Directly indecomposible multialgebras

Abstract: The aim of this paper is the study directly indecomposible multialgebras. In this regards, ﬁrst the

isomorphism theorems and correspondence theorem for multialgebras. Then by applying congruences relation

on multialgebras factor multialgebras are constructed and some important properties of them are obtained. In

particular, it is shown that every ﬁnite multialgebra is isomorphic to a direct products of directly indecomposable of

multialgebras. Finally, subdirect products and subdirect irreducible of multialgebras are investigated and Birkoff’s

theorem is extended to multialgebras.

Key–Words: Multialgebra, Fundamental relation, Isomorphism, Congruence relation, Factor congruence, Directly

indecomposible, Subdirectly irreducible.

Received: October 14, 2022. Revised: August 17, 2023. Accepted: September 21, 2023. Published: October 6, 2023.

1 Introduction

A multialgebra can be considered as a relational sys-

tems which generalize the universal algebras. In [17]

Schweigert studied the congruence of multialgebras.

R. Ameri et al. introduced and studied hyperalgebraic

system in [2]; some more properties of multialge-

bras such as identities, fundamental relation and direct

limit and etc. has been studied by C. Pelea (for more

details see [12], [13], [14]). In this paper we follow

[16] to study isomorphism theorems, directly inde-

composible and subdirect products of multialgebras.

This paper is organized in 5sections. In Section 2,we

gather the deﬁnition and basic properties of multial-

gebras which we need to development our paper. In

Section 3the isomorphism theorems and correspon-

dence theorem for multialgebras has been proved. In

Section 4, by using the notions of congruence, factor

congruence and direct product of multialgebras it is

shown that every ﬁnite multialgebra is isomorphic to

a direct product of directly indecomposable multial-

gebras. Finally, in Section 5subdirectly irreducible

of multialgebras are introduced and a necessary and

sufﬁcient condition that a multialgebra is subdirectly

irreducible is obtained. Finally, the Birkoff’s theorem

has been extended to multialgebras.

2 Preliminaries

In this section we gather all deﬁnitions and results of

multialgebras, which we need to development our pa-

per. In the sequel His a ﬁxed nonvoid set, P∗(H)

is the family of all nonvoid subsets of H, and for a

positive integer n Hndenotes the set of all n−tuples

elements of H.

For a positive integer nan−ary hyperoperation βon

His a function β:Hn→P∗(H). We say that

nthe arity of β. A subset Sof His closed under the

n-ary hyperoperation βif (x1, . . . , xn)∈Snimplies

that β(x1, . . . , xn)⊆S. A nullary hyperoperation on

His just an element of P∗(H); i.e. a nonvoid subset

of H.

An n-ary relation ρon His a subset of Hn. We

also say that the arity of ρis n. Orders and equivalence

relations on Hare the best examples of binary (i.e. 2-

array) relations on H. Henceforth sometimes we use

hyperoperation instead of the n-ary hyperoperation. A

hyperalgebraic system or a multialgebra ⟨H, (βi,|i∈

I)⟩is the set Hwith together a collection (βi,|i∈I)

of hyperoperations on H.

A subset Sof a multialgebra H=⟨H, (βi,|

i∈I)⟩is a submultialgebra of Hif Sis closed

under each hyperoperation βi, for all i∈I, that is

βi(a1, ..., an)⊆S, whenever (a1, ..., an)∈Sn. The

type of His the map from Iinto the set N∗of non-

negative integers assigning to each i∈Ithe arity of

βi.

A binary relation ρon a set Mis called compati-

ble (resp. strong compatible ) with an n-ary hyperop-

eration βif x1ρy1, ..., xnρynimplies that

β(x1, ..., xn)ρβ(y1, ..., yn),

1MAHSA DAVODIAN, 1MOHSEN ASGHARI-LARIMI, 2REZA AMERI

1Department of Mathematics, University of Golestan, Gorgan, IRAN

2Department of Mathematics, Faculty of Basic Science, University of Tehran, IRAN

PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

E-ISSN: 2732-9941

Volume 3, 2023

(β(x1, ..., xn)ρSβ(y1, ..., yn)),

where for nonempty subsets Aand Bof M,

AρB ⇐⇒ (∀a∈A∃b∈B:aρb

and ∀b∈B , ∃a∈A:bρa),

and

AρSB⇐⇒ ∀a∈A, ∀b∈B aρb.

Let ⟨H, (βi,|i∈I)⟩be a multialgebra. A binary

relation ρon Mis called (resp. strong) congruence

if ρis an equivalence relation and (resp. strongly)

compatible with every βi, i ∈I.

For n > 0we extend an n−ary hyperoperation β

on Hto an n−ary operation βon P∗(H)by setting

for all A1, ..., An∈P∗(H)

β(A1, ..., An) = {β(a1, ..., an)|(1)

ai∈Ai(i= 1, ..., n)}

It is easy to see that ⟨P∗(H),(βi,|i∈I)⟩is an

algebra. whenever possible we write ainstead of the

the singleton {a}; e.g. for a binary hyperoperation ◦

and a, b, c ∈Hwe write a◦(b◦c)for

{a} ◦ ({b} ◦ {c}) = {a◦u|u∈b◦c}.

An equivalence relation on Acompatible (resp.

strongly compatible) with a multialgebra Hon Ais

congruence (resp. strong congruence) of H. Denote

by Con(H)(resp.Cons(H)) the set of all congru-

ences (resp. strong congruences ) of H.

Let H=⟨A, (βi,|i∈I)⟩be a multialgebra and

let θ∈Con(H). Let A/θ ={Bj|j∈J}be the set

of blocks of θ. For every i∈Ideﬁne βion A/θ as

follows:

Let j1, ..., jmi∈Jbe arbitrary and let al∈Bjl

for l= 1, ..., mi. Deﬁne

βi(Bj1, ..., Bjmi) = {Bj|j∈J, (2)

Bjmeets βi(a1, ..., ami)}

Since θ∈Con(H), it can be veriﬁed that βiis

well deﬁned mi-ary hyperoperation on A/θ. Call

H/θ =⟨A/θ, (β|j∈J)⟩a factor multialgebra of

H. If, moreover, θ∈Con(H), then every βiis sin-

gleton valued, i.e. an operation on A/θ, and H/θ is

an algebra. For semihypergroups this fact are in [1]

the general case is in [11].

We view binary relation on Aas subsets of A2

and so for a multialgebra Hon Athe sets Con(H)

and Cons(H)are naturally ordered by set inclusion.

First we characterize the poset (Con(H, ⊆). Recall

that for a binary relations ρand σon Athe relation

product (also called de Morgan product) is

ρ◦σ={(x, y)∈A2|(x, u)∈ρ, (u, y)∈σ

for some u∈A}.

It is well known and easy to show that the re-

lation product is associative with the unital element

ω={(a, a)|a∈A}.

Example 1.(i)A hypergroupoid is a multialgebra of

type (2), that is a set Htogether with a (binary) hy-

peroperation ◦. A hypergroupoid (H, ◦), which is as-

sociative, that is x◦(y◦z)=(x◦y)◦zfor all

x, y, z ∈His called a semihypergroup.

(ii)A hypergroup is a semihypergroup such that

for all x∈Hwe have x◦H=H=H◦x(called

the reproduction axiom).

An element ein a hypergroup H= (H, ◦)is

called an identity of Hif for all x∈H, on has

x∈(e◦x)∩(x◦e).

(iii)A polygroup (or multigroup) is a semihy-

pergroup H= (H, ◦)with e∈Hsuch that for all

x, y ∈H

(i)e◦x=x=x◦e;

(ii)there exists a unique element, x−1∈Hsuch

that

e∈(x◦x−1)∩(x−1◦x), x ∈

z∈x◦y

(z◦y−1),

y∈

z∈x◦y

(x−1◦z).

In fact, a polygroup is a multialgebra of type

(2,1,0).

Deﬁnition 2. Let H=⟨H, (βi,|i∈I)⟩and H=

⟨H, (βi,|i∈I)⟩be two similar multialgebras. A

map hfrom Hinto His called a

(i)A homomorphism if for every i∈Iand all

(a1, ..., ani)∈Hniwe have that

h(βi((a1, ..., ani)) ⊆βi(h(a1), ..., h(ani));

(ii)a good homomorphism if for every i∈Iand

all (a1, ..., ani)∈Hniwe have

h(βi((a1, ..., ani)) = βi(h(a1), ..., h(ani)).

For a map h:H−→ Hset

ker h={(a, a′)|a, a′∈H, and h(a) = h(a′)}.

PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

E-ISSN: 2732-9941

Volume 3, 2023

It is well known and it can be easily seen that

kerh is an equivalence relation on H. If his a good

homomorphism, then it can be easily seen that θis

a strong congruence on H. Setting for all a∈H,

ϕ(a) = a/θ.

Deﬁnition 3. A universal algebra is a pair < A, (fi:

i∈I)>where Ais a nonempty set and (fi:i∈I)

is a family of ﬁnitary operations on Aindexed by I. A

ﬁnitary operation is an n−ary operation for some n,

and n−ary operation on Ais any function ffrom An

to A,nis the rank of f. In above we assume for every

i∈I,niis the rank of fi, and < ni, i ∈I > is called

tape of A.

Deﬁnition 4. Let τ=< ni:i∈I > be a sequence

over N={1,2,...}. By a multialgebra of tape τ, we un-

derstand a pair < H, (fi:i∈I)>, where His a

nonempty set and fiis an ni-ary hyper operation on

H,i.e, a map fi:Hni→P∗(H), for each i∈I.

Remark 5. Let < A, (fi:i∈I)>be a universal

multialgebra . Ainduces an algebra < P ∗(A),(fi:

i∈I)>with the operations:

fi(A0, ..., Ani) = {fi(a0, ..., ani−1)|ai∈Ai,∀i∈

{0, ..., ni−1}

for A0, ..., Ani−1∈P∗(A). We denote this algebra

by P∗(A).

Deﬁnition 6.Let Abe a multialgebra. The funda-

mental relation α∗on Ais the smallest equivalence

relation on Asuch that A/α∗is a universal algebra.

Lemma 7 [1]. If ρand σare binary relations on A

compatible with H, then τ=ρ◦σis compatible with

Lemma 8 [1]. (i)The relation ω={(a, a)|a∈A}

is compatible with Hand

(ii)the relation A2is strongly compatible with

Lemma 9 [1]. Let Hbe a multialgebra on H. Let

h > 0and let {σj|j∈J}be a set of h-ary relations

on Hstrongly compatible with H. Then σ=

j∈J

σj

is strongly compatible with H.

3 Isomorphism theorems of multial-

gebras

Theorem 10 [1]. Let H=⟨H, (βi,|i∈I)⟩and

H′=⟨H′,(β′

i,|i∈I)⟩be similar multialgebras, let

hbe a good homomorphism from Honto H′, and let

ϕbe the quotient map corresponding θ= ker h. Then

(i)θis a congruence relation on H;

(ii)ϕis a good homomorphism from Honto

H/θ;

(iii)the unique function ffrom H/θ onto H′

satisfying ϕ◦f=his a good isomorphism from H/θ

onto H′.

Proposition 11.Let Hbe a multialgebra and let θbe

the least element of Cons(H). Then (Cons(H),⊆)

is lattice isomorphic to the congruence lattice of the

algebra H/θ.

Deﬁnition 12.Suppose His an multialgebra and

ϕ, θ ∈Con(H)with θ⊆ϕ. Then let

ϕ/θ =⟨a/θ, b/θ⟩ ∈ (H/θ)2:⟨a, b⟩ ∈ ϕ.

Lemma 13.If ϕ, θ ∈Con(H)and θ⊆ϕ, then ϕ/θ is

congruence on H/θ [6].

Theorem 14 (Second Isomorphism Theorem). If

ϕ, θ ∈Con(H)and θ⊆ϕ, then the map

α: (H/θ)(ϕ/θ)→H/ϕ

α(a/θ)(ϕ/θ)=a/ϕ

is an isomorphism from H/θ(ϕ/θto H/ϕ.

Proof. Let a, b ∈A.

Then if (a/θ)(ϕ/θ)=(b/θ)(ϕ/θ)then it is equal

to a/θ, b/θ∈(ϕ/θi.e (a, b)∈ϕthen a/ϕ =b/ϕ

i.e αis well-deﬁned.

Now for βan n-ary function symbol and

a1, . . . , an∈Hwe have

αβ(H/θ)(ϕ/θ)(a1/θ)(ϕ/θ), . . . , (an/θ)(ϕ/θ)

=αβH/θa1/θ, . . . , an/θ)(ϕ/θ)

(by deﬁnition of factor multialgebra.)

then

=αβH(a1, . . . , an)θ(ϕ/θ)

=βH(a1, . . . , an)ϕ=βH/ϕa1/ϕ, . . . , an/ϕ

=βH/ϕα(a1/θ)(ϕ/θ), . . . , α(an/θ)(ϕ/θ).

□

Deﬁnition 15.Suppose H′is subset of Hand θis a

congruence on H. Let H′θ={a∈H:H′∩a/θ =

∅}. Let H′θbe the submultialgebra of Hgenerated by

H′θ. Also, deﬁne θ|H′be θ∩H′2, the restriction of θ

on H′.

PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

E-ISSN: 2732-9941

Volume 3, 2023

Lemma 16.If H′is a submultialgebra of Hand θ∈

Con(H)(resp. cons(H)), then

(i) The universe of H′θis H′θ.

(ii) θ|H′is a congruence (resp. strong congruence)

on H′.

Proof.

(i) Let βbe any n−ary hyperoperation and

a1, . . . , an∈H′θ. By deﬁnition, there ex-

ist b1, . . . , bn∈H′such that ⟨ai, bi⟩ ∈ θ,

i= 1, . . . , n. Because of congruency of θon

Hand H′is submultialgebra of H, we have

β(a1, . . . , an)¯

θβ(b1, . . . , bn).

Let a∈β(a1, . . . , an).

So there exist b∈β(b1, . . . , bn)s.t a/θ =b/θ.

Then H′∩(a/θ)=∅or a∈H′θ. Therefore

β(a1, . . . , an)⊆H′θ.

(ii) Proof is straightforward.

□

Theorem 17 (Third Isomorphism Theorem).If H′is

a submultialgebra of Hand θ∈Con(H), then

H′θ|′

H∼

=H′θθ|H′θ.

Proof. We prove that the mapping α:H′θ|H′→

H′θθ|H′θis deﬁned by αb/θ|H′=b/θ|H′θis Iso-

morphism. First we prove αis one-to-one:

Let b1θ|H′θ=b2θ|H′θ. Then ⟨b1, b2⟩ ∈

θ|H′θ=θ∩(H′θ)2.

So ⟨b1, b2⟩ ∈ θand ⟨b1, b2⟩ ∈ (H′θ)2. Therefore,

b1/θ ∩H′=∅and b2/θ ∩H′=∅. Consequently

there exist b′

1, b′

2∈H′, such that b1/θ =b′

1/θ and

b2/θ =b′

2/θ. Thus

⟨b′

1, b′

2⟩ ∈ θ

⟨b′

1, b′

2⟩ ∈ (H′)2,

Consequently ⟨b′

1, b′

2⟩ ∈ θ∩H′2=θ|H′,consequently

⟨b1, b2⟩ ∈ θ∩H′2=θ|H′. then b1θ|H′=b2θ|H′.

αis onto because, if bθ|H′θ∈H′θθ|H′θ, such that

b∈H′θ\H′then H′∩b/θ =∅i.e there exist b1∈H′

such that

αb1θ|H′=bθ|H′θ.

At last we prove αis a homomorphism.

αβb1θ|H′, . . . , bnθ|H′

=αβ(b1, . . . , bn)

θ|H′

=β(b1, . . . , bn)/θ|H′θ

=β(b1

θ|H′θ

, . . . , bn

θ|H′θ

)

=βα(b1

θ|H′

), . . . , α(bn

θ|H′

)

=βαb1

θ|H′, . . . , αbn

θ|H′.

□

Note that if Lis a lattice and a, b ∈Lwith a≤b, then

the interval [a, b]is subuniverse interval of a lattice

L, where a≤b, by [a, b], we mean the corresponding

sublattice of L.

Theorem 18 (Correspondence Theorem).Let H

be an multialgebra and let θ∈con(H)(res.

cons(H)). Then the mapping ψon [θ, ∇H]deﬁned

by ψ(ϕ) = ϕ/θ is a lattice isomorphism from [θ, ∇H]

to con(H/θ)(resp. cons(H/θ)), where [θ, ∇H]is a

sublattice of con(H)(resp cons(H))

Proof. ψis one to one. Because: for ϕ, ϕ′∈[θ, ∇H]

with ϕ=ϕ′. Then without loss of generality, we can

assume that there are a, b ∈Hwith (a, b)∈ϕ−ϕ′.

Then (a/θ, b/θ)∈(ϕ/θ)\(ϕ′/θ)so ψ(ϕ)=ψ(ϕ′).

To show that ψis onto, suppose ρ∈con(H/θ)

(resp. cons(H/θ)) and deﬁne φto be ker(πρπθ),

where πρ, πθare canonical projections. Then for

a, b ∈H,⟨a/θ, b/θ⟩ ∈ φ/θ

if and only if ⟨a, b⟩ ∈ φ= ker(πρπθ)if and only

if πρπθ(a) = πρπθ(b)if and only if and only if

(a/θ)/ρ = (b/a)/ρ)if and only if ⟨a/θ, b/θ⟩ ∈ ρ.

So φ/θ =ρ. let fis n-ary hyperoperation on Hand

a1, . . . , an∈H.

we have

ψfH(a1, . . . , an) = fH(a1, . . . , an)/θ

=fH/θ(a1/θ, . . . , an/θ)

=fH/θ(ψa1, . . . , ψan).

so ψis strong homomorphism. □

4 Directly indecomposible multialge-

bras

Deﬁnition 19.Let H1and H2be two mulialgebras of

the same type F.

PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

E-ISSN: 2732-9941

Volume 3, 2023

Deﬁne the (direct) product of multialgebra H1×H2to

be the multialgebra whose universe is the set H1×H2

and for f∈Fnand ai∈H1and a′

i∈H2;1≤i≤n:

fH1×H2(a1, b1), . . . , (an, bn)

=fH1(a1, . . . , an), fH2(b1, . . . , bn)

{(a, b)|a∈fH1(a1, . . . , an), b ∈fH2(b1, . . . , bn)}.

Note that in general neither H1nor H2is embedable

in H1×H2, although in special case like hypergroups,

it is possible because there is always a trivial subhy-

peralgebra.

However, both H1and H2are homomorphic image of

H1×H2.

Deﬁnition 20.The mapping πi:H1×H2→Hi,i∈

{1,2};

deﬁned by

πi(a1, a2)=ai,

is called the projection map on the i′th coordinate of

H1×H2.

Theorem 21.For i= 1,2the mapping πi:H1×

H2→Hiis a surjective strong homomorphism from

H=H1×H2to Hi. Furthermore, in con(H1×H2)

we have

ker π1∩ker π2= ∆,

ker π1and ker π2premute,

and

ker π1∨ker π2=∇.

Proof. It is easy to check that πiis a strong surjective

homomorphism (epimorphism). Now

(a1, a2),(b1, b2)∈ker πi

iff πi(a1, a2)=πib1, b2)

iff ai=bi

Thus

ker π1∩ker π2= ∆.

Now, consider (a1, b1),(b1, b2)are any two element of

H1×H2, then

(a1, a2)∈ker π1(a1, b2) ker π2(b1, b2),

and

(a1, a2)∈ker π2(b1, a2) ker π1(b1, b2),

hence

∇= ker π1◦ker π2.

But then ker π1and ker π2permute, and their join is

∇.□

Deﬁnition 22.A congruence ρon His a factor con-

gruence if there is a congruence σon Hsuch that

ρ∧σ= ∆,

and

ρ∨σ=∇.

and ρpermute with σ.

The pair ρand σis called a pair of factor congru-

ence on H.

Theorem 23.If ρand σis a pair of factor congruence

on H, then

H≃H/ρ ×H/σ

under the map

α(a) = (a/ρ, a/σ).

Proof. It is straight forward to see αis injective.

For every (a/ρ, b/σ)∈H1/ρ ×H2/σ,⟨a, b⟩ ∈ H.

So, there exist c∈H1such that aρcσb. Then

α(c) = ⟨c/ρ, c/σ⟩ ∈ ⟨a/ρ, b/σ⟩.

Thus αis onto. Finally for β∈Fnand a1, . . . , an∈

αβH(a1, . . . , an)

=βH(a1, . . . , an)/ρ, βH(a1, . . . , an)/σ

=βH/ρ(a1/ρ, . . . , an/ρ), βH/σ(a1/σ, . . . , an/σ)

=βH/ρ×H/σ(a1/ρ, a1/σ),...,(an/ρ, an/σ)

=βH/ρ×H/σαa1, . . . , αan;

hence αis indeed an isomorphism. □

Deﬁnition 24.A multialgebra His (directly) inde-

composable if His not isomorphic to a direct product

of two nontrivial multialgebras.

Example 25.Any ﬁnite multialgebra Hwith |H|a

prime number must be directly indecomposable.

By theorem 4.3 and 4.5 we have:

Corollary 26.His directly indecomposable if and

only if the only factor congruences on Hare ∆and

∇.

Proof. Suppose ∆and ∇be the only factor congru-

ences on H. Then

H/∆ = {a/∆|a∈H}={a}

PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

E-ISSN: 2732-9941

Volume 3, 2023

and

H/∇={a/∇|a∈H}=H.

So |a/∆|= 1 i.e H/∆is trivial. So by 4.5

H≃H/∆×H/∇,

i.e His directly indecomposible.

Now suppose Hbe directly indecomposible. So His

not isomorphic to a direct product of two nontrivial

multialgebras. If Hhave any factor congruence ex-

cept ∆,∇, then by 4.5

H∼

=H/ρ ×H/σ

that is conﬂict.

We can easily generalized the binary product H1×H2

as follows. □

Deﬁnition 27.Let (Hi)i∈Ibe a family of multialge-

bras of type F. The (direct) product H=Hiis

a multialgebra with universe 

i∈I

Hiand such that for

β∈Fnand a1, . . . , an∈i∈IHi,

βH(a1, . . . , an)(i) = βHia1(i), . . . , an(i)

for i∈I, i.e βHis deﬁned coordinate-wise.

The empty product ϕis the trivial multialgebra with

universe {ϕ}. As before we have projection maps

πj:

i∈I

Hi→Hj

for j∈Ideﬁned by

πj(a) = a(j).

which give surjective strong homomorphisms

πj:

i∈I

Hi→Hj.

If i={1,2, . . . , n}, we also write H1×· · ·×Hn. If I

is arbitrary but Hi=Hfor all i∈I, then we usually

write HIfor the direct product, and call it a (direct)

power of H.Hϕis a trivial multialgebra.

Theorem 28.If H1,H2and H3are of type F, then

(a) H1×H2≃H2×H1under α(a1, a2)=

(a2, a1).

(b) H1×(H2×H3)≃H1×H2×H3under

α(a1,(a2, a3)= (a1, a2, a3).

Proof. (a) For any β∈ F and a1∈H1,a2∈H2

deﬁne

α:H1×H2→H2×H1

α(a1, b1) = (b1, a1)

αis well-deﬁned and one-to-one because:

α(a1, b1) = α(a′

1, b′

iff (b1, a1) = (b′

1, a′

So b1=b′

1,a1=a′

1. Thus (a1, b1) = (a′

1, a′

2).

Now, we prove αis homomorphism. For any a1, b1∈

H1and a2, b2∈H2,

αβH1×H2(a1, a2),(b1, b2)

=αβH1(a1, b1), βH2(a2, b2)

=βH2(a2, b2), βH1(a1, b1)

=βH2×H1(a2, a1),(b2, b1)

=βH2×H1α(a1, a2), α(b1, b2).

(b) For H1, H2, H3of type F, deﬁned

α:H1×(H2×H3)→H1×H2×H3

αa1,(a2, a3)= (a1, a2, a3)

αis homomorphism because:

βH1×H2×H3α(a1,(a2, a3), α(a′

1,(a′

2, a′

3))

=βH1×H2×H3(a1, a2, a3),(a′

1, a′

2, a′

3)

=β(a1, a′

1), β(a2, a′

2), β(a3, a′

3)

=αβ(a1, a′

1),(β(a2, a′

2), β(a3, a′

3))

=αβ(a1, a′

1), β(a2, a3),(a′

2, a′

3))

=αβ((a1,(a2, a3)),(a′

1,(a′

2, a′

3)).

□

Deﬁnition 29.(i) If αi:H→Hi,i∈Iare maps,

then the natural map α:H→

i∈I

Hiis deﬁned by

(αa)(i) = αia.

(ii) If we are given maps αi:Hi→H′

i,i∈I, then

the natural map

α:

i∈I

Hi→

i∈I

H′

is deﬁned by

(αa)(i) = αi(a(i)).

PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

E-ISSN: 2732-9941

Volume 3, 2023

Theorem 30.(i)If αi:H→Hi,i∈Iis a family

of homomorphism, then the natural map αis a homo-

morphism from Hto H∗=

i∈I

Hi.

(ii)If αi:Hi→Hi,i∈I, is an indexed family of

(resp. strong) homomorphism, then the natural map α

is a (resp. strong) homomorphism from H∗=

i∈I

to H′∗ =

i∈I

H′

Proof. Suppose αi:H→Hiis a homomorphism

for i∈I. Then for a1, . . . , an∈Hand β∈ Fnwe

have

αβH(a1, . . . , an)(i)

=αiβH(a1, . . . , an)

=βHi(αia1, . . . , αian)

=βHi(αa1)(i), . . . , (αan)(i)

=β∏Hiαa1, . . . , αan(i);

Hence,

αβH(a1, . . . , an) = β∏Hi(αa1, . . . , αan),

so αis a homomorphism in (a).

Now (b) is consequence of (a) because the maps

Hiπ

−→ Hi

αi

−→ H′

βi

−→ H′

indeed that βi◦(αi◦πi)is homomorphism. □

Deﬁnition 31.If a1, a2∈Hand α:H→H′is a

map we say αseparate a1and a2if

αa1=αa2.

The maps αi:H→Hi,i∈I, separate points if for

each a1, a2∈Hwith a1=a2there is an αisuch that

αi(a1)=αi(a2).

Lemma 32.For an indexed family of maps αi:H→

Hi, the following are equivalent:

(i) The map αisperate points.

(ii) αis injective.

(iii) 

i∈I

ker αi= ∆.

Proof. Routine. □

Theorem 33.Let αi:H→Hi,i∈I, be a family

of (resp. strong) homomorphisms, i∈I, then natural

(resp.strong) homomorphism α:H→

i∈I

Hiis an

(resp. strong) embedding if and only if 

i∈I

ker αi= ∆

if and only if the maps αiseparate points.

Proof. Immediate from 4.15. □

5 Subdirect products of multialge-

bras

Deﬁnition 34.A multialgebra His a subdirect prod-

uct of an indexed family (Hi)i∈Iof multialgebras if

(i) H≤

i∈I

Hi, and

(ii) πi(H) = Hifor each i∈I.

An (resp.strong) embedding α:H→

i∈I

Hiis

(resp.strong) subdirect if α(H)is a (resp.strong) sub-

direct product of the Hi.

Note that if I=ϕ, the His a subdirect of ϕif and

only if H=ϕ, is trivial multialgebra.

Lemma 35 [16]. If θi∈con(H)for i∈Iand



i∈I

θi= ∆, then the natural homomorphism

φ:H→

i∈I

H/θi

deﬁned by

φ(a)(i) = a/θi

is a subdirect (strong) embedding.

Proof. Proof is similar to the proof for algebras and

omitted. □

Deﬁnition 36.A multialgebra His subdirectly irre-

ducible if for every subdirect embedding

α:H→

i∈I

there is an i∈Isuch that

πi◦α:H→Hi

is an isomorphism.

The following result give a characterization of subdi-

rectly irreducible multialgebras is most useful in prac-

tice.

Theorem 37.A multialgebra His subdirectly irre-

ducible if and only if His trivial or there is a mini-

mum congruence in conH − {∆}.

Proof. (=⇒) If His not trivial and conH − {∆}has

no minimum element then ∩(con(H− {∆}) = ∆.

Let I=conH − {∆}. Then the natural map

ν:H→

θ∈I

H/θ is a subdirect embedding by

Lemma 5.2, and as the natural map H→H/θ is not

PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

E-ISSN: 2732-9941

Volume 3, 2023

injective for θ∈I, it follows that His not subdirectly

irreducible.

(⇐=) If His trivial and ν:H→

i∈I

Hiis a

subdirect embedding then each Hiis trivial; hence

each πi◦νis an isomorphism. So suppose His not

trivial, and let θ=∩conH − {∆}= ∆. Choose

⟨a, b⟩ ∈ θ,a=b. If ν:H→

i∈I

Hiis a subdirect

embedding then for some i,(νa)(i)= (νb)(i), hence

(πi◦ν)(a)= (πi◦ν)(b).

Thus, ⟨a, b⟩/∈ker(πi◦ν)so θ⊈ker(πi◦ν). But this

implies ker(πi◦ν) = ∆. So πi◦ν:H→Hiis an

isomorphism. □

In the latter case the minimum element in

∩(con(H)− {∆}), a principal congruence and the

congruence lattice of Hlooks like the following dia-

gram

Theorem 38 (Birkhoff).Every multialbera His iso-

morphism to a subdirect product of subdirectly irre-

ducible.

Proof. By previous theorem we know trivial multi-

algebras are subdirectly irreducible. Then we only

need to consider the case of nontrivial H, that proof

is easily by Zorn’s Lemma. □

Some important question about multialgebras:

1. Let Abe a multialgebra and A∗be its funda-

mental algebra. Under what conditions the cor-

responding fundamental algebra and Aare sat-

isfying the same identities?.

2. Let Vbe a variety of multialgebras, what

identities hold in the variety generated by

({V(A∗)|A ∈ V}?.

It is convenient to introduce the following notation:

If Vis variety of multialgebras, by V∗we denote the

variety generated by {V∗(A)|A ∈ V}, where V∗is

denoted the corresponding elements of variety Vin

V∗.(See theorem 2 in Ivica Bosnjak et. all, 2003)

There is another subject that has attracted attention

of algebraists; If Kis a class of multialgebras, and

Aand Bare isomorphic multialgebras from K, then

A∗∼

=B∗. It is natural to ask whether the converse it

is true, i.e. is it true that for any A,Bfrom Kit holds:

A∗∼

=B∗⇒A∼

=B?

Clearly the implication is not true always (for example

let Aand Bbe two total hypergroups)

(x◦y=A∀x, y ∈A). Clearly, A∗∼

=B∗= (e).

Now if we choose sets Aand Bsuch that |A| =|B|,

then A≇B

References:

PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

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Volume 3, 2023

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PROOF

DOI: 10.37394/232020.2023.3.8

Mahsa Davodian, Mohsen Asghari-Larimi, Reza Ameri

E-ISSN: 2732-9941

Volume 3, 2023

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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