On powers of relational and algebraic systems

Acta Math. Hungar., 139 (3) (2013), 195–207DOI: 10.1007/s10474-012-0257-9

First published online September 5, 2012

ON POWERS OF RELATIONAL ANDALGEBRAIC SYSTEMS

N. CHAISANSUK1 and S. LEERATANAVALEE2,∗

1Department of Mathematics, Chiang Mai University, Faculty of Science, Chiang Mai, Thailande-mail: [email protected]

2Materials Science Research Center, Chiang Mai University, Faculty of Science, Chiang Mai,Thailand

e-mail: [email protected]

(Received January 5, 2012; revised March 7, 2012; accepted March 8, 2012)

Abstract. We extend the well-known Birkhoff’s operation of cardinal powerfrom partially ordered sets onto n-ary relational systems. The extended power isthen studied not only for n-ary relational systems but also for some of their spe-cial cases, namely partial algebras and total algebras. It turns out that a conceptof diagonality plays an important role when studying the powers.

1. Introduction

In the pioneering paper [1], G. Birkhoff introduced the operation of a car-dinal power of partially ordered sets and showed that it behaves analogouslyto the power of natural numbers. In particular, he proved that, given par-tially ordered sets G, H and K, the following three laws are fulfilled (× and∏

denote the direct product,∑

the direct sum and ∼= the isomorphism ofpartially ordered sets, respectively):

(GH

)K ∼= GH×K – the first exponential law,∏

i∈I GHi

∼= (∏

i∈I Gi)H – the second exponential law,

∏i∈I GH

i∼= G

P

i∈I Hi (if Hi, i ∈ I , are pair-wise disjoint) – the third ex-ponential law.

∗ Corresponding author.This research was supported by the Royal Golden Jubilee Ph.D. Program of the Thailand ResearchFund., the Graduate School and the Faculty of Science, Chiang Mai University, Thailand. Thecorresponding author thanks the National Research University Project under Thailand’s Officeof the Higher Education Commission for financial support. The authors would like to thank theanonymous referee for many useful comments and suggestions.

Key words and phrases: relational system, partial algebra, total algebra, diagonality, medi-ality, the first exponential law.

Mathematics Subject Classification: 08A02, 08A55.

0236-5294/$20.00 c© 2012 Akademiai Kiado, Budapest, Hungary

mailto:[email protected]

mailto:[email protected]

196 N. CHAISANSUK and S. LEERATANAVALEE

In [6], V. Novak extended the concept of direct (i.e., cardinal) powerfrom partially ordered sets onto n-ary relational systems. The direct powerof n-ary relational systems was then studied by J. Slapal in [10] (cf. also[12] and [9]) and, in [11], the same author studied the direct power of n-aryalgebras.

The aim of the present note is to continue the study of powers of n-ary re-lational systems and to improve and complete the results from [10] and [11].We will introduce and study a power of n-ary relational systems which gen-eralizes the direct power investigated in [10]. The power introduced will beshown to behave better than the direct power from [10] and sufficient con-ditions under which both powers coincide will be found. We will especiallyfocus on investigating powers of n-ary partial algebras which will be viewedas special (n + 1)-ary relational systems. Powers of n-ary algebras will bedealt with, too.

2. Relational systems

We recall first some definitions of basic concepts used throughout thispaper. Let n be a positive integer. A pair G = (G, p) is called an n-aryrelational system if G is a set, the so-called underlying set of G, and p is ann-ary relation on G, i.e., p � Gn (= G × G × . . . × G︸︷︷︸

n-times

). An n-ary relational

system G is said to be reflexive provided that (x1, x2, . . . , xn) ∈ p wheneverx1 = x2 = · · · = xn.

Let H = (H, q) and G = (G, p) be a pair of n-ary relational systems.A map f : H → G is called a homomorphism of H into G provided that(x1, x2, . . . , xn) ∈ q implies

(f(x1), f(x2), . . . , f(xn)

)∈ p. If f : H → G is a

bijection and both f : H → G and f −1 : G → H are homomorphisms, thenf is called an isomorphism of H onto G. If there exists an isomorphismof H onto G, then we write G ∼= H . We denote by Hom (H,G) the set ofall homomorphisms of H into G.

Given a pair G = (G,p) and H = (H, q) of n-ary relational systems, G issaid to be a relational subsystem of H provided that G � H and p = q ∩ Gn.

The direct product of a family Gi = (Gi, pi), i ∈ I of n-ary relational sys-tems is the n-ary relational system

∏i∈I Gi = (

∏i∈I Gi, q) where

∏i∈I Gi

denotes the cartesian product and, for any f1, . . . , fn ∈∏

i∈I Gi, (f1, . . . , fn)∈ q if and only if

(f1(i), . . . , fn(i)

)∈ pi for each i ∈ I . If Gi = G for each

i ∈ I , then we write GI instead of∏

i∈I Gi.

Definition 2.1. Let H = (H, q) and G = (G, p) be n-ary relationalsystems. The power of G and H is the n-ary relational system GH =(Hom (H,G), r

)where, for any f1, . . . , fn ∈ Hom (H,G), (f1, . . . , fn) ∈ r

Acta Mathematica Hungarica 139, 2013

ON POWERS OF RELATIONAL AND ALGEBRAIC SYSTEMS 197

if and only if (x1, . . . , xn) ∈ q implies(f1(x1), . . . , fn(xn)

)∈ p whenever

x1, . . . , xn ∈ H .

Remark 2.2. Let H = (H, q) and G = (G, p) be n-ary relational sys-tems.

(a) In [10], the direct power of G and H is defined to be the relationalsubsystem of the direct product GH whose underlying set is Hom (H,G),i.e., the n-ary relational system

(Hom (H,G), r

)where, for any f1, . . . , fn

∈ Hom(H,G), (f1, . . . , fn) ∈ r if and only if(f1(x), . . . , fn(x)

)∈ p for every

x ∈ H .(b) It may easily be seen that the power GH is always reflexive.

The power of n-ary relational systems does not fulfill the first exponen-tial law in general. The following statement gives a sufficient condition forthe validity of that law.

Theorem 2.3. Let G, H, K be n-ary relational systems. If K is re-flexive, then

(GH

)K ∼= GH×K.

Proof. Let H = (H, q), G = (G, p), K = (K, s), GH =(Hom(H,G),

R), H × K = (H × K,V ),

(GH

)K = (Hom(K,Hom (H,G)

), T ) and

GH×K =(Hom (H × K,G), U

)be n-ary relational systems. Define a

map ϕ : Hom(K,Hom(H,G)

)→ Hom(H × K,G) by ϕ(g)(y, z) = g(z)(y)

whenever g ∈ Hom(K,Hom (H,G)

), z ∈ K and y ∈ H . We will show

first that the map ϕ is well defined. Let g ∈ (Hom(K,Hom (H,G)

)

and((y1, z1), . . . , (yn, zn)

)∈ V . Then (y1, . . . , yn) ∈ q and (z1, . . . , zn) ∈ s.

Since(GH

)K is reflexive, we get (g, g, . . . , g︸︷︷︸

n-terms

) ∈ T . As (z1, . . . , zn) ∈ s, we

have(g(z1), . . . , g(zn)

)∈ R. Thus, (y1, . . . , yn) ∈ q implies

(g(z1)(y1), . . . ,

g(zn)(yn))

∈ p. Therefore,(ϕ(g)(y1, z1), . . . , ϕ(g)(yn, zn)

)=

(g(z1)(y1), . . . ,

g(zn)(yn))

∈ p. We have shown ϕ(g) ∈ Hom (H × K,G), so that ϕ is welldefined.

Further, we define a map α : Hom(H × K,G) → Hom(K,Hom(H,G)

)

by α(h)(z)(y) = h(y, z) whenever h ∈ Hom (H × K,G), z ∈ K and y ∈ H .We will show that α is well defined, too. Let h ∈ Hom (H × K,G),z ∈ K and (y1, . . . , yn) ∈ q. Since K is reflexive, we get (z, z, . . . , z

︸︷︷︸n-times

) ∈ s.

Consequently,((y1, z), . . . , (yn, z)

)∈ V . As h is a homomorphism, we

have(h(y1, z), . . . , h(yn, z)

)∈ p. Thus,

(α(h)(z)(y1), . . . , α(h)(z)(yn)

)=(

h(y1, z), . . . , h(yn, z))

∈ p. Therefore, α(h)(z) ∈ Hom (H,G) for each



z ∈ K. Further, let (z1, . . . , zn) ∈ s and (y1, . . . , yn) ∈ q. Then((y1, z1), . . . ,

(yn, zn))

∈ V and, since h is a homomorphism, we get(h(y1, z1), . . . ,

h(yn, zn))

∈ p. So

(α(h)(z1)(y1), . . . , α(h)(zn)(yn)

)=

(h(y1, z1), . . . , h(yn, zn)

)∈ p.

Consequently,(α(h)(z1), . . . , α(h)(zn)

)∈ R. Hence,

α(h) ∈ Hom(K,Hom (H,G)

),

so that α is well defined.We have ϕ · α(h)(y, z) = α(h)(z)(y) = h(y, z) and α · ϕ(g)(z)(y) =

ϕ(g)(y, z) = g(z)(y) for all h ∈ Hom(H × K,G), g ∈ Hom(K,Hom(H,G)

),

y ∈ H and z ∈ K.Finally, we will show that both ϕ and α are homomorphisms. Let

(g1, . . . , gn) ∈ T and((y1, z1), . . . , (yn, zn)

)∈ V . Then (y1, . . . , yn) ∈ q and

(z1, . . . , zn) ∈ s. Hence,(g1(z1), . . . , gn(zn)

)∈ R and, consequently,(

g1(z1)(y1), . . . , gn(zn)(yn))

∈ p. Therefore,

(ϕ(g1)(y1, z1), . . . , ϕ(gn)(yn, zn)

)=

(g1(z1)(y1), . . . , gn(zn)(yn)

)∈ p.

Thus, ϕ is a homomorphism.Let (h1, . . . , hn) ∈ U , (z1, . . . , zn) ∈ s and (y1, . . . , yn) ∈ q. Then we have(

(y1, z1), . . . , (yn, zn))

∈ V and, consequently,(h1(y1, z1), . . . , hn(yn, zn)

)

∈ p. Therefore,(α(h1)(z1)(y1), . . . , α(hn)(zn)(yn)

)=

(h1(y1, z1), . . . , hn(yn, zn)

)∈ p.

Hence,(α(h1)(z1), . . . , α(hn)(zn)

)∈ R, so that

(α(h1), . . . , α(hn)

)∈ T .

Thus, α is a homomorphism.Therefore, ϕ : Hom

(K,Hom(H,G)

)→ Hom(H × K,G) is an isomor-

phism (with ϕ−1 = α), i.e.,(GH

)K ∼= GH×K. �

Proposition 2.4. Let Gi, i ∈ I , be a family of n-ary relational systemsand let H be an n-ary relational system. Then

∏

i∈I

GHi

∼=( ∏

i∈I

Gi

)H

.

Proof. Let H = (H, q) and Gi = (Gi, pi) for every i ∈ I . We define amap ϕ : Hom (H,

∏i∈I Gi) →

∏i∈I Hom(H,Gi) by ϕ(f) = (pri · f ; i ∈ I)



whenever f ∈ Hom (H,∏

i∈I Gi) where pri :∏

i∈I Gi → Gi is the i-th pro-jection for every i ∈ I . Further, we define a map α :

∏i∈I Hom (H,Gi)

→ Hom (H,∏

i∈I Gi) by α((fi; i ∈ I)

)= 〈fi; i ∈ I〉 where 〈fi; i ∈ I〉(z)

=(fi(z); i ∈ I

)for each z ∈ H . It may easily be seen that ϕ is an isomor-

phism with ϕ−1 = α. �Let Hi = (Hi, qi), i ∈ I be a family of pair-wise disjoint n-ary relational

systems (i.e., n-ary relational systems with Hi ∩ Hj = ∅ whenever i, j ∈ I ,i = j). The direct sum of the family Hi, i ∈ I is the n-ary relational system∑

i∈I Hi given by∑

i∈I

Hi =( ⋃

Hi,⋃

qi

)

.

Proposition 2.5. Let G be an n-ary relational system and let Hi, i ∈ I ,be a family of pair-wise disjoint n-ary relational systems. Then

∏

i∈I

GHi

∼= GP

i∈I Hi .

Proof. Let G = (G, p) and Hi = (Hi, qi) for every i ∈ I . Define themap ϕ :

∏i∈I Hom (Hi,G) → Hom (

∑i∈I Hi,G) by

ϕ(fi; i ∈ I) = h where h(t) = fi(t) if t ∈ Hi (and i ∈ I).

Further, define the map α : Hom (∑

i∈I Hi,G) →∏

i∈I Hom (Hi,G) by

α(h) = (fi; i ∈ I) whenever h ∈ Hom( ∑

i∈I

Hi,G)

where fi = h|Hi

for every i ∈ I . It may easily be seen that ϕ is an isomorphism with ϕ−1 = α.�

Definition 2.6. An n-ary relational system G = (G, p) is said tobe diagonal provided that, whenever (xij) is an n × n-matrix over G,from (x1j , . . . , xnj) ∈ p for each j = 1, . . . , n and (xi1, . . . , xin) ∈ p for eachi = 1, . . . , n it follows that (x11, . . . , xnn) ∈ p.

Theorem 2.7. Let G,H be n-ary relational systems. If G is diagonaland H is reflexive, then the power GH is a diagonal relational subsystem ofthe direct product GH .

Proof. Let GH =(Hom (H,G), T

), G = (G, p), H = (H, q) and let

(fij) be an n × n-matrix over Hom (H,G) such that (fi1, . . . , fin) ∈ T foreach i = 1, . . . , n and (f1j , . . . , fnj) ∈ T for each j = 1, . . . , n. We will



show that (f11, . . . , fnn) ∈ T . Let (y1, . . . , yn) ∈ q. As (fi1, . . . , fin) ∈ Tfor each i = 1, . . . , n and (f1j , . . . , fnj) ∈ T for each j = 1, . . . , n, we have(fi1(y1), . . . , fin(yn)

)∈ p for each i = 1, . . . , n and

(f1j(y1), . . . , fnj(yn)

)∈ p

for each j = 1, . . . , n. Since G is diagonal, we get(f11(y1), . . . , fnn(yn)

)∈ p.

This yields (f11, . . . , fnn) ∈ T . Hence, GH is diagonal.To show that GH is a relational subsystem of the direct product GH , let

GH = (GH ,R). Then R = {(f1, . . . , fn) ∈ (GH)n;(f1(x), . . . , fn(x)

)∈ p for

each x ∈ H }. We will show that T = R ∩(Hom (H,G)

)n. Let (f1, . . . , fn)∈ T . Since H is reflexive, we get

(f1(x), . . . , fn(x)

)∈ p for each x ∈ H .

Therefore, (f1, . . . , fn) ∈ R. Consequently, T � R ∩(Hom (H,G)

)n. Let(g1, . . . , gn) ∈ R ∩

(Hom(H,G)

)n and (y1, . . . , yn) ∈ q. Then(g1(yi), . . . ,

gn(yi))

∈ p for each i = 1, . . . , n. Since gj is a homomorphism, we get(gj(y1), . . . , gj(yn)

)∈ p for each j = 1, . . . , n. Thus, the diagonality of G

implies(g1(y1), . . . , gn(yn)

)∈ p. Therefore, (g1, . . . , gn) ∈ T . �

3. Partial algebras

An n-ary partial algebra is an (n + 1)-ary relational system G = (G, p)such that (x1, . . . , xn, y) ∈ p and (x1, . . . , xn, z) ∈ p imply y = z (cf. [2]). The(n + 1)-ary relation p is then called an n-ary partial operation on G (notethat p : D → G is a map where D � Gn is a subset, the so-called domainof p) and we write y = p(x1, . . . , xn) instead of (x1, . . . , xn, y) ∈ p.

Let H = (H, q) and G = (G, p) be a pair of n-ary partial algebras.Clearly, a map f : H → G is a homomorphism of H into G if and onlyif q(x1, x2, . . . , xn) = x implies p

(f(x1), f(x2), . . . , f(xn)

)= f(x) whenever

x1, x2, . . . , xn, x ∈ H .Given a pair G = (G, p) and H = (H, q) of n-ary partial algebras,

G is called a partial subalgebra of H provided that G � H and, wheneverx1, . . . , xn ∈ G and x ∈ H , p(x1, . . . , xn) = x ⇔ q(x1, . . . , xn) = x (i.e., pro-vided that G is a relational subsystem of H and, whenever x1, . . . , xn ∈ Gand x ∈ H , from q(x1, . . . , xn) = x it follows that x ∈ G).

Clearly, the direct product of a family Gi = (Gi, pi), i ∈ I of n-ary partialalgebras is the n-ary partial algebra

∏i∈I Gi = (

∏i∈I Gi, q) where, for any

f1, . . . , fn, f ∈∏

i∈I Gi, q(f1, . . . , fn) = f if and only if pi

(f1(i), . . . , fn(i)

)

= f(i) for each i ∈ I . It is also obvious that the direct sum of a pair of n-arypartial algebras is an n-ary partial algebra.

An n-ary partial algebra G = (G, p) is said to be idempotent if G isreflexive, i.e., if p(x, x, . . . , x

︸︷︷︸n-times

) = x whenever x ∈ G.

Of course, if G, H are n-ary partial algebras, then the power GH is notnecessarily an n-ary partial algebra – see the following example:



Example 3.1. Let H = (H, q) and G = (G,p) be ternary relational sys-tems where H = {1, 2}, G = {a, b}, q =

{(1, 1, 2), (1, 2, 2), (2, 1, 2), (2, 2, 2)

}

and p ={

(a, a, a), (a, b, a), (b, a, a), (b, b, a)}

. Clearly, both H and G arebinary partial algebras. Let GH =

(Hom (H,G), r

). If f : H → G is

a map given by f(1) = a, f(2) = a and g : H → G is a map given byg(1) = b, g(2) = a, then f, g ∈ Hom (H,G). It may easily be seen that{

(f, f, f), (f, f, g)}

� r but f = g, so that GH is not a binary partial al-gebra. �

The next theorem gives a sufficient condition for the power GH of n-arypartial algebras to be an n-ary partial algebra.

Theorem 3.2. Let G, H be n-ary partial algebras. If H is idempotent,then the power GH is an n-ary partial algebra.

Proof. Let H = (H, q), G = (G,p) and GH =(Hom(H,G), r

)and let

(f1, . . . , fn, f) ∈ r and (f1, . . . , fn, g) ∈ r. As H is idempotent,(f1(x), . . . ,

fn(x), f(x))

∈ p and(f1(x), . . . , fn(x), g(x)

)∈ p for each x ∈ H . Since G

is an n-ary partial algebra, we get f(x) = g(x) for each x ∈ H . Therefore,f = g, so that GH is an n-ary partial algebra. �

Let H = (H, q) and G = (G,p) be n-ary partial algebras, H idempotent.By Theorem 3.2, the power of G and H is the n-ary partial algebra GH =(Hom (H,G), r

)where, for any f1, . . . , fn, f ∈ Hom(H,G), r(f1, . . . , fn)

= f if and only if q(x1, . . . , xn) = x implies p(f1(x1), . . . , fn(xn)

)= f(x)

whenever x1, . . . , xn, x ∈ H . It is also evident that GH is a partial subal-gebra of the direct product GH provided that r(f1, . . . , fn) = f if and onlyif p

(f1(x), . . . , fn(x)

)= f(x) for each x ∈ H . By Remark 2.2(b), GH is

idempotent.

Theorem 3.3. Let G, H, K be n-ary partial algebras. If H, K areidempotent, then GH,

(GH

)K and GH×K are n-ary partial algebras and

(GH

)K ∼= GH×K.

Proof. Since H, K are idempotent, GH and(GH

)K are n-ary partialalgebras by Theorem 3.2. In consequence of Theorem 2.3, we have

(GH

)K

∼= GH×K. �Theorem 3.2 and Propositions 2.4 and 2.5 result in the following two

statements:

Proposition 3.4. Let Gi, i ∈ I , be a family of n-ary partial algebrasand let H be an n-ary partial algebra. If H is idempotent, then GH

i is an



n-ary partial algebra for every i ∈ I and so are∏

i∈I GHi and (

∏i∈I Gi)

H

and, moreover,

∏

i∈I

GHi

∼=( ∏

i∈I

Gi

)H

.

Proposition 3.5. Let G be an n-ary partial algebra and let Hi, i ∈ I ,be a family of pair-wise disjoint n-ary partial algebras. If Hi is idempotentfor every i ∈ I , then GH

i is an n-ary partial algebra for every i ∈ I and sois G

P

i∈I Hi and, moreover,∏

i∈I

GHi

∼= GP

i∈I Hi .

The diagonality has already been defined for relational systems, hencealso for partial algebras. Clearly, an n-ary partial algebra (G, p) is diagonalif and only if, whenever (xij) is an n × n-matrix over G and x ∈ G, from

p(p(x11, . . . , x1n), . . . , p(xn1, . . . , xnn)

)

=(p(x11, . . . , xn1), . . . , p(x1n, . . . , xnn)

)= x

it follows that p(x11, . . . , xnn) = x.Theorems 2.7 and 3.2 immediately result in

Proposition 3.6. Let G, H be n-ary partial algebras, G diagonal andH idempotent. Then GH is a diagonal n-ary partial algebra which is a rela-tional subsystem of the direct power GH .

In general, given n-ary partial algebras H = (H, q) and G = (G, p),H idempotent, such that GH is a relational subsystem of the direct productGH , the power GH need not be a partial subalgebra of the direct productGH . Clearly, the power GH is a partial subalgebra of the direct productGH if and only if GH is a relational subsystem of the direct product GH

having the property that, if f1, . . . , fn ∈ Hom (H,G) are homomorphismssuch that, for every y ∈ H , there exists xy ∈ G with p

(f1(y), . . . , fn(y)

)

= xy, then the map f : G → H given by f(y) = xy for every y ∈ H satisfiesf ∈ Hom (H,G). To give a sufficient condition for the power GH of n-arypartial algebras to be a partial subalgebra of the direct product GH , wedefine:

Definition 3.7. An n-ary partial algebra (G, p) is called medial pro-vided that, whenever (xij) is an n × n-matrix over G and x1, . . . , xn, x ∈ G,from p

(p(x11, . . . , x1n), . . . , p(xn1, . . . , xnn)

)= x and p(x1j , . . . , xnj) = xj for

j = 1, . . . , n it follows that p(x1, . . . , xn) = x.



Theorem 3.8. Let G,H be n-ary partial algebras and let G be medial.Then there exists a partial subalgebra K of the direct product GH such thatK = Hom (H,G).

Proof. Let H = (H, q), G = (G, p) and GH = (GH , R) and letR(f1, . . . , fn) = f where f1, . . . , fn ∈ Hom(H,G), f ∈ GH . Then f(y) =p(f1(y), . . . , fn(y)

)for each y ∈ H . Let q(y1, . . . , yn) = y. Then

f(y) = f(q(y1, . . . , yn)

)

= p(f1

(q(y1, . . . , yn)

), . . . , fn

(q(y1, . . . , yn)

))

= p(p(f1(y1), . . . , f1(yn)

), . . . , p

(fn(y1), . . . , fn(yn)

)).

Since p(f1(yi), . . . , fn(yi)

)= f(yi) for each i = 1, . . . , n, the mediality of G

implies p(f(y1), . . . , f(yn)

)= f(y). Hence, f ∈ Hom (H,G). �

Corollary 3.9. Let G,H be n-ary partial algebras. If G is both diag-onal and medial and H is idempotent, then the power GH is a medial n-arypartial subalgebra of the direct product GH .

Proof. Let H = (H, q), G = (G, p) and GH =(Hom (H,G), r

). By

Theorem 2.7, GH is a relational subsystem of the direct product GH . ByTheorem 3.8, GH is a partial subalgebra of the direct product GH . We willshow that GH is medial. Let (fij) be an n × n-matrix over Hom(H,G) andlet f, f1, . . . , fn ∈ Hom (H,G) be such that

r(r(f11, . . . , f1n), . . . , r(fn1, . . . , fnn)

)= f

and r(f1i, . . . , fni) = fi for each i = 1, . . . , n. Since r(r(f11, . . . , f1n), . . . ,

r(fn1, . . . , fnn))

= f , we have

p(p(f11(y), . . . , f1n(y)

), . . . , p

(fn1(y), . . . , fnn(y)

)) = f(y)

for every y ∈ H . As r(f1i, . . . , fni) = fi for each i = 1, . . . , n, we getp(f1i(y), . . . , fni(y)

)= fi(y) for every y ∈ H . So, p

(f1(y), . . . , fn(y)

)= f(y)

for each y ∈ H because G is medial. Consequently, r(f1, . . . , fn) = f . It fol-lows that the power GH is a medial. �

4. Total algebras

An n-ary (total) algebra is an (n + 1)-ary relational system G = (G, p)such that, for any x1, . . . , xn ∈ G, there exists exactly one y ∈ G such that



(x1, . . . , xn, y) ∈ p (cf. [4]). The (n + 1)-ary relation p, is then a map p :Gn → G, which is called an n-ary operation on G. We write y = p(x1, . . . , xn)instead of (x1, . . . , xn, y) ∈ p.

Let H = (H,q) and G = (G,p) be a pair of n-ary algebras. Clearly, a mapf : H → G is a homomorphism of H into G provided that p

(f(x1), f(x2),

. . . , f(xn))

= f(q(x1, x2, . . . , xn)

)whenever x1, x2, . . . , xn ∈ H . Clearly, ev-

ery n-ary algebra is an n-ary partial algebra. A partial subalgebra of a totalalgebra G is clearly a total algebra, which is called a (total) subalgebra of G.

Obviously, the direct product of a family Gi = (Gi, pi), i ∈ I of n-ary al-gebras is the n-ary algebra

∏i∈I Gi = (

∏i∈I Gi, q) where, for any f1, . . . , fn,

f ∈∏

i∈I Gi, q(f1, . . . , fn) = f if and only if pi

(f1(i), . . . , fn(i)

)= f(i) for

each i ∈ I .The power GH of n-ary algebras need not be an n-ary algebra even if H

is idempotent – see the following example:

Example 4.1. Let H = (H, q), G = (G, p) be ternary relational sys-tems where H = {1, 2}, G = {a, b}, q =

{(1, 1, 1), (1, 2, 2), (2, 1, 2), (2, 2, 2)

}

and p ={

(a, a, a), (a, b, a), (b, a, a), (b, b, b)}

. Then H, G are binary alge-bras and H is idempotent. Let f : H → G be a map given by f(1) = a,f(2) = a, g : H → G be a map given by g(1) = b, g(2) = b and h : H → Gbe a map given by h(1) = b, h(2) = a. Then Hom (H,G) = {f, g, h}. LetGH =

(Hom (H,G), r

). Then

{(g, h, f), (g, h, g), (g, h, h)

}∩ r = ∅. Thus,

GH is not a binary algebra. �An n-ary algebra (G, p) is diagonal provided that, whenever (xij) is an

n × n-matrix over G and x ∈ G, from p(p(x11, . . . , x1n), . . . , p(xn1, . . . , xnn)

)

= p(p(x11, . . . , xn1), . . . , p(x1n, . . . , xnn)

)= x it follows that p(x11, . . . , xnn)

= x. By Definition 3.7, an n-ary algebra (G, p) is medial provided that,whenever (xij) is an n × n-matrix over G,

p(p(x11, . . . , x1n), . . . , p(xn1, . . . , xnn)

)

= p(p(x11, . . . , xn1), . . . , p(x1n, . . . , xnn)

).

Remark 4.2. Medial groupoids (i.e., binary algebras) are studied in [5].Diagonal and medial algebras are discussed in [8]. In [7], the n-ary idem-potent, diagonal and medial algebras (called briefly diagonal algebras) arestudied and it is shown that they are, up to isomorphisms, the n-ary al-gebras (X1 × . . . × Xn, p) where X1, . . . , Xn are sets and the operation p isdefined by p

((x1

1, . . . , x1n), . . . , (xn

1 , . . . , xnn)

)=

(x1

1, x22, . . . , x

nn

). Idempotent,

diagonal and medial groupoids are usually called rectangular bands (cf. [3]).

As a consequence of Theorem 3.8, we get

Theorem 4.3. Let G, H be n-ary algebras and let G be medial. Thenthere exists a subalgebra K of the direct product GH so that K = Hom(H,G).



Theorem 4.3 and Corollary 3.9 result in

Corollary 4.4. Let G, H be n-ary algebras. If G is both diagonal andmedial and H is idempotent, then the power GH is a medial n-ary subalgebraof the direct product GH .

Let H = (H, q) and G = (G, p) be n-ary algebras and let G be bothdiagonal and medial and H be idempotent. By Corollary 4.4, the power of Gand H is an n-ary algebra GH =

(Hom(H,G), r

)where, for any f1, . . . , fn,

f ∈ Hom(H,G), r(f1, . . . , fn) = f if and only if p(f1(x), . . . , fn(x)

)= f(x)

whenever x ∈ H . By Remark 2.2, GH is idempotent.As a consequence of Theorem 3.3 and Corollary 4.4, we get the following

statement which is proved also in [8]:

Theorem 4.5. Let G, H, K be n-ary algebras. If G is both diagonaland medial and H, K are idempotent, then GH,

(GH

)K and GH×K aren-ary algebras and, moreover,

(GH

)K ∼= GH×K.

Proposition 3.4 and Corollary 4.4 imply

Proposition 4.6. Let Gi, i ∈ I , be a family of n-ary algebras and let Hbe an n-ary algebra. If Gi is diagonal and medial for every i ∈ I and H isidempotent, then GH

i is an n-ary algebra for every i ∈ I and so are∏

i∈I GHi

and (∏

i∈I Gi)H and, moreover,

∏

i∈I

GHi

∼=( ∏

i∈I

Gi

)H

.

We have seen (in Corollary 4.4, Theorem 4.5 and Proposition 4.6) thatthe conjunction of diagonality and mediality is an important property re-sulting in well behavior of powers of n-ary algebras. We will therefore givetwo necessary and sufficient conditions for the validity of the conjunction.

Proposition 4.7. Let G = (G,p) be an n-ary algebra. Then the follow-ing conditions are equivalent:

(i) G is diagonal and medial;(ii) p

(p(x11, . . . , x1n), . . . , p(xn1, . . . , xnn)

)= p(x11, x22, . . . , xnn) for each

n × n-matrix (xij) over G;(iii) p(p(x11, . . . , x1n), x22, . . . , xnn) = p(x11, p(x21, . . . , x2n), x33, . . . , xnn)

= · · · = p(x11, . . . , xn−1,n−1, p(xn1, . . . , xnn)

)= p(x11, x22, . . . , xnn) for each

n × n-matrix (xij) over G.

Proof. (i) ⇒ (ii) is trivial.



(ii) ⇒ (i). Assume that (ii) holds. Then G is clearly diagonal. Wehave p

(p(x11, . . . , x1n), . . . , p(xn1, . . . , xnn)

)= p(x11, x22, . . . , xnn) and also

p(p(x11, . . . , xn1), . . . , p(x1n, . . . , xnn)

)= p(x11, x22, . . . , xnn) for each n × n-

matrix (xji) over G. Hence, G is medial.(ii) ⇒ (iii). Assume that (ii) holds. Let (xij) be an n × n-matrix over G.

Then

p(p(x11, . . . , x1n), . . . , p(xn1, . . . , xnn)

)= p(x11, x22, . . . , xnn)

and

p(p(xi1, . . . , xin), . . . , p(xi1, . . . , xin)

)= p(xi1, . . . , xin)

for each i = 1, . . . , n. We will show that p(p(x11, . . . , x1n), x22, . . . , xnn) =p(x11, x22, . . . , xnn). By (ii), we have

p(p(x11, . . . , x1n), x22, . . . , xnn)

= p(p(p(x11, . . . , x1n), . . . , p(x11, . . . , x1n)

),

p(x21, . . . , x2n), . . . , p(xn1, . . . , xnn))

= p(p(x11, . . . , x1n), p(x21, . . . , x2n), . . . , p(xn1, . . . , xnn)

)

= p(x11, x22, . . . , xnn).

By using the same argument as above, we get

p(x11, p(x21, . . . , x2n), x33, . . . , xnn

)

= . . . p(x11, . . . , xn−1,n−1, p(xn1, . . . , xnn)

)= p(x11, x22, . . . , xnn).

Hence (iii) holds.(iii) ⇒ (ii). Assume that (iii) holds. Let (xij) be an n × n-matrix over G.

Then

p(p(x11, . . . , x1n), p(x21, . . . , x2n), . . . , p(xn1, . . . , xnn)

)

= p(x11, p(x21, . . . , x2n), . . . , p(xn1, . . . , xnn)

)

= p(x11, x22, p(x31, . . . , x3n), . . . , p(xn1, . . . , xnn)

)

= . . . = p(x11, x22, . . . , xn−1,n−1, p(xn1, . . . , xnn)

)= p(x11, x22, . . . , xnn).

Thus, (iii) holds.



Example 4.8. By Proposition 4.7, a groupoid G = (G, ·) is diagonaland medial if and only if xyz = xz whenever x, y, z ∈ G. All diagonal andmedial groupoids G = (G, ·) with |G| � 4 are described in [9].

References

[1] G. Birkhoff, Generalized arithmetic, Duke Math. J., 9 (1942), 283–302.[2] P. Burmeister, A Model Theoretic Oriented Approach to Partial Algebras, Akademie-

Verlag (Berlin, 1986).[3] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. I, Amer-

ican Mathematical Society (Providence, Rhode Island, 1964).[4] G. Gratzer, Universal Algebra, Springer-Verlag (New York–Heidelberg–Berlin, 1979).[5] J. Jezek and T. Kepka, Medial Groupoids, Rozpravy CSAV, Rada Mat. a Prır. Ved.

93/1, Academia (Prague, 1983).[6] V. Novak, On a power of relational structures, Czech. Math. J., 35 (1985), 167–172.[7] J. Plonka, Diagonal algebras, Fund. Math., 58 (1966), 309–321.[8] J. Slapal, A note on diagonality of algebras, Math. Nachr., 158 (1992), 195–197.[9] J. Slapal, Cardinal arithmetic of general relational systems, Czech. Math. J., 43

(1993), 125–139.[10] J. Slapal, Direct arithmetic of relational systems, Publ. Math. Debrecen, 38 (1991),

39–48.[11] J. Slapal, On exponentiation of n-ary algebras, Acta Math. Hungar., 63 (1994), 313–

322.[12] J. Slapal, On the direct power of relational systems, Math. Slovaca, 39 (1989), 251–

255.


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