u ,.., u ) ∈ R R .
1 i j n
(ij)
1 jk
(ij)
2 1 i j n
(ij)
2 kj
(ij)
1
From which we obtain
( R=
1 ik R2 ) R=
(ij) (ij) (ij)
1 jk R2 R2(ij) kj R1(ij) . (17)
Hence we have proved the
2U , ( ik , I ik ) 2U , ( jk , I jk )
n n
Lemma 2. Monoids and are isomorphic, as well as monoids
2U , ( ij , I ij ) and 2U , ( ji , I ji ) .
n n
Let us set an algebraic structure 2U , ( ij , ik , I ij , I ik ) and then we can write the following logical
n
consequences:
( u1 ,.., ui ,.., u j ,.., uk ,.., un ) ∈ R1 ij ( R2 ik R3 ) ⇔ ∃u0 ( u1 ,.., u0 ,.., u j ,.., uk ,.., un ) ∈ R1 ∧
∧ ( u1 ,.., ui ,.., u0 ,.., uk ,.., un ) ∈ R2 ik R3 ⇔ ∃u0 ∃u0′ ( u1 ,.., u0 ,.., u j ,.., uk ,.., un ) ∈ R1 ∧
∧ ( u1 ,.., u0′ ,.., u0 ,.., uk ,.., un ) ∈ R2 ∧ ( u1 ,.., ui ,.., u0 ,.., u0′ ,.., un ) ∈ R3 ⇔
∃u0′ ∃u0 ( u1 ,.., u0 ,.., u j ,.., uk ,.., un ) ∈ R1 ∧ ( u1 ,.., u0′ ,.., u0 ,.., uk ,.., un ) ∈ R2 ∧
∧ ( u1 ,.., ui ,.., u0 ,.., u0′ ,.., un ) ∈ R3 ⇒
(
∃u0′ ∃u0 ( u1 ,.., u0 ,.., u j ,.., uk ,.., un ) ∈ R1 ∧ ( u1 ,.., u0′ ,.., u0 ,.., uk ,.., un ) ∈ R2 ∧ )
∧ ( ∃u0 ( u1 ,.., ui ,.., u0 ,.., u0′ ,.., un ) ∈ R3 ) ⇔ ∃u0′ ( u1 ,.., u0′ ,.., u j ,.., uk ,.., un ) ∈ R1 ij R2 ∧
(
∧ ∃u0 ( u1 ,.., ui ,.., u0 ,.., u0′ ,.., un ) ∈ R3 ∧ ( u1 ,.., u0 ,.., u j ,.., u0′ ,.., un ) ∈ 1R ⇔ )
⇔ ∃u0′ ( u1 ,.., u0′ ,.., u j ,.., uk ,.., un ) ∈ R1 ij R2 ∧ ( u1 ,.., ui ,.., u j ,.., u0′ ,.., un ) ∈ R3 ij 1R ⇔
⇔ ( u1 ,.., ui ,.., u j ,.., uk ,.., un ) ∈ ( R1 ij R2 ) ik ( R3 ij 1R ) .
This means that the following Lemma is true.
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Lemma 3. In an ordered algebra 2U , ( ij , ik , ⊆, I ij , I ik ,0 R ,1R ) , the pseudo distributive law holds
n
R1 ij ( R2 ik R3 ) ⊆ ( R1 ij R2 ) ik ( R3 ij 1R ) . (18)
According to [17], we use the notation 1R := U n and 0 R := ∅ .
Then look at composition
( u1 ,.., ui ,.., u j ,.., un ) ∈ R ij R (ij) ⇔ ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ R ∧ ( u1 ,.., ui ,.., u0 ,.., un ) ∈ R (ij) ⇔
(19)
⇔ ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ R ∧ ( u1 ,.., u0 ,.., ui ,.., un ) ∈ R .
Definition 1. The finitary relation R is called a function from i-th to j-th argument if
∀u1 ,..., ui ,.., u j , u′j ,.., un ( u1 ,.., ui ,.., u j ,.., un ) ∈ R ∧ ( u1 ,.., ui ,.., u′j ,.., un ) ∈ R → u j =
u′j . (20)
We can obtain from (19) - (20) the following set inclusion
( u1 ,.., ui ,.., u j ,.., un ) ∈ R ij R (ij) ⇒ ui = u j ⇔ ( u1 ,.., ui ,.., u j ,.., un ) ∈ I ij ⇔ R ij R (ij) ⊆ I ij . (21)
Definition 2. The finitary relation R is called a surjection from i-th argument if
∀u1 ,..., ui −1 , ui +1 ,.., u j ,.., un ∃u0 ( u1 ,.., u0 ,.., u j ,.., un ) ∈ R . (22)
From (21) - (22) we can get the reverse set inclusion
I ij ⊆ R ij R (ij) . (23)
Thus, in the case of R is a surjective function from i-th to j-th argument we have the equality
R ij R (ij) = I ij . (24)
Similarly, in the case of R is a surjective function from j-th to i-th argument we have the equality
R (ij) ij R = I ij . (25)
Let us denote the set of surjective functions from both (i-th to j-th and j-th to i-th) arguments as Fij .
It is easy that Fij is closed by ij , and hence we have proved the
Lemma 4. Fij , ( ij , I ij ) is a subgroup of the monoid 2U , ( ij , I ij ) .
n
As well as binary relations, finitary relations have the following properties [17]
R3 ) ( R1 ij R2 ) ∪ ( R1 ij R3 ) ,
R1 ij ( R2 ∪= (26)
( R2 ∪ R3 ) ij R1 = ( R2 ij R1 ) ∪ ( R3 ij R1 ) , (27)
R1 ij ( R2 ∩ R3 ) ⊆ ( R1 ij R2 ) ∩ ( R1 ij R3 ) , (28)
( R2 ∩ R3 ) ij R1 ⊆ ( R2 ij R1 ) ∩ ( R3 ij R1 ) , (29)
Fij , ( ij , I ij ) , 2U , ( ∪, ∩, ij , ik ,(ij) , ⊆,0 R ,1R , I ij , I ik )
n
and so we can set an algebraic structures 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
and sometimes called a finitary relation R ∈ 2U by a (n-uniform) hypergraph. The operation ij can
n
be called the “straightening the edges” or “deleting shared intermediate vertices”. Let us take an
example.
U = {u0 , u1 , u2 , u3 } , 2U , ( 23 , I 23 ) ,
3
Example 1 (algebraic). Let us set and
R = {( u1 , u0 , u3 ) , ( u1 , u2 , u0 )} . Now we can get
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( u0 , u0 , u0 ) , ( u0 , u1 , u1 ) , ( u0 , u2 , u2 ) , ( u0 , u3 , u3 ) ,
( u1 , u0 , u0 ) , ( u1 , u1 , u1 ) , ( u1 , u2 , u2 ) , ( u1 , u3 , u3 ) ,
I 23 = ,
( u2 , u0 , u0 ) , ( u2 , u1 , u1 ) , ( u2 , u2 , u2 ) , ( u2 , u3 , u3 ) ,
( u , u , u ) , ( u , u , u ) , ( u , u , u ) , ( u , u , u )
3 0 0 3 1 1 3 2 2 3 3 3
R 23 R = {( u1 , u2 , u3 )} ,
R 23 R 23 R = ∅ .
Despite its simplicity, operation 23 has some interesting applications. In examples 2, 3 we are
( )
going to denote 3-tuple ui1 , ui2 , ui3 as a word ui1 ui2 ui3 .
Example 2 (feature selection). Let R = {u1u0u0 , u1u1u0 , u1u2u0 , u1u2u1 , u1u2u3 , u1u3u0 } be a set of
words, and R f = {u1u0u3 } , R (23)
f = {u1u3u0 } are filters. First, apply the filter R f
R f 23 R = {u1u0u3 , u1u1u3 , u1u2u3 , u1u3u3 } .
(23)
Then apply the filter R f
R (23)
f 23 R f 23 R = {u1u0u0 , u1u1u0 , u1u2u0 , u1u3u0 } .
Example 3 (crossover). Let R = {u1u0u0 , u1u1u0 , u1u2u1 , u1u3u2 } be a population. Let us define the
evolution operator Ε ( R ) =R ∪ R 23 R and start a first step of evolution
Ε(R) =
{u1u0u0 , u1u1u0 , u1u2u0 , u1u2u1 , u1u3u1 , u1u3u2 } .
( )
In example 4 we are going to denote 3-tuple ui1 , ui2 , ui3 as an implies ui1 → ui2 → ui3 . ( )
Example 4 (AI). Let R = {u1 → ( u1 → u1 ) , u1 → ( u1 → u2 )} be a base set of AI premises. Let us
( R ∪ R ( ) ) , where R
∞
[ R]
k
k +1
define the semantic closure of R as = 23
= R k 23 R and R1 = R . By
k =1
definition we have
[ R ] = {u1 → ( u1 → u1 ) , u1 → ( u1 → u2 ) , u1 → ( u2 → u1 ) , u1 → ( u2 → u2 )} .
Note that ( ( u1 → ( u0 → u3 ) ) ∧ ( u1 → ( u2 → u0 ) ) ) → ( u1 → ( u2 → u3 ) ) is tautology, so the inference
rule u1 → ( u0 → u3 ) , u1 → ( u2 → u0 ) ђ u1 → ( u2 → u3 ) preserves truth.
We also note that ( ( ( u1 → u0 ) → u3 ) ∧ ( ( u1 → u2 ) → u0 ) ) → ( ( u1 → u2 ) → u3 ) is tautology, too.
It makes perfect sense to use an indicator function χ R : U n {0,1} for R ∈ 2U , that is defined as
n
1, ( u1 ,.., un ) ∈ R
χ R ( u1 ,.., un ) = .
0, ( u1 ,.., un ) ∉ R
In the case of finite set U = {u1 ,.., um } , we can use this function to define a join-vertices logical
array ψ R : (1..m ) { false, true} for (n-uniform) hypergraph. Let f :1..m U be a total bijection
n
and R ∈ 2U . We define
n
true, χ R ( f −1 ( k1 ) ,.., f −1 ( kn ) ) = 1
ψ
= ( k1 ,.., kn )
ψ= R R
.
( ( ) ( ) )
k1 ,..,kn
false, χ R f −1
k1 ,.., f −1
k n = 0
Let us denote { false, true} as D and a set of logical array defined above as D(1..m ) .
n
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We also can set a logical algebra that generalized adjacency matrices algebra. In this way we define
a binary operation ∗ij on D(1..m )
n
m
ψ k1 ,..,k=
∗ij ψ k2 ,..,k ∨ψ ∧ψ k21 ,..,ki ,..,k j −1 ,s ,k j +1 ,..,kn .
1
1 n 1 n k1 ,..,ki −1 , s ,ki +1 ,..,k j ,..,kn
s =1
By our construction semigroups 2 , ( ij ) and D(1..m ) , ( ∗ij ) are isomorphic.
n
Un
∞ n
More interesting is the case of algebraic structures on an underlying set n =1 2U and operations
n m
from 2U to 2U . For example, let us define the operations “gluing edges” g and “replacing chains”
r .
=R1 g R2 {( u ,.., u
1 m −1 , u2′ ,.., un′ ) | ∃u0 ( u1 ,.., um −1 , u0 ) ∈ R1 ∧ ( u0 , u2′ ,.., un′ ) ∈ R2 } ,
=R1 r R2 {( u ,.., u , u′ ,.., u′ ) | ∃u ∃i ∃j ( u ,.., u , u ,.., u ) ∈ R ∧ ( u′,.., u , u′ ,.., u′ ) ∈ R } .
1 i −1 j +1 n 0 1 i −1 0 m 1 1 0 j +1 n 2
For the finitary relation R = {( u1 , u0 , u3 ) , ( u1 , u2 , u0 )} from Example 1 we can get
R (13) = {( u3 , u0 , u1 ) , ( u0 , u2 , u1 )} ,
R g R (13) = {( u1 , u0 , u0 , u1 ) , ( u1 , u2 , u2 , u1 )} ,
R r R = {( u1 ) , ( u0 , u3 ) , ( u1 , u0 ) , ( u1 , u2 ) , ( u1 , u3 ) , ( u2 , u0 ) , ( u1 , u2 , u3 )} .
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.
4. References
[1] Hein M, Setzer S, Jost L and Rangapuram S S 2013 The total variation on hypergraphs-learning
on hypergraphs revisited Advances in Neural Information Processing Systems 2427-2435
[2] Ricatte T, Gilleron R and Tommasi M 2014 Hypernode graphs for spectral learning on binary
relations over sets Joint European Conference on Machine Learning and Knowledge Discovery
in Databases 662-677
[3] Louis A 2015 Hypergraph markov operators, eigenvalues and approximation
algorithms Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of
Computing 713-722
[4] Zhang C, Hu S, Tang Z G and Chan T H 2017 Re-revisiting Learning on Hypergraphs:
Confidence Interval and Subgradient Method Proceedings of the 34th International Conference
on Machine Learning, in PMLR 70 4026-4034
[5] Pu L, Faltings B 2012 Hypergraph learning with hyperedge expansion Machine Learning and
Knowledge Discovery in Databases 410-425
[6] Yu J, Tao D and Wang M 2012 Adaptive hypergraph learning and its application in image
classification IEEE Transactions on Image Processing 21(7) 3262-3272
[7] Panagopoulos A, Wang C, Samaras D and Paragios N 2013 Simultaneous cast shadows,
illumination and geometry inference using hypergraphs IEEE Transactions on Pattern Analysis
and Machine Intelligence 35(2) 437-449
[8] Wang M, Wu X 2015 Visual classification by l1-hypergraph modeling IEEE Transactions on
Knowledge and Data Engineering 27(9) 2564-2574
[9] Zhou D, Huang J and Scholkopf B 2007 Learning with Hypergraphs: Clustering, Classification,
and Embedding Advances in Neural Information Processing Systems: Proceedings of the 2006
Conference 19 1601-1608
[10] Ghoshdastidar D, Ambedkar D 2014 Consistency of Spectral Partitioning of Uniform
Hypergraphs under Planted Partition Model Advances in Neural Information Processing
Systems 27: Annual Conference on Neural Information Processing Systems 397-405
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 124
Data Science
V P Tsvetov
[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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