Algebras of finitary relations

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. R1◦R2 = R2◦R1 = {(u1,u2) |Ǝ u0(u0,u2)ЄR1^(u1,u0) Є R2} are isomorphic monoids, where I is identity relation on U . By

Keywords

• 2U×U
• (∪
• ∩
• ∅ U ×U ) and 2U n (∪ ∩ ∅ U n ) are well known to us

-

(2) (6)

Note that the way, we can define operations

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

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 =u1 {(u2 , ) | (u1, u2 ) ∈ R} , (1) R  R2 1 ={(u1, u2 ) | ∃u0 (u1, u0 ) ∈ R ∧ (u0 , u2 ) ∈ R2},

1 R1 2 R2 = R  R2−1 ,

1 (7) R1 3 R2 = R1−1  R2−1, (8) (9)

This makes it possible to set the following pairs of isomorphic magmas. 2U×U,(1,I)  2U×U,(1,I) 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 {Rn}n∞=0,(,I) , {Rn}∞n=0,(,I) , {Rn}n∞=0,(i,I) , {Rn}n∞=0,(i,I) (i∈1..3) are equal.

The monogenic monoid {Rn}n∞=0,(,I) and distributive algebraic structure {Rn}n∞=0,(,∪,I,∅) 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

Let us consider the underlying set of finitary relations 2Un , and define the following unary and binary operations for i ≠ j

R(ij) =R(ji) ={(u1,..,uj,..,ui,..,un)|(u1,..,ui,..,uj,..,un)∈R}, (10)

R1 ij R2 ={(u1,..,ui,..,uj,..,un)|∃u0 (u1,..,u0,..,uj,..,un)∈R1 ∧(u1,..,ui,..,u0,..,un)∈R2}.(11) Obviously, the operation (10) is an involution.

(R(ij))(ij) = R . (12)

Moreover,

R1 ij R2 = R2 ji R1. (13)

It is easy to prove that operation (11) is associative. Actually, (u1,..,ui,..,uj,..,un)∈R1 ij (R2 ij R3) ⇔ ∃u0 (u1,..,u0,..,uj,..,un)∈R1 ∧(u1,..,ui,..,u0,..,un)∈R2 ij R3 ⇔ ⇔ ∃u0 (u1,..,u0,..,uj,..,un)∈R1 ∧(∃u0′ (u1,..,u0′,..,u0,..,un)∈R2 ∧(u1,..,ui,..,u0′,..,un)∈R3) ⇔ ⇔ ∃u0′ (∃u0 (u1,..,u0,..,uj,..,un)∈R1 ∧(u1,..,u0′,..,u0,..,un)∈R2)∧(u1,..,ui,..,u0′,..,un)∈R3 ⇔ ⇔ ∃u0′ (u1,..,u0′,..,uj,..,un)∈R1 ij R2 ∧(u1,..,ui,..,u0′,..,un)∈R3 ⇔ ⇔ (u1,..,ui,..,uj,..,un)∈(R1 ij R2)ij R3.

Then we set

Iij = {(u1,..,ui,..,uj,..,un)|k ∈1..n ∧uk ∈U ∧uj = ui}∈2Un .

It is easy to see

(u1,..,ui,..,uj,..,un)∈Iij ij R ⇔ ∃u0 (u1,..,u0,..,uj,..,un)∈Iij ∧(u1,..,ui,..,u0,..,un)∈R ⇔ ⇔ ∃u0 (u1,..,ui,..,u0,..,un)∈R ∧uj = u0 ⇔ (u1,..,ui,..,uj,..,un)∈R, and similarly (u1,..,ui,..,uj,..,un)∈Rij Iij ⇔ ∃u0 (u1,..,u0,..,uj,..,un)∈R ∧(u1,..,ui,..,u0,..,un)∈Iij ⇔ ⇔ ∃u0 (u1,..,u0,..,uj,..,un)∈R ∧ui = u0 ⇔ (u1,..,ui,..,uj,..,un)∈R.

Thus, Iij ij R =Rij Iij =R. (15)

Hence we have just proved the

Lemma 1. 2Un,(ij,Iij) is a monoid.

Note that (u1,..,ui,..,uj,..,un)∈(R1 ij R2)(ij) ⇔ (u1,..,uj,..,ui,..,un)∈R1 ij R2 ⇔ ⇔ ∃u0 (u1,..,u0,..,ui,..,un)∈R1 ∧(u1,..,uj,..,u0,..,un)∈R2 ⇔ ⇔ ∃u0 (u1,..,ui,..,u0,..,un)∈R1(ij) ∧(u1,..,u0,..,uj,..,un)∈R2(ij) ⇔ ⇔ (u1,..,ui,..,uj,..,un)∈R1(ij) ji R2(ij) ⇔ (u1,..,ui,..,uj,..,un)∈R2(ij) ij R1(ij) .

In that way

(R1 ij R2)(ij) =R1(ij)ji R2(ij) =R2(ij)ij R1(ij) . (16)

Hence the bijective function f (R):= R(ij) is an isomorphism of monoids 2Un,(ij,Iij) and 2Un,(ji,Iji) .

Moreover, (u1,..,ui,..,uk,..,uj,..,un)∈(R1 ik R2)(ij) ⇔ (u1,..,uj,..,uk,..,ui,..,un)∈R1 ik R2 ⇔ ⇔ ∃u0 (u1,..,u0,..,uk,..,ui,..,un)∈R1 ∧(u1,..,uj,..,u0,..,ui,..,un)∈R2 ⇔ ⇔ ∃u0 (u1,..,ui,..,uk,..,u0,..,un)∈R1(ij) ∧(u1,..,ui,..,u0,..,uj,..,un)∈R2(ij) ⇔ ⇔ (u1,..,ui,..,uj,..,un)∈R1(ij) jk R2(ij) ⇔ (u1,..,ui,..,uj,..,un)∈R2(ij) kj R1(ij).

From which we obtain (R1 ik R2)(ij) =R1(ij)jk R2(ij) =R2(ij)kj R1(ij) . (17)

Hence we have proved the

Lemma 2. Monoids 2Un,(ik,Iik ) and 2Un,(jk,Ijk ) are isomorphic, as well as monoids 2Un,(ij,Iij) and 2Un,(ji,Iji) .

Let us set an algebraic structure 2Un,(ij,ik,Iij,Iik ) and then we can write the following logical consequences: (u1,..,ui,..,uj,..,uk,..,un)∈R1 ij (R2 ik R3) ⇔ ∃u0 (u1,..,u0,..,uj,..,uk,..,un)∈R1 ∧ ∧(u1,..,ui,..,u0,..,uk,..,un)∈R2 ik R3 ⇔ ∃u0 ∃u0′ (u1,..,u0,..,uj,..,uk,..,un)∈R1 ∧ ∧(u1,..,u0′,..,u0,..,uk,..,un)∈R2 ∧(u1,..,ui,..,u0,..,u0′,..,un)∈R3 ⇔ ∃u0′ ∃u0 (u1,..,u0,..,uj,..,uk,..,un)∈R1 ∧(u1,..,u0′,..,u0,..,uk,..,un)∈R2 ∧ ∧(u1,..,ui,..,u0,..,u0′,..,un)∈R3 ⇒ ∃u0′ (∃u0 (u1,..,u0,..,uj,..,uk,..,un)∈R1 ∧(u1,..,u0′,..,u0,..,uk,..,un)∈R2)∧ ∧(∃u0 (u1,..,ui,..,u0,..,u0′,..,un)∈R3) ⇔ ∃u0′ (u1,..,u0′,..,uj,..,uk,..,un)∈R1 ij R2 ∧ ∧(∃u0 (u1,..,ui,..,u0,..,u0′,..,un)∈R3 ∧(u1,..,u0,..,uj,..,u0′,..,un)∈1R) ⇔ ⇔ ∃u0′ (u1,..,u0′,..,uj,..,uk,..,un)∈R1 ij R2 ∧(u1,..,ui,..,uj,..,u0′,..,un)∈R3 ij 1R ⇔ ⇔ (u1,..,ui,..,uj,..,uk,..,un)∈(R1 ij R2)ik (R3 ij 1R).

This means that the following Lemma is true. Lemma 3. In an ordered algebra 2Un ,(ij ,ik , ⊆, Iij , Iik ,0R ,1R ) , the pseudo distributive law holds

R1 ij ( R2 ik R3 ) ⊆ ( R1 ij R2 ) ik ( R3 ij 1R ) .

According to [ 17 ], we use the notation 1R := U n and 0R := ∅ .

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) ⇔ ⇔ ∃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 ) ∈ Iij ⇔ R ij R(ij) ⊆ Iij . (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

Iij ⊆ 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) = Iij . (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 = Iij . (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 , Iij ) is a subgroup of the monoid 2Un ,(ij , Iij ) .

As well as binary relations, finitary relations have the following properties [ 17 ]

R1 ij ( R2 ∪ R3 ) =(R1 ij R2 ) ∪ ( R1 ij R3 ) , ( R2 ∪ R3 ) ij R1 = ( R2 ij R1 ) ∪ ( R3 ij R1 ) , R1 ij ( R2 ∩ R3 ) ⊆ ( R1 ij R2 ) ∩ ( R1 ij R3 ) , ( R2 ∩ R3 ) ij R1 ⊆ ( R2 ij R1 ) ∩ ( R3 ij R1 ) , and so we can set an algebraic structures have properties (12)-(18), (24)-(29).

Fij ,(ij , Iij ) , 2Un ,(∪,∩,ij ,ik ,(ij) , ⊆,0R ,1R , Iij , Iik ) that

3. Conclusion and examples

We have defined algebraic structures of finitary relations as a common case of well-known algebraic structures of binary relations. We have considered the algebraic structures on an underlying set 2Un and sometimes called a finitary relation R ∈ 2Un 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.

Example 1 (algebraic). Let us set U = {u0,u1,u2 ,u3}, 2U3 ,(23, I23 ) , and R = {(u1,u0,u3 ),(u1,u2 ,u0 )} . Now we can get (18) (19) (26) (27) (28) (29) (u0,u0,u0),(u0,u1,u1),(u0,u2,u2),(u0,u3,u3),   I23 = ((uu12,,uu00,,uu00)),,((uu12,,uu11,,uu11)),,((uu12,,uu22,,uu22)),,((uu12,u,u3,3u,u33)),,

, (u3,u0,u0),(u3,u1,u1),(u3,u2,u2),(u3,u3,u3) 

R23 R ={(u1,u2,u3)}, 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 χR :Un {0,1} for R∈2Un , that is defined as 1, (u1,..,un)∈R χR (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)n {false,true} for (n-uniform) hypergraph. Let f :1..m U be a total bijection and R∈2Un . We define ψkR1,..,kn =(k1,..,kn ψ R ) =true, χR ( f −1(k1),.., f −1(kn )) =1 .

 false, χR ( f −1(k1),.., f −1(kn )) = 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 Technologyand Nanotechnology" (ITNT-2019) r .

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 ψ k11,..,kn ∗ijψ k21,..,kn

m =∨ψ1

k1,..,ki−1,s,ki+1,..,k j ,..,kn ∧ψ k21,..,ki ,..,k j−1,s,k j+1,..,kn .

s=1 By our construction semigroups 2Un ,(ij ) and D(1..m)n ,(∗ij ) are isomorphic.

More interesting is the case of algebraic structures on an underlying set n∞=12Un and operations from 2Un to 2Um . For example, let us define the operations “gluing edges” g and “replacing chains” R1 g R2

={(u1,..,um−1,u2′ ,..,un′ ) | ∃u0 (u1,..,um−1,u0 ) ∈ R1 ∧ (u0,u2′ ,..,un′ ) ∈ R2}, R1 r R2

={(u1,..,ui−1,u′j+1,..,un′ ) | ∃u0∃i∃j (u1,..,ui−1,u0,..,um ) ∈ R1 ∧ (u1′,..,u0,u′j+1,..,un′ ) ∈ R2}. 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.

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