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) V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) Data Science V P Tsvetov (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, V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 120 Data Science V P Tsvetov 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. V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 121 Data Science V P Tsvetov 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 V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 122 Data Science V P Tsvetov ( 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 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 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. 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