Then we study the power ofn-ary relational systems obtained as the subsystem of their direct power carried by strong homomorphisms

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Vol. 19 (2018), No. 1, pp. 483–490 DOI: 10.18514/MMN.2018.2383

ON A POWER OF n-ARY RELATIONAL SYSTEMS CARRIED BY STRONG HOMOMORPHISMS

N. PHROMMARAT Received 11 July, 2017

Abstract. First, we introduce and discuss a new operation of product ofn-ary relational systems which lies, as for generality, between their direct product and the direct product of their reflexive hulls. Then we study the power ofn-ary relational systems obtained as the subsystem of their direct power carried by strong homomorphisms. We show that, with respect to the new operation of product, the power ofn-ary relational systems studied satisfies week forms of the first and second exponential laws and, with respect to the direct product and direct sum, it satisfies the third exponential law.

2010Mathematics Subject Classification: 08A02; 08A05; 08A99

Keywords: relational systems, direct product, strong homomorphism, the first, second and third exponential laws

1. INTRODUCTION

In his pioneering papers [2] and [3], G. Birkhoff introduced and studied the operation of a cardinal power of partially ordered sets and showed the validity of the first, second and third exponential laws for the operation, i.e.,

.A^B/^CŠA^BC-the first exponential law, Q

i2IA^B_i Š.Q

i2IAi/^B-the second exponential law, Q

i2IA^Bⁱ ŠA^Pⁱ²^I^Bⁱ (ifBi,i 2I, are pair-wise disjoint) -the third exponential law.

Birkhoff’s arithmetic of ordered sets has been generalized by several authors, see, e.g., [5] and [9]. The concept of a cardinal power was extended to relational systems in [7]. In this note, we will continue the study of the arithmetic of relational system.

In [8], M. Novotn´y and J. ˇSlapal introduced and studied new operations of product and power of relational systems which satisfy the first exponential law. The new product was defined by combining the direct (i.e., cardinal) operations of sum and product, and the new power was defined as the intersection of the direct power and another power of relational systems. In [1], Frantiˇsek Bednaˇr´ık and Josef ˇSlapal

This research was supported by Chiang Rai Rajabhat University, Thailand.

c 2018 Miskolc University Press

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studied the subsystem of the cardinal power of relational systems carried by the set of strong homomorphisms.

In the present paper, we will introduce a new operation of product ofn-ary relational systems which generalizes the concept of combined product in [8]. Moreover, we will study the subsystem of the direct power carried by the set of strong homomorphisms. We will prove the validity of the third exponential law and weak forms of the first and second exponential laws.

2. COMBINED PRODUCT OFn-ARY RELATIONAL SYSTEMS

Throughout this paper, ndenotes a positive integer. By an n-ary relationp on a setAwe mean a subsetpAⁿ. The pair.A; p/is called ann-ary relational system.

LetAD.A; p/;BD.B; q/be a pair ofn-ary relational systems. A mapping f WB!Ais said to be a homomorphismof Binto A if .x1; : : : ; xn/2q implies .f .x1/; : : : ; f .xn//2p. A homomorphismf ofBD.B; q/intoAD.A; p/is said to be astrong homomorphismofBintoAif, for everyb₁; : : : ; b_n2B,

.f .b1/; : : : ; f .bn//2p implies.b1; : : : ; bn/2q. We denote by Hom.B;A/ the set of all homomorphisms ofBintoAand bySHom.B;A/the set of all strong homomorphisms ofBintoA.

Iff WB!Ais a bijection and bothf WB!Aandf ¹WA!Bare homomorphisms, thenf is called anisomorphismofBontoA. If there exists an isomorphism of Bonto A, we say thatB and Aare isomorphic, in symbolsB ŠA. An n-ary relational systemAD.A; p/is said to be asubsystemof ann-ary relational system BD.B; q/ifABandpDq\Aⁿ. Given a pair ofn-ary relational systemsA;B, we writeB4AifBcan beembedded intoA, i.e., if there is a relational subsystem A⁰ofAsuch thatBŠA⁰.

The direct productof a familyAi D.Ai; pi/; i2I;ofn-ary relational systems is then-ary relational systemQ

i2IAi D.Q

i2IAi; q/whereQ

i2IAi is the cartesian product of sets and, for any f1; : : : ; fn2 Q

i2IAi; .f1; : : : ; fn/2q if and only if .f₁.i /; : : : ; f_n.i //2p_i for eachi2I. If the setI is finite, sayID f1; : : : ; mg, then we writeA1: : :Aminstead ofQ

i2IAi. IfAi DAfor eachi2I, then we write A^I instead ofQ

i2IAi.

Thedirect sumof a familyAi D.Ai; pi/; i2I;ofn-ary relational systems is the n-ary relational system P

i2IAi D.S

i2IA_i;S

i2Ip_i/. If the set I is finite, say I D f1; : : : ; mg, then we writeA1]: : :]Aminstead ofP

i2IAi.

Ann-ary relational systemAD.A; p/is said to bereflexiveprovided that .a1; a2; : : : ; an/2 p whenever a1 Da2 D: : :Dan. Let AD.A; p/ be an n-ary relational system. We denote bypN the n-ary relational system onG such that, for everya1; : : : ; an2A,.a1; : : : ; an/2 Npif and only ifa1D: : :Danor.a1; : : : ; an/2p.

Then-ary relational systems.A;p/N is called thereflexive hullofAand is denoted by A:N

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LetAi D.Ai; pi/; i 2I; be a family ofn-ary relational systems. Thecombined productof the familyAi; i2I, is then-ary relational systemN

i2IAiD.Q

i2IAi; r/

given byN

i2IAi DP

i2I

Q

j2IAij where Aij D

ANj ifiDj;

Aj ifi¤j:

Thus, for any.aⁱ₁Ii2I /; : : : ; .aⁱ_nIi2I /2Q

i2IAiwe have.a₁ⁱIi2I /; : : : ; .a_nⁱIi2 I /2rif and only if there exists a subsetJ I;cardJ1;such that.a₁ⁱ; : : : ; a_nⁱ/2pi

for everyi2IXJ anda₁ⁱ Daⁱ₂D: : :Da_nⁱ for everyi2J:

If the set I is finite, say I D f1; : : : ; mg, we write A1˝: : :˝Am instead of N

i2IAi. We then clearly haveA1˝: : :˝AmD.A11A12: : :A1m/].A21 A22: : :A2m/]: : :].Am1Am2: : : ;Amm/D.AN1A2: : :Am/].A1AN2 : : :Am/]: : :].A1A2: : : NAm/.

In particular, ifI D f1; 2g, then, for any.a1; b1/; : : : ; .an; bn/2A1A2,

..a1; b1/; : : : ; .an; bn//2rif and only if one of the following three conditions is satisfied:

(i) .a1; : : : ; an/2p1and.b1; : : : ; bn/2p2; (ii) a1Da2D: : :Danand.b1; : : : ; bn/2p2; (iii) .a1; : : : ; an/2p1andb1Db2D: : :Dbn.

Example1. LetAD.A; p/;BD.B; q/be binary relational systems whereAD f1; 2g; BD fa; bg; pD f.1; 2/gandqD f.a; b/g. ThenA˝BD.AB; r/whererD f..1; a/; .2; b//; ..1; a/; .1; b//; ..2; a/; .2; b//; ..1; a/; .2; a//; ..1; b/; .2;

b//g.

Example2. LetAD.A; p/;BD.B; q/be binary relational systems whereAD f1; 2g; BD fag; pD f.1; 2/gandqD f.a; a/g. ThenA˝BD.AB; r/whererD f..1; a/; .2; a//; ..1; a/; .1; a//; ..2; a/; .2; a//g. Thus,A˝Bis reflexive.

Remark1. LetAi; i 2I;be a family ofn-ary relational systems. IfAi is reflexive for every i 2I, thenQ

i2IAi DN

i2IAi. If at most one of the n-ary relational systemsAi; i2I;is not reflexive, thenN

i2IGi is reflexive.

First, we will show that the combined product ofn-ary relational systems distrib- utes over their direct sum.

Proposition 1. LetAD.A; p/andBiD.B_i; q_i/,i2I, ben-ary relational systems. ThenA˝P

i2IBi DP

i2I.A˝Bi/:

Proof. LetP

i2IBi D.S

i2IBi; s/;A˝Bi D.ABi; ri/ for eachi 2I, A˝ P

i2IBiD.AS

i2IBi; u/andP

i2I.A˝Bi/D.S

i2I.ABi/; v/:We will show that, for every

.a1; b1/; : : : ; .an; bn/2A[

i2I

BiD[

i2I

.ABi/; ..a1; b1/; : : : ; .an; bn//2u

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if and only if ..a1; b1/; : : : ; .an; bn//2v . It is easy to see that the following five conditions are equivalent:

(1) ..a1; b1/; : : : ; .an; bn//2u;

(2) one of the following three cases occurs:

(i) .a1; : : : ; an/2pand.b1; : : : ; bn/2s;

(ii) a1Da2D: : :Danand.b1; : : : ; bn/2s;

(iii) .a1; : : : ; an/2pandb1Db2D: : :Dbn; (3) one of the following three cases occurs:

(i) .a1; : : : ; an/2pand.b1; : : : ; bn/2qi for somei2I;

(ii) a1Da2D: : :Danand.b1; : : : ; bn/2qi for somei2I;

(iii) .a1; : : : ; an/2pandb1Db2D: : :Dbn; (4) ..a1; b1/; : : : ; .an; bn//2ri for somei2I; (5) ..a1; b1/; : : : ; .an; bn//2v.

This proves the statement.

3. POWER OF RELATIONAL SYSTEMS WITH RESPECT TO STRONG

HOMOMORPHISMS

Definition 1. LetBD.B; q/andAD.A; p/ben-ary relational systems. Thedir- ect powerofAandBis then-ary relational system.Hom.B;A/; r/where, for any f1; : : : ; fn 2 Hom.B;A/, .f1; : : : ; fn/ 2 r if and only if .f1.x/; : : : ; fn.x//2p for each x2B:The subsystem.SHom.B;A/; r\.SHom.B;A//ⁿ/of the direct power.Hom.B;A/; r/ofAandBis called thedirect power of AandB with respect to strong homomorphisms.

For any pair ofn-ary relational systemsA,B, we denote byA^BandA^˘^Bthe direct power and the direct power with respect to strong homomorphisms, respectively, of AandB.

It is easy to see that the direct powerA^B ofn-ary relational systems is reflexive wheneverAis reflexive.

Example3. LetAD.A; p/;BD.B; q/be binary relational systems whereAD f1; 2g; B D fag; pDAA and q D f.a; a/g. Assume that f W A!B is a map given byf .1/Df .2/Da. Then,Hom.A;B/DSHom.A;B/D ffgandA^˘BD .SHom.B;A/; s/wheresD f.f; f /g.

Example4. LetAD.A; p/;BD.B; q/be binary relational systems whereAD f1; 2; 3g; BD fa; bg; pD f.1; 1/; .2; 2/; .3; 3/; .1; 2/; .2; 1/; .1; 3/; .3; 1/; .2; 3/gandqD BB. Then, every map ofAintoB is a homomorphism ofAintoB. Assume that f; g WA!B are maps given by f .1/Df .2/Da; f .3/Db andg.1/Dg.3/D a; g.2/Db. Then,SHom.A;B/D ff; ggandA^˘BD.SHom.B;A/; s/wheresD

SHom.A;B/SHom.A;B/.

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We will show now that, for the combined product of n-ary relational systems and their power with respect to strong homomorphisms, weak forms of the first and second exponential laws hold.

Theorem 1. LetA;B;Cben-ary relational systems. Then

A^˘^.B˝C/4.A^˘B/^˘C:

Proof. LetAD.A; q/;BD.B; p/;CD.C; s/;A^˘BD.SHom.B;A/; r/;

B˝AD.BA; v/; .A^˘B/^˘CD.SHom.C; SHom.B;A//; t /andA^˘^.B˝C/D .SHom.B˝C;A/; u/ben-ary relational systems.

We define a map˛WSHom.B˝C;A/!SHom.C; SHom.B;A//by˛.h/.c/.b/D h.b; c/wheneverh2SHom.B˝C;A/,c2C andb2B.

Leth2SHom.B˝C;A/,c2C and.b1; : : : ; bn/2q. Consequently, ..b1; c/; : : : ; .bn; c//2v. Sincehis a homomorphism, we have.h.b1; c/; : : : ;

h.bn; c//2p. Thus,.˛.h/.c/.b1/; : : : ; ˛.h/.c/.bn//D.h.b1; c/; : : : ; h.bn; c//2p.

Therefore,˛.h/.c/2Hom.B;A/for eachc2C.

Letb1; : : : ; bn2Band.˛.h/.c/.b1/; : : : ; ˛.h/.c/.bn//2p:Then.˛.h/.c/.b1/;

: : : ; ˛.h/.c/.bn//D.h.b1; c/; : : : ; h.bn; c//2p. Sincehis a strong homomorphism, we get..b1; c/; : : : ; .bn; c//2v. Thus,.b1; : : : ; bn/2q:Then,˛.h/.c/2SHom.B;A/

for eachc2C.

Next, let .c1; : : : ; cn/2s. Then ..b; c1/; : : : ; .b; cn//2v. Since h is a homomorphism, we get .h.b; c1/; : : : ; h.b; cn//2 p. So .˛.h/.b/.c1/; : : : ; ˛.h/.b/.cn//

D.h.b; c1/; : : : ; h.b; cn//2p. Consequently,.˛.h/.c1/; : : : ; ˛.h/.cn//2r. Hence,

˛.h/2Hom.C; Hom.B;A//.

Letc1; : : : ; cn2C and.˛.h/.c1/; : : : ; ˛.h/.cn//2r:Then.˛.h/.c1/b; : : : ;

˛.h/.cn/b/2p, hence.˛.h/.b/.c1/; : : : ; ˛.h/.b/.cn//D.h.b; c1/; : : : ; h.b; cn//2p:

Now, ashis a strong homomorphism, it follows that..b; c1/; : : : ; .b; cn//2v:Hence .c1; : : : ; cn/2s. Therefore,˛.h/2SHom.C; Hom.B;A//.

We will show that˛ is a homomorphisms. Let .h1; : : : ; hn/2u,.c1; : : : ; cn/2s and .b1; : : : ; bn/2q. Then we have..b1; c1/; : : : ; .bn; cn//2v and, consequently, .h1.b1; c1/; : : : ; hn.bn; cn// 2 p. Therefore, .˛.h1/.c1/.b1/; : : : ;

˛.hn/.cn/.bn// D .h1.b1; c1/; : : : ; hn.bn; cn// 2 p: Hence, .˛.h1/.c1/; : : : ;

˛.hn/.cn//2r, so that.˛.h1/; : : : ; ˛.hn//2t. Thus,˛is a homomorphism.

Finally, we will show that˛is an injection. Let˛.h1/D˛.h2/for a pairh1; h22 SHom.B˝C;A/:Then˛.h1/cD˛.h2/cfor everyc2C and we have˛.h1/.c/bD

˛.h2/.c/bfor everyc 2C andb2B. Henceh1.b; c/Dh2.b; c/for every.b; c/2 BC:Thus,h1Dh2and we have shown that˛ is an injection.

Therefore˛is an embedding ofA^˘^.B^˝^C/into.A^˘^B/^˘^Cand the proof is complete.

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Theorem 2. LetAi D.Ai; pi/; i2I;be a family ofn-ary relational systems and Bbe ann-ary relational system. Then

O

i2I

.A^˘B_i /4.O

i2I

Ai/^˘B: Proof. LetAi D.Ai; pi/for everyi 2I,BD.B; q/, N

i2IAi D.Q

i2IAi; r/, A^˘_i^B D.SHom.B;Ai/; ui/ for every i 2I, N

i2IAi˘B D.Q

i2ISHom.B;Ai/;

s/and.N

i2IAi/^˘BD.Hom.B;N

i2IAi/; t /.

We define a map˛WQ

i2ISHom.B;Ai/!.Q

i2IAi/^H by˛.fⁱI i2I /.b/D .fⁱ.b/I i2I /for eachb2B.

Let.fⁱIi2I /2Q

i2ISHom.B;Ai/and.b1; : : : ; bn/2q. Sincefⁱ2Hom.B;Ai/, we have.fⁱ.b1/; : : : ; fⁱ.bn//2pi for everyi2I. Then..fⁱ.b1/Ii

2I /; : : : ; .fⁱ.bn/Ii2I //2rand we have.˛.fⁱIi2I /.b1/; : : : ; ˛.fⁱIi2I /.bn//2 r. Therefore,˛.fⁱI i2I /2Hom.B;N

i2IAi/.

Let.fⁱI i2I /2Q

i2ISHom.B;Ai/and

.˛.fⁱI i2I /.b1/; : : : ; ˛.fⁱIi 2I /.bn//2r:

Consequently,..fⁱ.b1/Ii2I /; : : : ; .fⁱ.bn/Ii2I //2r. Therefore,

.fⁱ.b1/; : : : ; fⁱ.bn//2pifor everyi2I. Asfⁱis a strong homomorphism for every i2I, we have.b1; : : : ; bn/2q. We have shown that˛mapsQ

i2ISHom.B;Ai/into

SHom.B;N

i2IAi/.

We will show that ˛ is a homomorphism. Let .f₁ⁱI i 2I /; : : : ; .f_nⁱI i 2 I /2 Q

i2ISHom.B;Ai/and..f₁ⁱIi 2I /; : : : ; .f_nⁱIi 2I //2s: LetJ I, card J 1.

Consequently, .f₁ⁱ; : : : ; f_nⁱ/2ui for every i 2IXJ andf₁ⁱ Df₂ⁱ D: : :Df_nⁱ for every i 2J: Hence, .f₁ⁱ.b/; : : : ; f_nⁱ.b//2 pi for every i 2 IXJ and every b 2 B and f₁ⁱ.b/Df₂ⁱ.b/D: : :Df_nⁱ.b/ for every i 2J and every b 2B. Hence, ..f₁ⁱ.b/I i 2I /; : : : ; .f_nⁱ.b/I i 2I //2r for every b 2B. Therefore, .˛.f₁ⁱI i 2 I /.b/; : : : ; ˛.f_nⁱIi2I /.b//2rfor everyb2Band then.˛.f₁ⁱIi2I /; : : : ; ˛.f_nⁱIi2 I //2t. Thus,˛is a homomorphism ofN

i2I.A^˘B_i /into.N

i2IAi/^˘B:

Suppose that ˛.fⁱI i 2 I /D˛.gⁱI i 2 I / where .fⁱI i 2I /; .gⁱI i 2I / 2 Q

i2ISHom.B;Ai/:Then.fⁱ.b/I i 2I /D˛.fⁱI i2I /.b/D˛.gⁱI i 2I /.b/D .gⁱ.b/I i 2I / for every b 2 B. Therefore, fⁱ.b/Dgⁱ.b/ for every i 2 I and every b 2 B: Hence, fⁱ Dgⁱ for every i 2I: Thus, ˛: Q

i2ISHom.B;Ai/!

SHom.B;N

i2IAi/is an injection. We have shown thatN

i2I.A^˘_i^B/4.N

i2IAi/^˘^B: Next, we will show that, for the powers ofn-ary relational systems with respect to strong homomorphisms, the third exponential law holds.

Theorem 3. LetAbe ann-ary relational system and letBi,i2I, be a family of pair-wise disjointn-ary relational systems. Then

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Y

i2I

A^˘Bⁱ ŠA^˘^Pⁱ²^I^Bⁱ: Proof. LetAD.A; p/, Bi D.Bi; qi/for everyi 2I,P

i2IBi D.P

i2IBi; v/, A^˘Bⁱ D.SHom.Bi;A/; ri/ for every i 2 I, Q

i2IA^˘Bⁱ D .Q

i2ISHom.Bi;A/;

t /andA^˘^Pⁱ²^I^Bⁱ D.SHom.P

i2IBi;A/; u/:

We define a map 'WQ

i2ISHom.Bi;A/!SHom.P

i2IBi;A/ by '.fⁱI i 2 I /Dhwhere h.t /Dfⁱ.t /whenevert2Hi (andi 2I /: Let.b1; : : : ; bn/2qi for any i 2I. Since fi is a homomorphism, we have .'.fⁱI i 2I /b1; : : : ; .fⁱI i 2 I /bn/D.h.b1/; : : : ; .h.bn//D.fⁱ.b1/; : : : ; .fⁱ.bn//

2rfor anyi2I. Then'.fⁱIi 2I /2Hom.P

i2IBi; A/:

Let.'.fⁱIi 2I /b1; : : : ; .fⁱI i2I /bn/for everyb1; : : : ; bn2B:Thus,

..fⁱ.b₁/; : : : ; .fⁱ.b_n//D.h.b₁/; : : : ; .h.b_n//D'.fⁱIi2I /b₁; : : : ; .fⁱIi2I /b_n/2 r for anyi2I. Sincefⁱ is a strong homomorphism, we have.b1; : : : ; bn/2qi for anyi2I. Therefore,'.fⁱI i2I /2SHom.P

i2IBi; A/.

Further, we define a map˛WSHom.P

i2IBi;A/!Q

i2ISHom.Bi;A/by

˛.h/ D .fⁱI i 2 I /wheneverh 2 SHom.P

i2IBi;A/wherefⁱ D hjBi

for everyi 2I: It is easy to see that fi is a strong homomorphism of Bi into A for everyi2I. It follows that˛.h/2Q

i2ISHom.Bi;A/.

We will show that both ' and ˛ are homomorphisms. Let .f₁ⁱI i 2 I /; : : : ; .f_nⁱI i 2I / 2Q

i2ISHom.Bi;A/ and ..f₁ⁱIi 2 I /; : : : ; .f_nⁱIi 2I // 2T. Con- sequently,.f₁ⁱ; : : : ; f_nⁱ/2ri for eachi2I. Then

.f₁ⁱ.b/; : : : ; f_nⁱ.b//D.h1.b/; : : : ; hn.b//2p for each i 2I andb 2P

i2IBi. Therefore, .'.f₁ⁱIi 2I /; : : : ; '.f_nⁱIi 2I //2u.

Thus,'is a homomorphism.

Let h1; : : : ; hn 2 SHom.P

i2IBi;A/ and .h1; : : : ; hn/ 2 U. Then .h1jBi.b/;

: : : ; hnj^Bi.b//D.f₁ⁱ.b/; : : : ; f_nⁱ.b//2p for anyi 2I andb2P

i2IBi. Therefore, .f₁ⁱ; : : : ; f_nⁱ/2ri. Thus,..f₁ⁱIi2I /; : : : ; .f_nⁱIi2I //2T and then

.˛.h1/; : : : ; ˛.hn//2t.

It follows that'is an isomorphism with' ¹D˛.

Remark2. Ann-ary hyperalgebrais an.nC1/-ary relational systemGD.G; p/

such that, for anyx1; : : : ; xn2G;there existsy2Gwith.x1; :::; xn; y/2p(cf. [10]).

The.nC1/-ary relationp onGmay then be considered to be a mappWGⁿ!exp G and it is called ann-ary hyperoperationonG; we writey2p.x₁; : : : ; x_n/instead of.x1; : : : ; xn; y/2p. If suchyis unique wheneverx1; : : : ; xn2G, then the.nC1/- ary relational systemG is nothing but ann-ary algebra (cf. [6]). The .nC1/-ary relationp may then be considered to be a mappWGⁿ!G and it is called ann-ary operationon G; we writeyDp.x1; : : : ; xn/instead of.x1; : : : ; xn; y/2p. Powers of hyperalgebras were studied in [10] and those of algebras in [4]. Of course, if

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A;B aren-ary algebras (hyperalgebras), then the power A^˘B need not be ann-ary algebra (hyperalgebra). So, it is an open problem to find conditions under which the three exponential laws or at least their week forms are satisfied for n-ary algebras (hyperalgebras).

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Author’s address

N. Phrommarat

Department of Mathematics, Faculty of Science and Technology, Chiang Rai Rajabhat University, 57100 Chiang Rai, Thailand

E-mail address:nitimachaisansuk@gmail.com