Ratio Mathematica

Connections between ideals of semisimple
EMV-algebras and set-theoretic filters

Xiaoxue Zhang∗

Hongxing Liu†

Abstract

In this paper, we mainly study connections between ideals of the
semisimple EMV-algebra M and filters on some nonempty set Ω. We
show that there is a bijection between the set of all closed ideals of
M and the set of all filters on Ω. We get that this correspondence also
holds between the set of all closed prime ideals of M and the set of
all weak ultrafilters on Ω. We prove that the topological space of all
closed prime ideals of M and the topological space of all weak ultra-
filters on Ω are homeomorphic.
Keywords: Semisimple EMV-algebra; Ideal; Filter; Closure opera-
tion; Closed ideal
2020 AMS subject classifications: 06D99 1

∗School of Mathematics and Statistics, Shandong Normal University, 250014, Jinan, P. R. Chi-
na; zhangxiaoxuexz@163.com.
†School of Mathematics and Statistics, Shandong Normal University, 250014, Jinan, P. R. Chi-

na; lhxshanda@163.com.
1

is published under the CC-BY licence agreement.

Received on January 12, 2022. Accepted on May 12, 2022. Published on September 25, 2022.

Volume 43, 2022

doi: 10.23755/rm.v43i0.786. ISSN: 1592-7415. eISSN: 2282-8214. © The Authors. This paper



Xiaoxue Zhang, Hongxing Liu

1 Introduction
An MV-algebra is an algebra (M;⊕,∗, 0) of type (2, 1, 0, 0) which has the

top element 1. The study of MV-algebras is very in-depth and comprehensive,
which has important applications in other areas of mathematical research. There
are close connections between ideals of a semisimple MV-algebra and filters on
some associated nonempty set. Moreover, there exists a bijection between the set
of all closed ideals of a semisimple MV-algebra and the set of all filters on some
nonempty set. For more details about it, we recommend the monographs Cignoli
et al. [2013], Lele et al. [2021].

An EMV-algebra is an algebra (M;∨,∧,⊕, 0) of type (2, 2, 2, 0), which is a
new class of algebraic structures. EMV-algebras cannot guarantee the existence
of the top element 1, which are the generalizations of MV-algebras. MV-algebras
are termwise equivalent to EMV-algebras with the top element, Dvurečenskij and
Zahiri [2019].

We shall mainly study connections between ideals of a semisimple EMV-
algebra M and filters on Ω, where M ⊆ [0, 1]Ω and [0, 1]Ω is an EMV-clan of
fuzzy functions on some nonempty set Ω. This paper is organized as follows. In
Section 2, we give some basic notions and theorems on EMV-algebras, which will
be used in the paper. In Section 3, we start by introducing the limits of f ∈ M
along a filter F on Ω. We study the connections between ideals of M and filters
on Ω. In Section 4, we define a closure operation on M. We exhibit a one-to-one
correspondence between the set of all closed ideals of M and the set of all filters
on Ω. We show that there is a homeomorphism between the topological space of
all closed prime ideals of M and the topological space of all weak ultrafilters on
Ω. In addition, there is an example of an ideal that is a non-closed ideal, and some
properties of closed ideals are listed.

2 Preliminaries
In this section, we introduce some basic notions and theorems on an EMV-

algebra, which will be used in the following sections.
A filter F on a nonempty set Ω is a collection of subsets of Ω satisfying (i)

the intersection of two elements in F again belongs to it and (ii) for all S ∈ F ,
S ⊆ T ⊆ Ω implies that T ∈ F . By (ii), we have Ω ∈ F for any filter F on
Ω. A filter F is called proper if ∅ /∈ F . It is obvious that if F1 and F2 are filters
on Ω, F1 ∩ F2 is also a filter of Ω. In fact, for all S1,S2 ∈ F1 ∩ F2, we get
S1 ∩ S2 ∈ F1 ∩ F2. Moreover, for any S ∈ F1 ∩ F2 and S ⊆ T ⊆ Ω, which
implies T ∈ F1 and T ∈ F2. So T ∈ F1 ∩F2. We have shown that F1 ∩F2 is a
filter on Ω.



Connections between ideals of semisimple EMV-algebras and set-theoretic filters

Definition 2.1. ([Cignoli et al., 2013, Definition 1.1.1]) An MV-algebra is an
algebra (M;⊕,∗, 0, 1) of type (2, 1, 0, 0) such that (M;⊕, 0) is a commutative
monoid, and for all x,y ∈ M satisfying the following axioms:
(MV1) x∗∗ = x;
(MV2) x⊕ 0∗ = 0∗;
(MV3) (x∗ ⊕y)∗ ⊕y = (y∗ ⊕x)∗ ⊕x.

For all x,y ∈ [0, 1], the real interval [0, 1] with the operations x⊕y = min{x+
y, 1} and x∗ = 1 −x is an MV-algebra. Let (M; +, 0) be a monoid. An element
a ∈ M is called idempotent if it satisfies the equation a + a = a. We denote
the set of all idempotent elements of M by I(M). We recommend Cignoli et al.
[2013] for MV-algebras.

EMV-algebras as the generalizations of MV-algebras have many important
properties. We recommend Dvurečenskij and Zahiri [2019] for EMV-algebras.

Definition 2.2. ([Dvurečenskij and Zahiri, 2019, Definition 3.1]) An EMV-algebra
is an algebra (M;∨,∧,⊕, 0) with type (2, 2, 2, 0) satisfying the followings:
(EMV1) (M;∨,∧, 0) is a distributive lattice with the least element 0;
(EMV2) (M;⊕, 0) is a commutative ordered monoid with the neutral element 0;
(EMV3) for all a,b ∈ I(M) with a ≤ b and for each x ∈ [a,b], the element
λa,b(x) = min{y ∈ [a,b] | x⊕y = b} exists in M, and ([a,b];⊕,λa,b,a,b) is an
MV-algebra;
(EMV4) for any x ∈ M, there is a ∈I(M) such that x ≤ a.

EMV-algebras cannot guarantee the existence of the top element 1. An ideal I
of an EMV-algebra M is a nonempty subset satisfying (i) for all x,y ∈ I, x⊕y ∈ I
and (ii) for each y ∈ I and x ∈ M, x ≤ y can deduce x ∈ I. Let Ideal(M) to
denote the set of all ideals of M. An ideal I of M is proper if I 6= M. A proper
ideal I is called prime if for any x,y ∈ M, x ∧ y ∈ I implies that x ∈ I or
y ∈ I. We use P(M) to denote the set of all prime ideals of M. An ideal I of M
is maximal if for all x ∈ M\I, we have 〈I ∪{x}〉 = M, where 〈I ∪{x}〉 = {z ∈
M | z ≤ a⊕n.x for some a ∈ I and some n ∈ N}. The set of all maximal ideals
of M is denoted by MaxI(M). It is well known that any maximal ideal of M must
be prime ([Dvurečenskij and Zahiri, 2019]). An EMV-algebra M is semisimple if
and only if Rad(M) = {0}, where Rad(M) , ∩{I | I ∈ MaxI(M)}. The set
Rad(M) is called the radical of M.

For two EMV-algebras (M1;∨,∧,⊕, 0) and (M2;∨,∧,⊕, 0), a mapping Φ :
M1 −→ M2 is called an EMV-homomorphism if Φ preserves the operations
∨,∧,⊕ and 0, and for each b ∈I(M1) and for each x ∈ [0,b], we have Φ(λb(x)) =
λΦ(b)(Φ(x)). Every MV-homomorphism is also an EMV-homomorphism, but the
converse is not necessarily true ([Dvurečenskij and Zahiri, 2019]). A mapping s :
M −→ [0, 1] is said a state-morphism on M if s is an EMV-homomorphism from



Xiaoxue Zhang, Hongxing Liu

the EMV-algebra M into the EMV-algebra of the real interval ([0, 1];∨,∧,⊕, 0)
with top element, such that there exists an element x ∈ M with s(x) = 1. The
set Ker(s) = {x ∈ M | s(x) = 0} is called the kernel of the state-morphism s
([Dvurečenskij and Zahiri, 2019]).

Theorem 2.1. ([Dvurečenskij and Zahiri, 2019, Theorem 4.2 (ii)]) Let M be an
EMV-algebra and s be a state-morphism on M. Then Ker(s) is a maximal ideal
of M. In addition, there is a unique maximal ideal I of M such that s = sI, where
sI : x 7−→ x/I for all x ∈ M.
Definition 2.3. ([Dvurečenskij and Zahiri, 2019, Definition 4.9]) Let Ω be a
nonempty set. A system T ⊆ [0, 1]Ω is called an EMV-clan if it satisfies the fol-
lowing conditions:
(1) 0 ∈ T such that 0(w) = 0 for all w ∈ Ω;
(2) if a ∈ T is a 0-1-valued function, then a − f ∈ T for each f ∈ T with
f(w) ≤ a(w) for all w ∈ Ω, and if f,g ∈ T with f(w),g(w) ≤ a(w) for all
w ∈ Ω, then f ⊕ g ∈ T , where (f ⊕ g)(w) = min{f(w) + g(w),a(w)} for all
w ∈ Ω;
(3) for each f ∈ T , there exists a 0-1-valued function a ∈ T such that f(w) ≤
a(w) for all w ∈ Ω;
(4) for given w ∈ Ω, there exists f ∈ T such that f(w) = 1.

From Dvurečenskij and Zahiri [2019, Proposition 4.10], we see that any EMV-
clan can be organized into an EMV-algebra. That is, every EMV-clan on some
Ω 6= ∅ is an EMV-algebra, see Dvurečenskij and Zahiri [2019].

3 Ideals of semisimple EMV-algebras and filters on
associated nonempty sets

Let M be a semisimple EMV-algebra. By Dvurečenskij and Zahiri [2019,
Theorem 4.11], there is an EMV-clan [0, 1]Ω on some Ω 6= ∅ such that M is
an EMV-subalgebra of [0, 1]Ω. In this section, for a semisimple EMV-algebra
M ⊆ [0, 1]Ω, we shall define the notion of limits along a filter. The connections
between ideals of M and filters on Ω are studied. For each f ∈ M and for all ε > 0,
we denote D(f,ε) = {x ∈ Ω | f(x) < ε}.
Definition 3.1. Let M be a semisimple EMV-algebra and F be a filter on Ω such
that M ⊆ [0, 1]Ω. For any f ∈ M and t ∈ [0, 1], we call that f converges to t
along F if for every ε > 0, there is S ∈ F such that | f(S) − t |< ε.
Proposition 3.1. Let M be a semisimple EMV-algebra and F be a proper filter on
Ω such that M ⊆ [0, 1]Ω. Then for each f ∈ M, there has at most one limit along
F.



Connections between ideals of semisimple EMV-algebras and set-theoretic filters

Proof. The proof is similar to Lele et al. [2021, Proposition 2.2].2
For any f ∈ M, the limit of f along a proper filter F on Ω does not necessarily

exist. But it would be unique if it exists by Proposition 3.1. We denote it by
limFf.

Let I be an ideal of M and F be a filter on Ω. We define
FI = {S ⊆ Ω | D(f,ε) ⊆ S for some f ∈ I and ε > 0}

and
IF = {f ∈ M | f converges to 0 along F}={f ∈ M | D(f,ε) ∈ F for all

ε > 0}.

Proposition 3.2. Let M be a semisimple EMV-algebra and F be a filter on Ω such
that M ⊆ [0, 1]Ω. For all f,g ∈ M:
(1) If limFf and limFg exist, then limF (f⊕g) exists and limF (f⊕g) = limFf⊕
limFg.
(2) If limFf exists, then limFλa(f) exists and limFλa(f) = λa(limFf), where a
is an idempotent element of M such that f ∈ [0,a].

Proof. (1) Suppose that f,g ∈ M, limFf and limFg exist. There exist-
s an idempotent element a ∈ I(M) such that f,g ∈ [0,a]. Also, we have
limFf,limFg ≤ a(x) for all x ∈ Ω. In the MV-algebra ([0,a];⊕,λa, 0,a), limFf
and limFg also exist. By Lele et al. [2021, Lemma 2.4], we have limF (f ⊕ g)
exists and limF (f ⊕g) = limFf ⊕ limFg.

(2) Recall that λa(f) = min{z ∈ [0,a] | z ⊕ f = a}, where a ∈ I(M) with
f ∈ [0,a]. Since ([0,a];⊕,λa, 0,a) is an MV-algebra, the result follows from Lele
et al. [2021, Lemma 2.4].2

Recall that an ultrafilter U on Ω is a filter which is maximal, in other words,
any filter that contains it is equal to it. An ultrafilter U on Ω is equally a collection
of subsets of Ω satisfying (i) U is proper, (ii) the intersection of two subsets in the
collection belongs to it and (iii) for any subset V , V ∈ U if and only if Ω\V /∈ U,
see Garner [2020, Definition 2]. From (iii), we see that Ω ∈ U for any ultrafilter
U on Ω. We shall show that the limits along an ultrafilter exist.

Proposition 3.3. Let M be a semisimple EMV-algebra and U be an ultrafilter on
Ω such that M ⊆ [0, 1]Ω. Then, for any f ∈ M, there has a unique limit along U.

Proof. Suppose that there is no t ∈ [0, 1] such that limUf = t. That is, for any
t ∈ [0, 1], there exists ε0 > 0 such that f−1(Ot) /∈ U, where Ot = (t−ε0, t + ε0).
In fact, if for all ε > 0, there exists t0 ∈ [0, 1] such that f−1(Ot0 ) ∈ U, where
Ot0 = (t0 − ε,t0 + ε). It follows that limUf = t0, which is a contradiction.
Since [0, 1] is compact, for each open covering {Ot | t ∈ [0, 1]} of [0, 1], where
Ot = (t − ε,t + ε), there exists a finite subset {Ot1,Ot2, ......,Otn} such that
[0, 1] =

⋃n
i=1 Oti . Since U is an ultrafilter on Ω, we have

⋃n
i=1 f

−1(Oti ) =



Xiaoxue Zhang, Hongxing Liu

f−1(
⋃n
i=1 Oti ) = f

−1([0, 1]) = Ω ∈ U. By Garner [2020, Definition 2], there
is j ∈ {1, 2, ......,n} such that f−1(Otj ) ∈ U, which is a contradiction. Hence, f
has at least one limit along U.

By Proposition 3.1, the uniqueness of the limit is clear.2

Theorem 3.1. Let M be a semisimple EMV-algebra and U be an ultrafilter on
Ω such that M ⊆ [0, 1]Ω. Consider the mapping ΦU : M −→ [0, 1] given
by ΦU (f) = limUf, where f ∈ M. Then ΦU is an EMV-homomorphism with
Ker(ΦU ) = IU .

Proof. Let ΦU : M −→ [0, 1] be a mapping defined by ΦU (f) = limUf,
where f ∈ M. By Proposition 3.3, the limit of f along U is unique. So ΦU
is well-defined. For all f,g ∈ M, there is a ∈ I(M) such that f,g ∈ [0,a]
and ([0,a];⊕,λa, 0,a) is an MV-algebra. Now we consider the restriction of ΦU
on [0,a]. From Lele et al. [2021, Proposition 2.6] we see that ΦU |[0,a] is an MV-
homomorphism. Clearly, ΦU (0) = 0. Also, we have ΦU (f⊕g) = ΦU (f)⊕ΦU (g),
ΦU (f ∨g) = ΦU (f)∨ΦU (g) and ΦU (f ∧g) = ΦU (f)∧ΦU (g). That is, ΦU is an
EMV-homomorphism. In addition, Ker(ΦU ) = {f ∈ M | limUf = 0} = IU .2

Theorem 3.2. Let M be a semisimple EMV-algebra such that M ⊆ [0, 1]Ω. We
have the followings:
(1) For each ideal I of M, FI is a filter on Ω. Moreover, if I is proper, then FI is
proper.
(2) For each filter F on Ω, IF is an ideal of M. Moreover, if F is proper, then IF is
proper.

Proof. (1) Let I be an ideal of M.
(i) For all ε > 0 and f ∈ I, we have D(f,ε) = {x ∈ Ω | f(x) < ε} ⊆ Ω.

Then Ω ∈ FI.
(ii) Let S1 ⊆ S2 ⊆ Ω and S1 ∈ FI. There exist f ∈ I and ε > 0 such that

D(f,ε) ⊆ S1 ⊆ S2. This implies that S2 ∈ FI.
(iii) Suppose that S1,S2 ∈ FI. There exist f,g ∈ I and ε,δ > 0 such that

D(f,ε) ⊆ S1 and D(g,δ) ⊆ S2. It follows that D(f,ε) ∩D(g,δ) ⊆ S1 ∩S2. In
addition, since D(f ⊕g, min(ε,δ)) ⊆ D(f,ε) ∩D(g,δ) and f ⊕g ∈ I, we have
D(f,ε)∩D(g,δ) ∈ FI. By (ii), it now follows that S1∩S2 ∈ FI. So FI is a filter
on Ω.

Let I be a proper ideal. Suppose that FI is not proper. Then ∅ ∈ FI. So there
exist f ∈ I and ε > 0 such that f(x) ≥ ε for all x ∈ Ω. We choose N ≥ 1 such
that f(x) ≥ ε ≥ 1

N
. Then Nf ∈ I and Nf(x) ≥ 1. It implies that 1 ∈ I and

I = M, which is a contradiction. Therefore, FI is proper.
(2) Let F be a filter on Ω.
(i) Since 0 ∈ IF , we have IF 6= ∅.



Connections between ideals of semisimple EMV-algebras and set-theoretic filters

(ii) For all f,g ∈ IF , by Proposition 3.2, we have limF (f ⊕ g) = limFf ⊕
limFg = 0. So f ⊕g ∈ IF .

(iii) Suppose that f ∈ M, g ∈ IF and f ≤ g. We have limFf ≤ limFg = 0.
Then f ∈ IF . Therefore, IF is an ideal of M.

Let F be a proper filter. If IF is not proper, then IF = M. For all f ∈ IF = M,
for all ε > 0, we have D(f,ε) ∈ F . There exists a ∈ I(M) such that f ≤ a and
a ∈ M = IF . So for any x ∈ Ω, there is g(x) > 0 such that a(x) ≥ g(x), where
g ∈ [0,a]. It follows that ∅ = D(a,g(x)) ∈ F , which is a contradiction. Hence,
IF is proper.2

Proposition 3.4. Let M be a semisimple EMV-algebra such that M ⊆ [0, 1]Ω.
Then we have the followings:
(1) For each ideal I of M, I ⊆ IFI .
(2) For each filter F on Ω, FIF ⊆ F .
(3) For each filter F on Ω, FIF = F if {0, 1}

Ω ⊆ M.

Proof. The proof is similar to Lele et al. [2021, Proposition 2.8].2

Proposition 3.5. Let M be a semisimple EMV-algebra such that M ⊆ [0, 1]Ω. We
have the followings:
(1) If {0, 1}Ω ⊆ M, then for each maximal ideal K of M, FK is an ultrafilter on
Ω.
(2) IU is a maximal ideal of M if U is an ultrafilter on Ω.
(3) If {0, 1}Ω ⊆ M, the converse of (2) is true.

Proof. (1) Let K be a maximal ideal of M and S ⊆ Ω. Suppose S /∈ FK. We
will show that Ω\S ∈ FK.

We define f ∈ M by

f(x) =

{
0 x ∈ S,
1 x /∈ S.

Then we have D(f, 0.5) = S /∈ FK. It follows that f /∈ K. Let b ∈ I(M) such
that f ∈ [0,b]. It follows from f /∈ K that f /∈ Kb, where Kb = K ∩ [0,b]. Since
K is a maximal ideal of M, by Dvurečenskij and Zahiri [2019, Proposition 3.22],
Kb is a maximal ideal of the MV-algebra ([0,b];⊕,λb, 0,b). By the maximality
of Kb, there exists n ≥ 1 such that λb(nf) ∈ Kb. Then λb(nf) ∈ K. Notice
that nf = f, which follows that λb(f) = λb(nf) ∈ K. In addition, we also have
Ω\S = Ω\D(f, 0.5) = D(λb(f), 0.5) ∈ FK. Hence, by Freiwald [2014, Chapter
IX, Theorem 3.5], FK is an ultrafilter on Ω.

(2) Let U be an ultrafilter on Ω. From Theorem 3.1, there is an EMV-homomor-
phism ΦU : M −→ [0, 1] defined by ΦU (f) = limUf. Since M ⊆ [0, 1]Ω is
semisimple, for given w ∈ Ω, there is f ∈ M such that f(w) = 1. So for



Xiaoxue Zhang, Hongxing Liu

{w} ⊆ Ω ∈ U and all ε > 0, we have f({w}) ⊆ (1 − ε, 1 + ε), which im-
plies that there exists f ∈ M such that ΦU (f) = limUf = 1. Hence, ΦU is a
state-morphism on M. By Theorem 2.1, Ker(ΦU ) = IU is a maximal ideal of M.

(3) If IU be a maximal ideal of M. Then FIU is an ultrafilter on Ω by (1). By
Proposition 3.4 (3), U = FIU is an ultrafilter.2

Proposition 3.6. Let M be a semisimple EMV-algebra and F be a filter on Ω such
that {0, 1}Ω ⊆ M ⊆ [0, 1]Ω. Then for any f ∈ M, F is an ultrafilter if and only if
f has a unique limit along F.

Proof. ⇒: If F is an ultrafilter. By Proposition 3.3 we see that f has a unique
limit along F.
⇐: Suppose that f has a unique limit along F , where f ∈ M. Consider the

mapping ΦF : M −→ [0, 1] defined by ΦF (f) = limFf. We have that ΦF is
well-defined. By the proof of Proposition 3.5, ΦF is a sate-morphism on M. So
Ker(ΦF ) = IF is a maximal ideal of M by Theorem 2.1. Therefore, F is an
ultrafilter on Ω by Proposition 3.5 (3).2

4 Closed ideals of semisimple EMV-algebras
In this section, we introduce the notions of closure operations and c-closed

ideals on EMV-algebras. We get a bijection between the set of all closed ideals
of M and the set of all filters on Ω. We exhibit a homeomorphism between the
topological space of all closed prime ideals of M and the topological space of all
weak ultrafilters on Ω.

Definition 4.1. A closure operation on an EMV-algebra M is a mapping c :
Ideal(M) −→ Ideal(M) satisfying the following conditions: for all I,J ∈
Ideal(M),
(C1) I ⊆ Ic;
(C2) if I ⊆ J, then Ic ⊆ Jc;
(C3) Icc = Ic; where Ic=c(I).

Proposition 4.1. Let M be a semisimple EMV-algebra and M ⊆ [0, 1]Ω. For each
ideal I of M, we denote Ic = IFI . Then c is a closure operation on M.

Proof. The proof is similar to Lele et al. [2021, Proposition 3.1].2
An ideal I of M is called c-closed if Ic = I. We frequently prefer to call an

ideal is closed instead of c-closed. The set of all closed ideals of M is denoted
by C(M). In the subsequent sections, we shall mainly study closed ideals of M,
where the closure operation is given by Proposition 4.1. Now we show that any
maximal ideal must be contained in C(M).



Connections between ideals of semisimple EMV-algebras and set-theoretic filters

Proposition 4.2. Let M be a semisimple EMV-algebra and M ⊆ [0, 1]Ω. Every
maximal ideal of M is a closed ideal.

Proof. Let I be a maximal ideal of M. IFI is a proper ideal by Theorem 3.2.
By Proposition 3.4 (1), we have I ⊆ IFI . Suppose I & IFI . For any f ∈ IFI\I,
by the maximality of I, we have M = 〈I ∪{f}〉⊆ IFI , which is a contradiction.
So I = IFI . We have shown that I is closed.2

Theorem 4.1. Let M be a semisimple EMV-algebra such that {0, 1}Ω ⊆ M ⊆
[0, 1]Ω. Then there is a bijection between the set of all closed ideals of M and the
set of all filters on Ω.

Proof. Let F(Ω) to denote the set of all filters on Ω. Define two mappings:
Θ : C(M) −→ F(Ω) by Θ(I) = FI and Υ : F(Ω) −→C(M) by Υ(F) = IF .

By Theorem 3.2 and Proposition 3.4(3), Θ and Υ are well-defined. For any I ∈
C(M) and F ∈ F(Ω), we get ΘΥ(F) = Θ(IF ) = FIF = F and ΥΘ(I) =
Υ(FI) = IFI = I. So ΘΥ and ΥΘ are identical mappings. Hence, Θ is a
bijection.2

Remark 4.1. From Theorem 4.1, we get a one-to-one correspondence between
the set of all closed ideals of M and the set of all filters on Ω. We shall study
the restriction of this correspondence. We define CM (M) = {I ∈ C(M) | I ∈
MaxI(M)} and FU (Ω) = {F | F is an ultrafilter on Ω}. Suppose that {0, 1}Ω ⊆
M ⊆ [0, 1]Ω. It is easy to verify that there is also a bijection between CM (M) and
FU (Ω).

In fact, define two mappings Ψ : FU (Ω) −→ CM (M) given by Ψ(U) = IU
and Ψ′ : CM (M) −→ FU (Ω) given by Ψ′(I) = FI. From Proposition 3.4 (3) and
Proposition 3.5 we see that Ψ and Ψ′ are well-defined. Similar to Theorem 4.1,
we can prove that Ψ is a bijection.

Next, we will study a special class of filters on Ω, which corresponds to closed
prime ideals of M. A filter F on Ω is called a weak ultrafilter if IF is a prime ideal
of M. We denote the set of all weak ultrafilters on Ω by W(Ω).

Proposition 4.3. Let M be a semsimple EMV-algebra and M ⊆ [0, 1]Ω. Every
ultrafilter on Ω is a weak ultrafilter.

Proof. Let F be an ultrafilter on Ω. Then IF is a maximal ideal of M by
Proposition 3.5 (2). So IF is prime ([Dvurečenskij and Zahiri, 2019]). Hence, F
is a weak ultrafilter.2

Proposition 4.4. Let M be a semisimple EMV-algebra and M ⊆ [0, 1]Ω. If I is a
prime ideal of M, FI is a weak ultrafilter on Ω.



Xiaoxue Zhang, Hongxing Liu

Proof. Let I be a prime ideal of M. Then FI is proper. It follows that IFI
is a proper ideal by Theorem 3.2. Suppose that f ∧ g ∈ IFI for f,g ∈ M. We
get D(f ∧ g,ε) ∈ FI for all ε > 0. Since D(f,ε),D(g,ε) ⊆ D(f ∧ g,ε) ∈ FI,
we have that at least one of D(f,ε) and D(g,ε) is nonempty. That is, f ∈ IFI or
g ∈ IFI . In fact, suppose that D(f,ε) and D(g,ε) are empty sets. It follows that
∅ = D(f ∧g,ε) ∈ FI, which is a contradiction. We have shown that FI is a weak
ultrafilter on Ω.2

Theorem 4.2. Let M be a semisimple EMV-algebra such that {0, 1}Ω ⊆ M ⊆
[0, 1]Ω. Then there is a bijection between the set of all closed prime ideals of M
and the set of all weak ultrafilters on Ω.

Proof. Let Pc(M) to denote the set of all closed prime ideals of M. Define
two mappings:

Φ : Pc(M) −→ W(Ω) defined by Φ(I) = FI and Γ : W(Ω) −→ Pc(M)
defined by Γ(F) = IF .
The mappings Φ and Γ are well-defined by Proposition 4.4, Proposition 3.4 (3)
and the definition of weak ultrafilters.

For any I ∈Pc(M) and F ∈ W(Ω), we have ΓΦ(I) = Γ(FI) = IFI = I and
ΦΓ(F) = Φ(IF ) = FIF = F . So ΦΓ and ΓΦ are identical mappings. Hence, Φ is
a bijection.2

Lemma 4.1. Let M be a semisimple EMV-algebra such that M ⊆ [0, 1]Ω. Then
there is a topology on the space W(Ω) which has Bw , {Uw(f) | f ∈ M} as a
basis, where Uw(f) = {F ∈ W(Ω) | f /∈ IF} for f ∈ M.

Proof. For any F ∈ W(Ω), there is f ∈ M\IF such that F ∈ Uw(f) ∈ Bw
since IF is prime.

Furthermore, for all f,g ∈ M, suppose that F ∈ Uw(f) ∩ Uw(g). Then
f /∈ IF and g /∈ IF . We have f ∧ g /∈ IF since IF is a prime ideal of M, which
follows that Uw(f) ∩Uw(g) ⊆ Uw(f ∧ g). For any F ∈ Uw(f ∧ g), we have
f ∧ g /∈ IF . It implies that D(f ∧ g,ε0) /∈ F for some ε0 > 0. It follows from
D(f,ε0),D(g,ε0) ⊆ D(f ∧ g,ε0) /∈ F and F ∈ W(Ω) that f /∈ IF and g /∈ IF .
Then Uw(f ∧g) ⊆Uw(f) ∩Uw(g). So Uw(f ∧g) = Uw(f) ∩Uw(g). That is, for
any F ∈ Uw(f) ∩Uw(g), there is Uw(f ∧ g) ∈ Bw such that F ∈ Uw(f ∧ g) ⊆
Uw(f) ∩Uw(g).

We have shown that the sets Uw(f) form a basis of the topology on W(Ω).2
From Lemma 4.1, we get a space W(Ω) whose topology is the topology gen-

erated by Bw. The open sets on W(Ω) are sets
⋃

Uw(f)∈Bw′
Uw(f), where Bw′ ⊆ Bw

and f ∈ M. When we refer to the topological space W(Ω), it will be with refer-
ence to the topology {

⋃
Uw(f)∈Bw′

Uw(f) | Bw′ ⊆Bw} ([Munkres, 2000]).



Connections between ideals of semisimple EMV-algebras and set-theoretic filters

Lemma 4.2. Let M be a semisimple EMV-algebra and M ⊆ [0, 1]Ω. The sets
Uc(f),f ∈ M form a basis of the topology on Pc(M), where Uc(f) = {I ∈
Pc(M) | f /∈ I} for f ∈ M.

Proof. We denote Bc = {Uc(f) | f ∈ M}.
For any I ∈ Pc(M), there is f ∈ M\I such that I ∈ Uc(f) ∈ Bc since I is

proper.
It is obvious that Uc(f) ∩Uc(g) ⊆ Uc(f ∧ g). Suppose that I ∈ Uc(f ∧ g).

Then f ∧ g /∈ I = IFI , where f,g ∈ M. Similar to Lemma 4.1, we have
f /∈ IFI = I and g /∈ IFI = I. It implies that Uc(f ∧ g) ⊆ Uc(f) ∩ Uc(g).
So Uc(f ∧ g) = Uc(f) ∩ Uc(g). That is, for any I ∈ Uc(f) ∩ Uc(g), there is
Uc(f ∧g) ∈Bc such that I ∈Uc(f ∧g) ⊆Uc(f) ∩Uc(g).

Hence, we have shown that Bc as the basis of the topology on Pc(M).2
By Lemma 4.2 and Munkres [2000], the topology on Pc(M) is the topology

generated by Bc where the open sets are sets
⋃

Uc(f)∈Bc′
Uc(f), where Bc′ ⊆ Bc and

f ∈ M.

Theorem 4.3. Let M be a semisimple EMV-algebra such that {0, 1}Ω ⊆ M ⊆
[0, 1]Ω. Then the two topological spaces Pc(M) and W(Ω) are homeomorphic.

Proof. Consider the two well-defined bijections Φ and Γ defined by Theorem
4.2.

(1) Φ is continuous. Without lost of generality, we shall prove that the preim-
age of any Uw(f) in W(Ω) is open in Pc(M). We have Φ−1(Uw(f)) = Γ(Uw(f)) =
{IF | f /∈ IF}. For any IF ∈ Γ(Uw(f)), where F ∈ W(Ω) and f /∈ IF , by Propo-
sition 3.4 (3), we have IF ∈ Pc(M). Then IF ∈ Uc(f). So Γ(Uw(f)) ⊆ Uc(f).
Moreover, for any I ∈ Uc(f), then I ∈ Pc(M) and f /∈ I. We have FI ∈ W(Ω)
and f /∈ I = IFI . It implies that I ∈ Γ(Uw(f)). So Uc(f) ⊆ Γ(Uw(f)). Hence,
Φ−1(Uw(f)) = Γ(Uw(f)) = Uc(f) is an open set in Pc(M).

(2) Γ is continuous. We shall prove Γ−1(Uc(f)) = Uw(f). We have Γ−1(Uc(f)) =
Φ(Uc(f)) = {FI | f /∈ I}. For any F ∈ Uw(f), we get F ∈ W(Ω) and f /∈ IF .
By Proposition 3.4 (3), we see that IF ∈ Pc(M) and F = FIF ∈ Φ(Uc(f)).
So Uw(f) ⊆ Φ(Uc(f)). For each FI ∈ Φ(Uc(f)), where I ∈ Pc(M) and
f /∈ I = IFI . It follows that FI ∈ Uw(f). So Φ(Uc(f)) ⊆ Uw(f). Thus
Γ−1(Uc(f)) = Φ(Uc(f)) = Uw(f) is an open set in W(Ω).

We have shown that Φ is a homeomorphism between Pc(M) and W(Ω).2

Example 4.1. There exist non-closed ideals.
Let M be a semisimple EMV-algebra such that M ⊆ [0, 1]Ω. Suppose that I is

an ideal of M. It is obvious that IFI = {f ∈ M | ∀ε > 0,∃δ > 0 and g ∈ I such
that g−1([0,δ)) ⊆ f−1([0,ε))}. In fact, for each f ∈ IFI , we have D(f,ε) ∈ FI



Xiaoxue Zhang, Hongxing Liu

for all ε > 0. So there exist g ∈ I and δ > 0 such that D(g,δ) ⊆ D(f,ε). It
follows that g−1([0,δ)) ⊆ f−1([0,ε)).

Let M = [0, 1]Z
+

, where all operations given by Definition 2.3 and Dvurečenskij
and Zahiri [2019, Proposition 4.10]. Let I = {f ∈ M | for all but finitely many
n ∈ Z+ such that f(n) = 0}. It follows from (f⊕g)(n) = min{f(n)+g(n),a(n)}
and simple exercises that I is an ideal of M, where f,g ∈ I and a ∈ M is a 0-1-
valued function such that f(n),g(n) ≤ a(n) for all n ∈ Z+.

Consider f given by f(n) = n+1
n2+1

(n ∈ Z+). Clearly, f ∈ M\I. It is easy to
see that f(n) → 0 when n → ∞. That is, for all ε > 0, there is N ∈ Z+ such
that f(n) < ε when n > N. Now we consider g ∈ M defined by

g(n) =

{
1
n

1 ≤ n ≤ N,
0 n > N.

Then g ∈ I and D(g,δ) ⊆ D(f,ε) for δ = min{ 1
N+1

,ε}. It implies that
g−1([0,δ)) ⊆ f−1([0,ε)). So f ∈ IFI . We have shown that I is a non-closed
ideal.

Definition 4.2. Let M be an EMV-algebra and I be an ideal of M. Then I is called
radical if I = Rad(M), where Rad(M) is the radical of M.

Proposition 4.5. Let M be a semisimple EMV-algebra such that M ⊆ [0, 1]Ω. The
following conditions are satisfied:
(1) The intersection of closed ideals of M is also a closed ideal.
(2) An ideal I of M is closed if I is radical.

Proof. (1) Let {Iα | α ∈ Λ} be a family of closed ideals of M. For each
β ∈ Λ, it follows from

⋂
α∈Λ Iα ⊆ Iβ that (

⋂
α∈Λ Iα)

c ⊆ Iβc = Iβ. Then
(
⋂
α∈Λ Iα)

c ⊆
⋂
β∈Λ Iβ =

⋂
α∈Λ Iα. Since

⋂
α∈Λ Iα ⊆ (

⋂
α∈Λ Iα)

c, we have
(
⋂
α∈Λ Iα)

c =
⋂
α∈Λ Iα. So

⋂
α∈Λ Iα ∈C(M).

(2) Suppose that I is radical. It implies that I = ∩{K | K ∈ MaxI(M)}. So
by Proposition 4.2 and (1), I is closed.2

5 Conclusion
For a semisimple EMV-algebra M such that M ⊆ [0, 1]Ω, we introduce the

notion of limits along a filter on Ω, which is unique if it exists. For all ultrafilters
U on Ω and for all f ∈ M, we give an EMV-homomorphism ΦU with kernel equal
to IU , which is defined by ΦU (f) = limUf. We study connections between ideals
of M and filters on Ω. We define closure operations and closed ideals on EMV-
algebras. We show that there is a bijection between the set of all closed ideals
of M and the set of all filters on Ω. We show that there is a homeomorphism



Connections between ideals of semisimple EMV-algebras and set-theoretic filters

between the topological space Pc(M) and the topological space W(Ω). We give
an example of a non-closed ideal and some properties of closed ideals.

Assume that F is a filter of the proper EMV-algebra M and I is an ideal of
M. We can show that IF = {λa(x) | x ∈ F,a ∈I(M),x ≤ a} is an ideal of M.
If F is a maximal filter of M, IF is a maximal ideal of M can be proved. We can
also get that FI = {λa(x) | x ∈ I,a ∈I(M)\I,x < a} is a filter of M under the
assumption that ∀a ∈I(M),a /∈ I =⇒ (∀b ∈I(M),a < b)λb(a) ∈ I.

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A. Dvurečenskij and O. Zahiri. On emv-algebras. Fuzzy Sets and Systems, 373:
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R. C. Freiwald. An introduction to set theory and topology. Washington University
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R. Garner. Ultrafilters, finite coproducts and locally connected classifying toposes.
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C. Lele, J. B. Nganou, and C. M. Oumarou. Ideals of semisimple mv-algebras and
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J. Munkres. Topology (2nd Edition). Prentice-Hall, Inc, London, 2000.