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)
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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 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,
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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 )()(∈ 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.
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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 ,, }. 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
R23 R= {}() uuu1,, 2 3 ,
RRR23 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) Rff23 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 RR23 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= RR23 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 inference 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 indicator function 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 finite set 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 .
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We also can set a logical algebra that generalized adjacency matrices algebra. In this way we define n ()1..m a binary operation ∗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 .
RR12112g={( uuuu,.., mn−− ,′′ ,..,)()() |∃ uuuu 0110102 ,.., m , ∈∧ Ruuu,′′ ,.., n ∈ R 2}, ′′ ′ ′′ RR12111r={( 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) Rg R= {()() uuuu1001,,, , uuuu 1221 ,,, },
Rr 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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