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Algebras of Finitary Relations Algebras of finitary relations V P Tsvetov1 1Samara National Research University, Moskovskoe Shosse 34А, Samara, Russia, 443086 e-mail: [email protected] 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 n 2UU× ,(∪∩ , ,, ∅ ,UU × ) and 2Un ,(∪∩ , ,, ∅ ,U ) are well known to us. 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: −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) V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) Data Science V P Tsvetov (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 algebraic structure 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) RR1ij 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, RRR1ij 22= ji R 1. (13) It is easy to prove that operation (11) is associative. Actually, (uuuu1,..,i ,.., j ,.., n )()(∈ R1ij R 23 ij R⇔∃ uuuuu 010,.., ,..,j ,.., n )∈ R11 ∧( uuuu,..,i ,.., 0 ,.., n )∈ R 23ij 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 )∈ RR1ij 2 ∧( uuuu 1,..,i ,.., 0′ ,.., n )∈ R3 ⇔ ⇔∈(uuuu1,..,i ,.., j ,.., n )() RR12ij 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, V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 120 Data Science V P Tsvetov 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 )()(∈ RR12ij ⇔ 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 ) R1ji R2( uuuu 1,..,i ,.., j ,.., n ) R21ij R. In that way (ij) (ij) (ij) (ij) (ij) ()RR12ij = R 1 ji R 2 = R 2ij R 1. (16) (ij) U n Hence the bijective function 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) ()RR12ik = 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 )∈ R1ij() 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 )∈ RR1ij 2 ∧ ′′ ∧∃( uuuuuu01( ,..,i ,.., 0 ,.., 0 ,.., n)∈ R3 ∧( uuuuu 1,.., 0 ,..,j ,.., 0 ,.., nR)∈ 1 ) ⇔ ⇔∃uuuuuu01′′( ,.., 0 ,..,j ,.., k ,.., n )∈ RR1ij 2 ∧( uuuuu 1,..,i ,.., j ,..,0′ ,..,n )∈ R3ij 1 R ⇔ ⇔∈(uuuu1,..,i ,.., j ,.., k ,.., u n )()() RR12ij 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 R1ij() 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 argument 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 )∈ Rij 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 equality (ij) RRij = Iij . (24) Similarly, in the case of R is a surjective function from j-th to i-th argument we have the equality (ij) Rij RI= 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 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] R1ij () RR 23∪=()() RR 12 ij ∪ RR 13ij , (26) ()RR23∪=ij R 1()() RR 21ij ∪ RR 31 ij , (27) R1ij () 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 ,, }.
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