This paper introduces EMV-algebras, a new algebraic structure extending MV-algebras without requiring a top element, and explores their properties, representations, and categorical equivalences.
Contribution
The paper defines EMV-algebras, investigates their properties, and establishes their relationship with MV-algebras, including embeddings, ideal structures, and categorical equivalences.
Findings
01
Every EMV-algebra can be embedded into an MV-algebra.
02
Semisimple EMV-algebras are isomorphic to EMV-clans of fuzzy functions.
03
The category of EMV-algebras is categorically equivalent to certain MV-algebra and $ ext{l}$-group categories.
Abstract
The paper deals with an algebraic extension of MV-algebras based on the definition of generalized Boolean algebras. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-algebra and every EMV-algebra contains an MV-algebra. First, we present basic properties of EMV-algebras, give some examples, introduce and investigate congruence relations, ideals and filters on this algebra. We show that each EMV-algebra can be embedded into an MV-algebra and we characterize EMV-algebras either as MV-algebras or maximal ideals of MV-algebras. We study the lattice of ideals of an EMV-algebra and prove that any EMV-algebra has at least one maximal ideal. We define an EMV-clan of fuzzy sets as a special EMV-algebra. We show any semisimple EMV-algebra is isomorphic to an EMV-clan of fuzzy functions on a set. We consider theβ¦
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Taxonomy
TopicsAdvanced Algebra and Logic Β· Rough Sets and Fuzzy Logic
The paper deals with an algebraic extension of MV-algebras based on the definition of generalized Boolean algebras. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-algebra and every EMV-algebra contains an MV-algebra. First, we present basic properties of EMV-algebras, give some examples, introduce and investigate congruence relations, ideals and filters on this algebra. We show that each EMV-algebra can be embedded into an MV-algebra and we characterize EMV-algebras either as MV-algebras or maximal ideals of MV-algebras. We study the lattice of ideals of an EMV-algebra and prove that any EMV-algebra has at least one maximal ideal. We define an EMV-clan of fuzzy sets as a special EMV-algebra. We show any semisimple EMV-algebra is isomorphic to an EMV-clan of fuzzy functions on a set. We consider the variety of EMV-algebra and we present an equational base for each proper subvariety of the variety of EMV-algebras. We establish a categorical equivalencies of the category of proper EMV-algebras, the category of MV-algebras with a fixed special maximal ideal, and a special category of Abelian unital β-groups.
*Acknowledgement: AD is thankful for the support by the grants VEGA No. 2/0069/16 SAV and GAΔR 15-15286S *
1 Introduction
MV-algebras were defined by Chang [Cha] as an algebraic counterpart of many-valued reasoning. The principal result of the theory of MV-algebras is a representation theorem by Mundici [Mun1] saying that there is a categorical equivalence between the category of MV-algebras and the category of unital Abelian β-groups. Today the theory of MV-algebras is very deep and has many interesting connections with other parts of mathematics with many important applications to different areas. For more details on MV-algebras, we recommend the monographs [CDM, Mun3].
GMV-algebras, called also pseudo MV-algebras [GeIo] or non-commutative MV-algebras [Rac], are a non-commutative
generalization of MV-algebras and the algebraic counterparts of non-commutative many valued logic. Moreover, Galatos and Tsinakis generalized the notion of an MV-algebra in the context of residuated lattices to include both commutative and unbounded structures in [GaTs] and introduced the notion of generalized MV-algebra.
Indeed, a pseudo MV-algebra is a bounded integral generalized MV-algebra. They extended the relation between unital β-groups and pseudo MV-algebras established by Mundici [Mun1] for MV-algebras and by
DvureΔenskij [Dvu2] for pseudo MV-algebras. Many other results in these structures can be find in [DDT, Dvu1, Dvu2, RaSa, ShLu]. We note that MV-algebras are studied in the last period also in the frames of involutive semirings, see [DiRu].
There is another way how to generalize the concept of MV-algebras considering the definition of generalized Boolean algebras.
In the paper, first we use the definition of generalized Boolean algebras to extend the concept of MV-algebras. We call this structure an EMV-algebra. These algebras generalize MV-algebras, and in any EMV-algebra M, the interval [0,a] forms an MV-algebra for each idempotent aβM.
The paper is organized as follows. After Preliminaries, Section 2, we introduce EMV-algebras in Section 3. We present some examples and find relations between generalized Boolean algebras and EMV-algebras. We exhibit basic properties of EMV-algebras, and in particular, we show a one-to-one relationship between ideals and congruences of EMV-algebras.
In Section 4, we introduce state-morphisms as analogues of finitely additive probability measures, and we show their intimate relationship with maximal ideals. We introduce EMV-clans as EMV-algebras of fuzzy sets where all algebraic operations are defined by points; they are exactly
semisimple EMV-algebras up to isomorphism. We also show that any semisimple EMV-algebra can be embedded into an MV-algebra.
In Section 5, we show a relationship between ideals and filters of EMV-algebras and we show that nevertheless an EMV-algebra has not necessarily a top element, it contains at least one maximal ideal. Finally, we prove that every EMV-algebra can be embedded into an MV-algebra. Moreover, every EMV-algebra is either an MV-algebra or it can be embedded into an MV-algebra as its maximal ideal. This allows us to study subvarieties of the variety of EMV-algebras and to present an equational base for each subvariety of EMV-algebras. In addition, in Section 6, we present mutual categorical equivalencies of the category of proper EMV-algebras with the special category of MV-algebras with a fixed maximal ideal having enough idempotents or with a special category of Abelian unital β-groups.
2 Preliminaries
In the section, we gather some basic notions relevant to MV-algebras which will be needed in the next sections. For more details,
we recommend to consult
[DiSe, CDM, Mun3] for MV-algebras.
An MV-algebra is an algebra (M;β,β²,0,1) (henceforth write simply M=(M;β,β²,0,1)) of type (2,1,0,0), where (M;β,0) is a
commutative monoid with the neutral element [math] and for all x,yβM, we have:
(i)
xβ²β²=x;
2. (ii)
xβ1=1;
3. (iii)
xβ(xβyβ²)β²=yβ(yβxβ²)β².
In any MV-algebra (M;β,β²,0,1),
we can define also the following operations:
[TABLE]
In addition, let xβM. For any integer nβ₯0, we set
[TABLE]
and
[TABLE]
Moreover, the relation xβ€yβxβ²βy=1 is a partial order on M and (M;β€) is a lattice, where
xβ¨y=(xβy)βy and xβ§y=xβ(xβ²βy). Note that, for each xβM, xβ² is the least element
of the set {yβMβ£xβy=1}.
We use MV to denote the category of MV-algebras whose objects are MV-algebras and morphisms are MV-homomorphisms.
A non-empty subset I of an MV-algebra (M;β,β²,0,1) is called an ideal of M if I is a down set which is closed under β.
The set of all ideals of M is denoted by I(M). It is well known that for each x,yβM, if IβI(M) and y,xβyβI, then xβI.
For each ideal I of M, the relation ΞΈIβ on M defined by (x,y)βΞΈIβ if and only if xβy,yβxβI is a congruence relation on M, and x/I and M/I will denote {yβMβ£(x,y)βΞΈIβ} and {x/Iβ£xβM}, respectively. A prime ideal is an ideal Iξ =M of M such that M/I is a linearly ordered MV-algebra, or equivalently, for all x,yβM, xβyβI or yβxβI. The set of all minimal prime ideals of M is denoted by Min(M).
An element a of an MV-algebra (M;β,β²,0,1) is called a
Boolean element if there is bβM such that aβ§b=0 and aβ¨b=1. The set of all Boolean elements of M forms a Boolean algebra; it is denoted by B(M).
Theorem 2.1**.**
[CDM, Thm. 1.5.3]*
For every element x in an MV-algebra M, the following conditions are equivalent:*
(i)
xβB(M);
(ii)
xβ¨xβ²=1;
(iii)
xβ§xβ²=0;
(iv)
xβx=x;
(v)
xβx=x;
(vi)
xβy=xβ¨y* for all yβM;*
(vii)
xβy=xβ§y* for all yβM.*
Let (M;+,0) be a monoid. An element aβM is called idempotent if a+a=a. The set of all idempotent elements of M
is denoted by I(M).
A monoid (G;+,0) is called partially ordered if it is equipped with a partial order relation β€ that is compatible with +, that is,
aβ€b implies x+a+yβ€x+b+y for all x,yβG.
A partially ordered monoid (G;+,0) is called a lattice ordered monoid or simply an β-monoid
if G with its partially order relation is a lattice. In a similar way,
a group (G;+,0) is said to be a partially ordered group if it is a partially ordered monoid. A partially ordered group (G;+,0) is called a lattice ordered group or simply an β-group
if G with its partially order relation is a lattice. An element xβG is called positive if 0β€x. An element u of an β-group (G;+,0) is called a strong unit of G if, for each gβG, there exists nβN such that gβ€nu. A couple (G,u), where G is an β-group and u is a fixed strong unit for G, is said to be a unitalβ-group.
If (G;+,0) is an Abelian β-group with strong unit u, then the interval [0,u]:={gβGβ£0β€gβ€u} with the operations xβy:=(x+y)β§u and xβ²:=uβx forms
an MV-algebra, which is denoted by Ξ(G,u)=([0,u];β,β²,0,u). Moreover, if (M;β,0,1) is an MV-algebra, then according to the famous theorem by Mundici, [Mun1], there exists a unique (up to isomorphism) unital Abelian β-group
(G,u) with strong u such that Ξ(G,u) and (M;β,0,1) are isomorphic (as MV-algebras).
Let A be the category of unital Abelian β-groups whose objects are unital β-groups and morphisms are unital β-group morphisms (i.e. homomorphisms of β-groups preserving fixed strong units). It is important to note that MV is a variety whereas A not because it is not closed under infinite products. Then Ξ:AβMV is a functor between these categories. Moreover, there is another functor from the category of MV-algebras to A sending M to a Chang β-group induced by good sequences of the MV-algebra M, which is
denoted by Ξ:MVβA. For more details relevant to these functors, please see [CDM, Chaps 2 and 7].
Theorem 2.2**.**
[CDM, Thms 7.1.2, 7.1.7]*
The composite functors ΞΞ and ΞΞ are naturally equivalent to the identity functors of
MV and A, respectively. Therefore, the categories A and MV are categorically equivalent.*
Recall that a residuated lattice is an algebra (L;β¨,β§,β ,β,/,e) of type (2,2,2,2,2,0) such that (L;β¨,β§) is a lattice,
(L;β ,e) is a monoid, and for all x,y,zβL,
[TABLE]
A residuated lattice is called commutative if it satisfies the identity xβ y=yβ x and is called integral if
it satisfied the identity xβ§e=x.
Galatos and Tsinakis, [GaTs], introduced the concept of a generalized MV-algebra (GMV-algebra) which is a
residuated lattice that satisfies the identities
[TABLE]
It is well known that bounded commutative integral GMV-algebras and MV-algebras coincide (see [JiTs, GaTs]).
3 EMV-algebras, Ideals, and Congruences
In the section, we define qEMV-algebras and EMV-algebras which form an important subclass of qEMV-algebras. We present some examples and we define subalgebras and homomorphisms. We show that EMV-algebras form a variety. Congruences on the class of EMV-algebras are in a one-to-one correspondence with the set of ideals. We show that every semisimple EMV-algebra can be embedded into an MV-algebra.
Definition 3.1**.**
An algebra (M;β¨,β§,β,0) of type (2,2,2,0) is called a quasi extended MV-algebra (qEMV-algebra in short) if it satisfies the
following conditions:
(EMV1)Β (M;β¨,β§,0) is a distributive lattice with the least element [math];
(EMV2)Β (M;β,0) is a commutative ordered monoid with neutral element [math];
(EMV3)Β for all a,bβI(M) such that aβ€b, the element
[TABLE]
exists in M for all xβ[a,b], and the algebra ([a,b];β,Ξ»a,bβ,a,b) is an MV-algebra.
We say that an qEMV-algebra (M;β¨,β§,β,0) has enough idempotent elements if, for each xβM, there is aβI(M) such that
xβ€a. An extended MV-algebra, an EMV-algebra in short, is a qEMV-algebra (M;β¨,β§,β,0) which has enough idempotent elements.
From now on, in this paper, we usually denote Ξ»0,bβ by Ξ»bβ.
Now we present some examples of qEMV-algebras and EMV-algebras, respectively.
Example 3.2**.**
(1)
Any MV-algebra (M;β,β²,0,1) is an EMV-algebra. Let a,bβB(M). By Proposition 2.1, for each
x,yβ[a,b], we have xβyβ₯aβy=aβ¨y=yβ₯a and xβyβ€xβb=xβ¨b=b, thus
[a,b] is closed under β. It can be easily seen that x:=(xβ²β¨a)β§b is the least element of the set
{zβ[a,b]β£xβz=b}. Moreover, for each xβ[a,b],
[TABLE]
Therefore, ([a,b];β,β,a,b) is an MV-algebra (for more details we refer to [DMN]), and so any MV-algebra is an EMV-algebra.
2. (2)
Any generalized Boolean algebra (M;β¨,β§,0) (studied also as a Boolean ring, see [LuZa, Kel]) forms an EMV-algebra (M;β¨,β§,β,0), where β=β¨ and if aβ€b, then Ξ»a,bβ(x) is the unique relative complement of x in the interval [a,b].
3. (3)
Let (B;β¨,β§) be a generalized Boolean algebra and (M;β,β²,0,1) be an MV-algebra. Then it can be easily
shown that MΓB is an EMV-algebra.
4. (4)
Any bounded qEMV-algebra is an MV-algebra. Note that if M is a qEMV-algebra with the greatest element 1, then
M=[0,1] and M is an MV-algebra.
5. (5)
Let G be a non-trivial β-group. The set of positive elements G+ of G with the natural operation + and natural ordering is a qEMV-algebra. Since [math] is the only idempotent element, so G+ is not an EMV-algebra.
6. (6)
Let {(Miβ;β,β²,0,1)}iβIβ be a family of MV-algebras and S={fββiβIβMiββ£Supp(f)\mboxisfinite}.
Clearly, S is closed under β¨, β§ and β. Moreover, if fβS, then u=(uiβ)iβIβ, where uiβ=1 for all
iβSupp(f) and uiβ=0 for all iβIβSupp(f), is an element of S which is idempotent and fβ€u.
It can be easily shown that S is a qEMV-algebra and so S is an EMV-algebra. We will denote this qEMV-algebra by
βiβIβMiβ.
7. (7)
Let (M;β,β²,0,1) be an MV-algebra and A be any
ideal of M. Then similarly to (1), we can see that
A is a qEMV-algebra.
8. (8)
Let J be an ideal of an MV-algebra (A;β,β²,0,1) and B be a generalized Boolean algebra. Then
BΓJ with the pointwise operations forms a qEMV-algebra.
9. (9)
Let (M;β¨,β§,β,0) be an EMV-algebra. Then it is straightforward to show that
(I(M);β¨,β§,0) is a generalized Boolean algebra. Moreover,
M is an MV-algebra if and only if I(M) is a Boolean algebra.
10. (10)
Every finite EMV-algebra is an MV-algebra.
11. (11)
(i) For each x,yβM, there exist a,bβI(M) such aβ€xβyβ€b and so
x,yβ[a,b]. Since ([a,b];β,Ξ»a,bβ,a,b) is an MV-algebra, then the element Ξ»a,bβ(Ξ»a,bβ(x)βy)βy is the supremum of x and y taken in the MV-algebra [a,b] and it coincides with xβ¨y. Similarly, xβ§y=Ξ»a,bβ(Ξ»a,bβ(x)βΞ»a,bβ(Ξ»a,bβ(x)βy)). In addition, if a0ββ€x,yβ€b0β for some a0β,b0ββI(M), then Ξ»a,bβ(Ξ»a,bβ(x)βy)=xβ¨y=Ξ»a0β,b0ββ(Ξ»a0β,b0ββ(x)βy).
(ii) If a,c,b are idempotents with aβ€cβ€b and xβM such xβ[a,b], then xβc=xβ¨c and xβ§c=Ξ»a,bβ(Ξ»a,bβ(x)βΞ»a,bβ(Ξ»a,bβ(x)βc)).
(iii) The Riesz Decomposition Theorem holds: If zβ€xβy, then there are xzββ€x and yzββ€y such that x=xzββyzβ. Or if x1ββx2β=y1ββy2β, there are four elements c11β,c12β,c21β,c22ββM such that x1β=c11ββc12β, x2β=c21ββc22β, y1β=c11ββc21β and y2β=c21ββc22β. These facts follow from the analogous properties in the MV-algebra [0,a], where aβ₯x,y,x1β,x2β,y1β,y2β.
Remark 3.4**.**
Let (L;β¨,β§,β ,β,/,e) be a residuated lattice such that (L;β ,e) is commutative and (L;β¨,β§) is a lattice with the least element [math].
Then y/x=xβy for all x,yβL, and xβy is usually written xβy (see [GaTs, p. 12]). We claim that L is not an EMV-algebra. Otherwise, since 0β0 is the greatest element of L, then by Example 3.2(4), L is an MV-algebra. Therefore, from Example 3.2 we get that there exists an EMV-algebra which is not a generalized MV-algebra in the sense of [GaTs].
The following proposition shows that the notions of a qEMV-algebra and of an EMV-algebra can be defined also in a simpler way.
Proposition 3.5**.**
Let (M;β¨,β§,β,0) be an algebra of type (2,2,2,0). Then
(i)
M* is a qEMV-algebra if and only if ([0,b];β,Ξ»bβ,0,b) is an MV-algebra for all bβI(M). In such a case, Ξ»a,bβ(x)=Ξ»bβ(x)β¨a.*
(ii)
M* is an EMV-algebra if and only if*
(EMVIβ1)* (M;β¨,β§,0) is a lattice with the least element 0;*
(EMVIβ2)* (M;β,0) is a commutative monoid with neutral element [math];*
(EMVIβ3)* for each xβM, there is bβI(M) with xβ€b such that ([0,b];β,Ξ»bβ,0,b) is an MV-algebra.*
Proof.
(i) Let, for all bβI(M), ([0,b];β,Ξ»bβ,0,b) be an MV-algebra. Take a,bβI(M) such that aβ€b.
We show that for each xβ[a,b], Ξ»a,bβ(x) exists and Ξ»a,bβ(x)=Ξ»bβ(x)β¨a.
Indeed, xβ(Ξ»bβ(x)β¨a)=(xβΞ»bβ(x))β¨(xβa)=b. Now, let zβ[a,b] such that xβz=b. Then
by definition of Ξ»bβ(x), we have Ξ»bβ(x)β€z and so Ξ»bβ(x)β¨aβ€zβ¨a=z. Now, we can easily see that ([a,b];β,Ξ»a,bβ,a,b) is an MV-algebra. Therefore, (M;β¨,β§,β,0) is a qEMV-algebra. The proof of the converse is clear.
(ii) Let (EMVIβ1)β(EMVIβ3) hold.
First we show that (M;β,0) is an ordered monoid. Let x,yβM be such that xβ€y. For each zβM, by the assumption,
there exists aβI(M) such that yβ¨zβ€a and ([0,a];β,Ξ»aβ,0,a) is an MV-algebra and so
xβzβ€yβz (since x,y,zβ[0,a]). That is, (M;β,0) is an ordered monoid. In a similar way, we can show that
(M;β¨,β§) is a distributive lattice. Now, by (i), it is enough to show that for all bβI(M), ([0,b];β,Ξ»bβ,0,b) is an MV-algebra.
Let b be an arbitrary idempotent element of M. By the assumption,
there is uβI(M) such that bβ€u and ([0,u];β,Ξ»uβ,0,u) is an MV-algebra.
It can be easily seen that Ξ»bβ(x)=bβ§Ξ»uβ(x) for all xβ[0,b], and similarly
to Example 3.2(1), ([0,b];β,Ξ»bβ,0,b) is an MV-algebra. Therefore, M is an EMV-algebra.
Clearly, the converse holds.
β
Let (M;β¨,β§,β,0) be an EMV-algebra. Then for all aβI(M), we have a well-known binary operation
[TABLE]
on the MV-algebra ([0,a];β,Ξ»aβ,0,a).
Inspired by the equivalence in Proposition 3.5(i), we can define the notion of a qEMV-subalgebra also in the following equivalent way.
(ii) Let (M1β;β¨,β§,β,0) and (M2β;β¨,β§,β,0) be qEMV-algebras. A map f:M1ββM2β is called a qEMV-homomorphism
if f preserves the operations β¨, β§, β and [math], and for each bβI(M1β) and for each xβ[0,b], f(Ξ»bβ(x))=Ξ»f(b)β(f(x)).
If M1β and M2β are two EMV-algebras, then each qEMV-homomorphism f:M1ββM2β is said to be an EMV-homomorphism.
Lemma 3.7**.**
Let M1β and M2β be two qEMV-algebras and f:M1ββM2β be a qEMV-homomorphism.
(i)
If B is a subalgebra of M2β, then fβ1(B) is a subalgebra of M1β.
(ii)
If M1β and M2β are EMV-algebras, f:M1ββM2β is an EMV-homomorphism, and A is an EMV-subalgebra of M1β, then f(A) is an EMV-subalgebra of M2β.
(i) Definition 3.6 yields that each MV-homomorphism is an EMV-homomorphism, but the converse is not true in general case.
Indeed, let (M;β¨,β§,β,0) be an EMV-algebra and aβI(M). Then ([0,a];β,Ξ»aβ,0,a) is an MV-algebra (and so an EMV-algebra). Clearly, the inclusion map i:[0,a]βM is an EMV-homomorphism. Now, if (M;β,β²,0,1) is an MV-algebra and aβMβ{0,1} is its Boolean element, then the inclusion map i:[0,a]βM is not an MV-homomorphism (since i(a)ξ =1).
An EMV-homomorphism f:MβN is said to be strong if, for each bβI(N), there exists aβI(M) such that bβ€f(a). Clearly, any MV-homomorphism is strong as an EMV-homomorphism. Moreover, if (M;β,β²,0,1) is an MV-algebra and f:MβN is an EMV-homomorphism, then N is an MV-algebra and f is an MV-homomorphism, since xβ€f(1) for all xβN.
(ii) Let f:MβN be a strong EMV-homomorphism and S be a full subset of I(M) (that is, for each bβI(M), there exists aβS such that bβ€a). For each aβI(M), set faβ:=fβ£[0,a]ββ. Then we have
(F1) {faββ£aβS} is a family of MV-homomorphisms;
(F2) if a,bβS such that aβ€b, then faβ=fbββ£[0,a]ββ;
(F3) {f(a)β£aβS} is a full subset of I(N).
Conversely, if (M;β¨,β§,β,0) and (N;β¨,β§,β,0) are two EMV-algebras, S is a full subset of M and {faβ:[0,a]βNβ£aβS} is a family of maps satisfying conditions (F1)β(F3), then the map f:MβN defined by f(x)=faβ(x), where aβS and xβ[0,a], is a strong EMV-homomorphism.
Proposition 3.9**.**
Let (M;β¨,β§,β,0) be a qEMV-algebra, a,bβI(M) such that aβ€b.
Then for each xβ[0,a], we have
(i)
Ξ»aβ(x)=Ξ»bβ(x)β§a;
(ii)
Ξ»bβ(x)=Ξ»aβ(x)βΞ»bβ(a);
(iii)
Ξ»a,bβ(x)=Ξ»bβ(x)β¨a;
(iv)
Ξ»aβ(x)β€Ξ»bβ(x);
(v)
Ξ»bβ(a)* is an idempotent, and Ξ»aβ(a)=0.*
Proof.
Since ([0,b];β,Ξ»bβ,0,b) is an MV-algebra, aβ[0,b] and aβa=a, we get that aβ¨x=aβx for all
xβ[0,b]. Let xβ[0,a].
(i) From Ξ»bβ(x)β§aβ[0,a] and (Ξ»bβ(x)β§a)βx=(Ξ»bβ(x)βx)β§(aβx)=bβ§(aβ¨x)=a,
it follows that Ξ»aβ(x)β€Ξ»bβ(x)β§a. Also,
b=aβΞ»bβ(a)=(xβΞ»aβ(x))βΞ»bβ(a)=xβ(Ξ»aβ(x)βΞ»bβ(a)), so
Ξ»aβ(x)βΞ»bβ(x)β₯Ξ»bβ(x). Hence
(Ξ»aβ(x)βΞ»bβ(a))β§aβ₯Ξ»bβ(x)β§a. Since a is a Boolean element of the MV-algebra [0,b], then so is
Ξ»bβ(a), which implies that
Ξ»bβ(x)β§aβ€(Ξ»aβ(x)βΞ»bβ(a))β§a=(Ξ»aβ(x)β¨Ξ»bβ(a))β§a=Ξ»aβ(x)β§a=Ξ»aβ(x).
Summing up the above results, we get that
Ξ»aβ(x)=Ξ»bβ(x)β§a.
(ii) By (i) we have
[TABLE]
On the other hand, x,aβ[0,b] and xβ€a, it follows that Ξ»bβ(a)β€Ξ»bβ(x) and so Ξ»bβ(x)β¨Ξ»bβ(a)=Ξ»bβ(x).
Therefore, Ξ»bβ(x)=Ξ»aβ(x)β¨Ξ»bβ(a).
(v) Since [0,b] is an MV-algebra and aβ[0,b] is an idempotent, Ξ»bβ(a) is the relative complement of a in [0,b], so it is also an idempotent. The rest statement Ξ»aβ(a)=0 follows from definition of Ξ»aβ.
β
Remark 3.10**.**
Let (M;β¨,β§,β,0) and (N;β¨,β§,β,0) be
EMV-algebras and f:MβN be a map preserving β and [math].
If for each xβM, there is bβI(M) such that xβ[0,b] and
f(Ξ»bβ(x))=Ξ»f(b)β(f(x)), then f is an EMV-homomorphism.
Indeed, if x,yβM, there is
bβI(M) such that x,yβ[0,b]. Since ([0,b];β,Ξ»bβ,0,b)
is an MV-algebra,
xβ¨y=Ξ»bβ(Ξ»bβ(x)βy)βy and xβ§y=Ξ»bβ(Ξ»bβ(x)β¨Ξ»bβ(y)). Hence, f preserves β¨ and
β§.
It follows that for each xβM, there is aβI(M) such that xβ€a and f:[0,a]β[0,f(a)] is a homomorphism of
MV-algebras.
Now, let a be an arbitrary idempotent element of M. Then there exists
uβI(M) such that aβ€u and
f:[0,u]β[0,f(u)] is a homomorphism of MV-algebras. By Proposition 3.9(v), for each xβ[0,a], we have
[TABLE]
It follows that f is an EMV-homomorphism.
Theorem 3.11**.**
Let EMV be the class of EMV-algebras. Then EMV is a variety.
Proof.
The class EMV is closed under HSP.
β
Definition 3.12**.**
Let (M;β¨,β§,β,0) be a qEMV-algebra. An equivalence relation ΞΈ on M is called a congruence relation or simply a congruence if
it satisfies the following conditions:
We denote by Con(M) the set of all congruences on M.
The next proposition makes our work easier when we want to verify that an equivalence relation on a qEMV-algebra is a congruence.
Proposition 3.13**.**
An equivalence relation ΞΈ on an EMV-algebra (M;β¨,β§,β,0) is a congruence if it is compatible with β¨, β§ and β, and
for all (x,y)βΞΈ, there exists bβI(M) such that x,yβ€b and (Ξ»bβ(x),Ξ»bβ(y))βΞΈ.
Let ΞΈ be a congruence relation on an EMV-algebra (M;β¨,β§,β,0) and M/ΞΈ={[x]β£xβM} (we usually use x/ΞΈ instead of [x]). Consider the induced operations β¨, β§ and β on M/ΞΈ defined by
First, we show that for all x/ΞΈβ[0/ΞΈ,a/ΞΈ], Ξ»aβ(x)/ΞΈ is the least element of the set
{z/ΞΈβ[0/ΞΈ,a/ΞΈ]β£z/ΞΈβx/ΞΈ=a/ΞΈ}. For each x/ΞΈβ[0/ΞΈ,a/ΞΈ], we have
x/ΞΈ=x/ΞΈβ§a/ΞΈ=(xβ§a)/ΞΈ and xβ§aβ[0,a]. So, we can assume that xβ[0,a].
If y/ΞΈβ[0/ΞΈ,a/ΞΈ] such that x/ΞΈβy/ΞΈ=a/ΞΈ, then (xβy,a)βΞΈ and x,y,aβ[0,a], thus
(xβy,a)βΞΈaβ, that is x/ΞΈaββy/ΞΈaβ=a/ΞΈaβ
(which implies that x/ΞΈβy/ΞΈ=a/ΞΈ).
Hence by (3.1), y/ΞΈaββ₯Ξ»aβ(x)/ΞΈaβ
and so y/ΞΈβ₯Ξ»aβ(x)/ΞΈ. Also, Ξ»aβ(x)/ΞΈβx/ΞΈ=a/ΞΈ. Thus Ξ»a/ΞΈβ(x/ΞΈ) exists and
is equal to Ξ»aβ(x)/ΞΈ. Now, it is straightforward to check that Ξ»a/ΞΈβ satisfies the conditions (1) and (3) in
definition of MV-algebras. It follows that ([0/ΞΈ,a/ΞΈ];β,Ξ»a/ΞΈβ,0/ΞΈ,a/ΞΈ) is an MV-algebra.
Therefore, by Proposition 3.5(ii), (M/ΞΈ;β¨,β§,β,0/ΞΈ) is an EMV-algebra, and the mapping xβ¦x/ΞΈ is an EMV-homomorphism from M onto M/ΞΈ.
Example 3.14**.**
(i) Let f:MβN be a qEMV-homomorphism. Then ker(f)={(x,y)βMΓNβ£f(x)=f(y)} is a congruence on M.
(ii) Let {(Miβ;β,β²,0,1)β£iβI} be a family of MV-algebras and ΞΈiβ be a congruence on Miβ for all iβI. Set
[TABLE]
Then clearly, ΞΈ is an equivalence relation on βiβIβMiβ which is compatible with β¨, β§ and β.
Let (f,g)βΞΈ, for some f,gββiβIβMiβ. Define h:IββiβIβMiβ by
[TABLE]
Clearly, h is an idempotent element of βiβIβMiβ and f,gβ€h. By Example 3.2(5), we know that
βiβIβMiβ is an EMV-algebra. Let bββiβIβMiβ be an idempotent.
Consider the MV-algebra ([0,b];β,Ξ»bβ,0,b).
It can be easily seen that for all Ξ±β[0,b],
[TABLE]
Since ΞΈiβ is a congruence on Miβ for all iβI, then
(Ξ»bβ(f),Ξ»bβ(g))βΞΈ and so by Proposition 3.13, ΞΈ is a congruence on the qEMV-algebra βiβIβMiβ.
Definition 3.15**.**
A non-empty subset I of a qEMV-algebra (M;β¨,β§,β,0) is called an ideal if for each x,yβM
(i)
xβyβI for all x,yβI;
(ii)
xβ€y and yβI implies that xβI.
The set of all ideals of M is denoted by Ideal(M). Clearly {0},MβIdeal(M). An ideal I of M is proper if Iξ =M.
Similarly as for MV-algebras, see [CDM, Prop 1.2.6], we have a one-to-one relationship between the set of ideals and the set of congruences on a qEMV-algebra.
Theorem 3.16**.**
If ΞΈ is a congruence on an EMV-algebra (M;β¨,β§,β,0), then IΞΈβ:=0/ΞΈ is an ideal of M.
Conversely, let I be an ideal of an EMV-algebra (M;β¨,β§,β,0). Then the relation ΞΈIβ defined by
[TABLE]
is a congruence on M. In addition, the mapping Iβ¦ΞΈIβ is a bijection from the set Ideal(M) onto the set of congruences on M.
Proof.
Let ΞΈ be a congruence on an EMV-algebra (M;β¨,β§,β,0). Then it can be easily shown that
IΞΈβ:=0/ΞΈ is an ideal of M.
Hence
(x,y),(y,z)βΞΈIuββ. Since ΞΈIuββ is a congruence on the MV-algebra ([0,u];β,Ξ»uβ,0,b), it proves that ΞΈIβ is a congruence on M.
In an analogous way, and using [CDM, Prop 1.2.6], we have that the mapping Iβ¦ΞΈIβ is a bijection in question.
β
Definition 3.17**.**
Let I be an ideal of an EMV-algebra (M;β¨,β§,β,0). We denote the EMV-algebra (M/ΞΈIβ;β¨,β§,β,0/ΞΈIβ) simply by M/I, and M/I is called the quotient EMV-algebra of M induced by I.
In Example 3.2(2), we showed that any generalized Boolean algebra is an EMV-algebra. Now, let (M;β¨,β§,β,0) be an EMV-algebra and I be an ideal of M such that for each xβM, there is aβI(M) such that xβ[0,a] and Ξ»aβ(x)β§xβI. Then M/I is a generalized Boolean algebra. By definition, it suffices to show that for each xβM, x/Iβx/I=x/I
(since in this case, the MV-algebra [x/I,y/I] is a Boolean algebra).
First, we note that for each xβM, xβI if and only if x/I=0/I.
Let xβM. Then by the assumption, there exists aβI(M) such that
xβ§Ξ»aβ(x)βI, so in the MV-algebra ([0/I,a/I];β,Ξ»a/Iβ,0/I) we have
x/Iβ§Ξ»a/Iβ(x/I)=x/Iβ§Ξ»aβ(x)/I=0/I.
Hence, x/I is a Boolean element of this MV-algebra and so x/Iβx/I=x/I.
We recall that a qEMV-algebra M is simple if M possesses only two congruences, and due to Theorem 3.16, this is equivalent to the condition Ideal(M)={{0},M}.
Theorem 3.18**.**
Any simple EMV-algebra is a simple MV-algebra.
Proof.
Let (M;β¨,β§,β,0) be a simple EMV-algebra. We claim that ([0,a];β,Ξ»aβ,0,a) is a simple MV-algebra for
all aβI(M). Otherwise, there are aβI(M)β{0} and an ideal I of the MV-algebra [0,a] such that Iξ =[0,a] and {0}ξ =I. So, I is an ideal of the EMV-algebra M different from {0} and M, which is a contradiction. Thus,
([0,a];β,Ξ»aβ,0,a) is a simple MV-algebra. We show that a=maxM. Put xβM. Then there exists bβI(M) such
that x,aβ€b. Since ([0,b];β,Ξ»bβ,0,b) is simple and aβ[0,b], then by [CDM, Thm. 3.5.1],
there is nβN such that n.a=b. From aβI(M) it follows that a=n.a, hence a=b. That is xβ€a.
Therefore, M=[0,a] and so it is a simple MV-algebra.
β
Example 3.19**.**
The qEMV-algebra in Example 3.2(5) is a simple qEMV-algebra if G=R.
By the proof of [CoDa, Thm. 2.2], the generalized Boolean algebra can be embedded into a Boolean algebra of subsets of MaxI(B). Since MaxI(B) is finite, then B is also a finite set. Therefore, the element β{aβ£aβB} is the top element of B as well as of M which implies M is an MV-algebra.
Assume that M is linearly ordered. If I1β and I2β are two different ideals, there are xβI1ββI2β and yβI2ββI1β. If xβ€y, then xβI2β, an absurd, and if yβ€x again we get an absurd. Hence, I1β=I2β, β£MaxI(M)β£=1, and by the first part, M is an MV-algebra.
β
Theorem 3.26**.**
Let I be a maximal ideal of an EMV-algebra (M;β¨,β§,β,0). Then M/I is an MV-algebra.
Proof.
We claim that M/I is a simple EMV-algebra. Let B be an ideal of the EMV-algebra M/I. Set
A:={xβMβ£x/IβB}. Clearly, 0βA. If x,yβA, then x/I,y/IβB and so (xβy)/I=x/Iβy/IβB.
Also, if x,yβM such that xβ€yβA, then clearly, x/Iβ€y/IβB and so x/IβB, that is xβA. So, A is an ideal of the EMV-algebra (M;β¨,β§,β,0) which clearly contains I. Hence I=A or A=M.
Therefore, M/I is a simple EMV-algebra. Now, from Theorem 3.18, it follows that M/I is an MV-algebra.
β
Due to [CDM, Thm 3.5.1], if A is a simple MV-algebra, then A is isomorphic to a unique MV-subalgebra of the MV-algebra of the real interval [0,1], hence in Theorem 3.26, M/I can be embedded in a unique way into the real interval [0,1] because there is an element xβMβI, so that x/I>0/I, and we can assume that the maximal value in M/I is equal to the real number 1.
4 State-morphisms, Maximal Ideals, and EMV-clans
In the section we introduce state-morphisms which in the case of MV-algebras are exactly extremal states. States are averaging of truth-values in Εukasiewicz logic and they correspond to an analogue of finitely additive measures in classical logic. We show that state-morphisms are in a one-to-one correspondence with maximal ideals. We present EMV-clans as EMV-algebras of fuzzy sets where all algebraic operations are defined by points. They are prototypes of semisimple EMV-algebras.
According to [Mun2], a mapping s on an MV-algebra M such that s:Mβ[0,1] is (i) a state if (a) s(1)=1 and (b) s(aβb)=s(a)+s(b) whenever aβb=0; (ii) a state-morphism if s is an MV-homomorphism from M into the MV-algebra of the real interval [0,1]; (iii) an extremal state if s=Ξ»s1β+(1βΞ»)s2β, where s1β,s2β are states on M and Ξ» is a real number such that 0<Ξ»<1, then s1β=s2β=s. Due to [Mun2] and [Dvu3], we have that (i) every non-degenerate MV-algebra possesses at leat one state; (ii) each state-morphism is a state, and it is an extremal state, and conversely, (iii) every extremal state is a state-morphism.
Inspired by the notion of a state-morphism on MV-algebras, we define a state-morphism on an EMV-algebra M as follows: A mapping s:Mβ[0,1] is a state-morphism if s is an EMV-homomorphism from M into the EMV-algebra of the real interval [0,1] such that there is an element xβM with s(x)=1. In the latter case, we can assume that there is an idempotent a such that s(a)=1. We define the set \mboxKer(s)={xβMβ£s(x)=0}, the kernel of a state-morphism s.
The basic properties of state-morphisms are as follows.
Proposition 4.1**.**
Let s be a state-morphism on an EMV-algebra M. Then
(i)
s(0)=0;
(ii)
s(a)β{0,1}* for each idempotent aβM;*
(iii)
if xβ€y, then s(x)β€s(y);
(iv)
s(Ξ»aβ(x))=s(a)βs(x)* for each xβ[0,a], aβI(M).*
(v)
\mboxKer(s)* is a proper ideal of M.*
Proof.
(i) It is trivial.
(ii) Let aβI(M) and assume 0<s(a). There is the least integer n0β such that n0β.s(a)=1 in the MV-algebra of the real interval [0,1]. Then s(a)=s(n0β.a)=1.
(iii) Let xβ€y. There is an idempotent aβM such that yβ€a. If s(a)=0, then the restriction saβ of s onto the MV-algebra [0,a] is the zero function, so that s(x)=s(y). If s(a)=1, then the restriction saβ is a state-morphism on the MV-algebra [0,a], and the monotonicity of saβ in [0,a] implies s(x)β€s(y).
(iv) We have Ξ»aβ(x)βx=a for each xβ[0,a]. If s(a)=0, the statement follows from (iii). If s(a)=1, the restriction saβ is a state-morphism on the MV-algebra [0,a], and for saβ we have saβ(Ξ»aβ(x))=1βs(x) which proves (iv).
(v) It follows easily from (i) and (iii).
β
Theorem 4.2**.**
(i)* If I is a maximal ideal of an EMV-algebra M, then M/I can be embedded in a unique way into the MV-algebra of the real interval [0,1] such that the mapping sIβ:xβ¦x/I, xβM, is a state-morphism.*
(ii)* If s is a state-morphism, then \mboxKer(s) is a maximal ideal of M.
In addition, there is a unique maximal ideal I of M such that s=sIβ.*
(iii)* If for state-morphisms s1β and s2β we have \mboxKer(s1β)=\mboxKer(s2β), then s1β=s2β.*
Proof.
(i) Let I be a maximal ideal. Due to Theorem 3.26 and Proposition 3.23, the mapping sIβ is in fact an EMV-homomorphism. Since I is maximal, there is xβMβI, so that x/I>0/I. Without loss of generality, we can assume that the greatest value in M/I is 1. Hence, sIβ is a state-morphism on M.
(iii) There is an idempotent a such that aβ/{xβMβ£s1β(x)=0}. Then for the restrictions of s1β and s2β onto the MV-algebra [0,a], we have by [Dvu3, Prop 4.5] that s1β(x)=s2β(x) for each xβ[0,a], then s1β(x)=s2β(x) for each xβM.
β
Let SM(M) denote the set of state-morphisms on an EMV-algebra M. In Theorem 5.6, it will be proved that every M contains a maximal ideal, so that by Theorem 4.2(i), SM(M) is non-void whenever Mξ ={0}.
Using Theorem 3.24, we show that every state-morphism on I(M) can be extended to a state-morphism on an EMV-algebra M.
Theorem 4.3**.**
Every state-morphism s on I(M) of an EMV-algebra M can be extended to a state-morphism s^ on M.
Let M be an EMV-algebra satisfying the general comparability property. Then every state-morphism on I(M) can be extended to a unique state-morphism on M.
Now, there is a natural question βunder which suitable condition on an ideal I of an EMV-algebra (M;β¨,β§,β,0), the
quotient EMV-algebra induced by I, M/I, is an MV-algebraβ?
Lemma 4.6**.**
Let (M;β¨,β§,β,0) be an EMV-algebra, I an ideal of M, b,dβI(M) such that bβ€d, and let Ibβ and Idβ be ideals of the MV-algebras [0,b] and [0,d], respectively. If x,yβ[0,b] such that
x/Idβ=y/Idβ, then x/Ibβ=y/Ibβ.
Proof.
Let x,yβ[0,b] such that x/Idβ=y/Idβ. Then Ξ»dβ(Ξ»dβ(x)βy),Ξ»dβ(Ξ»dβ(y)βx)βIdββI.
We will show that Ξ»bβ(Ξ»bβ(x)βy),Ξ»bβ(Ξ»bβ(y)βx)βIbβ.
[TABLE]
In a similar way, we can see that Ξ»bβ(Ξ»bβ(y)βx)βIbβ. Therefore, x/Ibβ=y/Ibβ.
β
The following equivalencies on the induced order for a quotient EMV-algebra are used in Theorem 4.7.
Let I be an ideal of an EMV-algebra (M;β¨,β§,β,0) and x,yβM. Then
[TABLE]
Theorem 4.7**.**
Let I be an ideal of EMV-algebra (M;β¨,β§,β,0). Then M/I is an MV-algebra if and only if
there exists aβI(M) such that Ξ»bβ(a)βI for all bβI(M) greater than a.
Proof.
Let M/I be an MV-algebra. Then there exists aβM such that x/Iβ€a/I for all xβM. Since M is an EMV-algebra,
there is bβI(M) such that aβ€b and so a/I=b/I. Thus, without loss of generality we can assume that aβI(M).
Let b be an arbitrary element of I(M) greater than a. Since a/I is the maximum of M/I, then a/I=b/I and so
there is dβI(M) such that a,bβ€d and Ξ»dβ(Ξ»dβ(a)βb),Ξ»dβ(Ξ»dβ(b)βa)βI.
Since a,bβ[0,b], by Lemma 4.6, we get that Ξ»bβ(a)=Ξ»bβ(Ξ»bβ(b)βa)βI.
Conversely, let xβM. Then there exists bβI(M) such that x,aβ[0,b]. Since
Ξ»bβ(Ξ»bβ(x)βa)β€Ξ»bβ(a)βI, then x/Iβ€a/I. Therefore, M/I is an MV-algebra. β
From Theorem 3.26 and Theorem 4.7 we get that each maximal ideal satisfies the condition in Theorem 4.7.
We say that an EMV-algebra M is semisimple if Rad(M):=β{Iβ£IβMaxI(M)}={0}; the set Rad(M) is said to be the radical of M.
In what follows, we show that every generalized Boolean algebra is semisimple.
Lemma 4.8**.**
Let Mξ ={0} be a generalized Boolean algebra. Then:
(i)
An ideal I of M is maximal if and only if, for each aβ/I and each bβM with a<b, Ξ»bβ(a)βI.
(ii)
M* is a semisimple EMV-algebra.*
Proof.
If M has the top element, M is a Boolean algebra and the statement is well-known from the theory of Boolean algebras. Thus let us assume that M has no top element. In generalized Boolean algebras we have xβy=xβ¨y.
(ii) First we have to note that every generalized Boolean algebra Mξ ={0} possesses at least one maximal ideal, as it will be proved in Theorem 5.6 below.
Let xβRad(M). If x>0, using Zornβs lemma, we have that there is a maximal filter F of M containing x. By Theorem 5.6 below, the set IFβ={Ξ»aβ(z)β£zβF,aβI(M),zβ€a} is a maximal ideal of M. Let b be an idempotent such that bβ/IFβ and xβ€b. Then Ξ»bβ(x)βIFβ and xβIFβ. Hence, b=xβΞ»bβ(x)βIFβ which is absurd. Consequently, x=0.
β
An important family of EMV-algebras is a family of EMV-clans of fuzzy sets which as we show below are only semisimple EMV-algebras.
Any EMV-clan T can be organized into an EMV-algebra of fuzzy sets where all operations are defined by points.
(i)* Let f,gβT and f,gβ€a,b, where a,b are characteristic functions from T. Then (fβg)(Ο)=max{f(Ο)+g(Ο)βa(Ο),0}=max{f(Ο)+g(Ο)βb(Ο),0} and fβgβT. Similarly, fβg=fβ(aβg)=fβ(bβg)βT.*
We assert that T is an EMV-algebra of fuzzy sets. By (iv) we have that if f,gβT, there is a characteristic function aβT such that f,gβ€a. If bβT is another characteristic function such that aβ€b, we have
(i) We can define fβaβg in the similar but dual way as we defined already fβaβg if f,gβ[0,a], and if f,gβ[0,b], then fβaβg=fβbβg.
(ii) Let f,gβT and fβ€g. There is a characteristic function aβT such that f,g belong to the MV-algebra [0,a]. Then gβf=gβ(aβf)β[0,a] and similarly fβaβ(aβg)=fβbβ(bβg), so we can define fβg:=fβaβ(aβg).
Then the mapping xβ¦x^ preserves 0,β¨,β§,β, if x^β€a^, then xβ€a, so that Ξ»^aβ(x)=a^βx^βM, and [0^,a^] is an MV-algebra. In other words, M is an EMV-clan of fuzzy sets, and the mapping xβ¦x^ is an EMV-isomorphism from M onto M.
Given a state-morphism s on M, there is an element xβM such that s(x)=1. Then x^(s)=1.
Let gβC0β(T)βT. Then g=1βg0β for some g0ββT. Whence, g0β=1β(1βg0β)βT and T is a maximal ideal of C0β(T).
β
Corollary 4.16**.**
Every proper semisimple EMV-algebra can be embedded into an MV-algebra as its maximal ideal.
Proof.
It follows from Theorem 4.15 and Theorem 4.11.
β
5 Filters, Ideals and Representation of EMV-algebras
One of the main purposes of this part is to show that any EMV-algebra has at least one maximal ideal. For this reason, first we define the notion of a filter of an EMV-algebra, showing that for each filter F there is an ideal related to it. Since any bounded EMV-algebra with top element 1 is an MV-algebra, the existence of a maximal ideal is an easy application of Zornβs lemma if 0ξ =1. Therefore, we will prove the existence of a maximal ideal in any proper EMV-algebraM, that is, M has no maximal element. Therefore, in a proper EMV-algebra M, for each xβM, we can find an idempotent element a such that x<a. In particular, we show that every EMV-algebra can be embedded into an MV-algebra, and we show a basic result saying that every EMV-algebra is either an MV-algebra or it can be embedded into an MV-algebra as its maximal ideal.
Lemma 5.1**.**
Let (M;β¨,β§,β,0) be an EMV-algebra. For all x,yβM, we define
[TABLE]
where aβI(M) and x,yβ€a. Then β:MΓMβM is an order preserving, associative well-defined binary operation on M which does not depend on aβI(M) with x,yβ€a.
In addition, if x,yβM, xβ€y, then
[TABLE]
for all idempotents a,b of M with x,yβ€a,b, and
[TABLE]
If x,yβ[0,a] for some idempotent aβM, then
[TABLE]
Moreover, a binary operation β on M defined by xβy=xβΞ»aβ(y) is correctly defined for all x,yβM.
An element xβM is idempotent if and only if xβx=x.
Proof.
It suffices to show that β is
well defined. Put x,yβM. We show that for all a,bβI(M) such that x,yβ€a,b, we have
Ξ»aβ(Ξ»aβ(x)βΞ»aβ(y))=Ξ»bβ(Ξ»bβ(x)βΞ»bβ(y)). That is xβaββy=xβbββy.
Indeed, take cβI(M) such that a,bβ€c. Then by Proposition 3.9, we have
[TABLE]
In a similar way, we can show that Ξ»cβ(Ξ»cβ(x)βΞ»cβ(y))=Ξ»bβ(Ξ»bβ(x)βΞ»bβ(y)).
To prove associativity, let x, y and z be elements of an EMV-algebra (M;β¨,β§,β,0). Put cβI(M) such that x,y,zβ€c. Then by definition of β, we have xβy=Ξ»cβ(Ξ»cβ(x)βΞ»cβ(y)), yβz=Ξ»cβ(Ξ»cβ(y)βΞ»cβ(z)) and
both belong to [0,c]. It follows that (xβy)βz=Ξ»cβ(Ξ»cβ(xβy)βΞ»cβ(z)) and
xβ(yβz)=Ξ»cβ(Ξ»cβ(x)βΞ»cβ(yβz)), which implies that (xβy)βz=xβ(yβz). Therefore, in any EMV-algebra (M;β¨,β§,β,0) the binary operation β is associative.
In a similar way, we can see that β is order preserving.
Now let xβ€y, x,yβ€a,b for some a,bβI(M). There is an idempotent c such that a,bβ€c. Check and use Proposition 3.9(ii)
[TABLE]
and
[TABLE]
because for yβ€aβ€c we have Ξ»cβ(a)β€Ξ»cβ(y). This implies yβΞ»aβ(x)=yβΞ»cβ(x). In the same way we have yβΞ»bβ(x)=yβΞ»cβ(x) establishing
yβΞ»aβ(x)=yβΞ»bβ(x).
To prove (5.2), it is enough to calculate it in the MV-algebra [0,a].
Now let x,yβ€a for some aβI(M). Then xβΞ»aβ(xβ§y)=xβ(Ξ»aβ(x)β¨Ξ»aβ(y))=(xβΞ»aβ(x))β¨(xβΞ»aβ(y))=xβΞ»aβ(y).
The property x is an idempotent of M iff xβx follows from definition of the operation β.
β
For any integer nβ₯1 and any x of an EMV-algebra M, we can define
[TABLE]
and if M has a top element 1, we define also x0=1.
Definition 5.2**.**
A non-empty subset F of an EMV-algebra (M;β¨,β§,β,0) is called a filter if it satisfies the following conditions:
(i)
for each x,yβM, if xβ€y and xβF, then yβF (formally F is an upset);
(ii)
for each x,yβF, xβyβF.
The set of all filters of M is denoted by Fil(M). Clearly, MβFil(M), and a filter F is proper if Fξ =M. A proper filter which cannot be a proper subset of another proper filter of M is said to be maximal, and we denote by MaxF(M) the set of maximal filters of M. By Zornβs lemma, MaxF(M)ξ =β .
Let (M;β¨,β§,β,0) be a proper EMV-algebra. Then there is a non-zero idempotent element aβM. We can easily see that βa is a filter of the EMV-algebra M, which is clearly a proper subset of M. In a similar way, we can see that βaβ{a} is also a
proper filter of M.
Proposition 5.3**.**
Let F be a filter of a proper EMV-algebra (M;β¨,β§,β,0). Then the set
[TABLE]
is an ideal of M.
Proof.
First, we note that for each xβM, we have
[TABLE]
Let x,yβM such that xβIFβ and yβ€x. Then there exists aβI(M) such that xβ€a and Ξ»aβ(x)βF.
Since x,yβ[0,a], then Ξ»aβ(x)β€Ξ»aβ(y) and so by the assumption, Ξ»aβ(y)βF. It follows that yβIFβ.
Now, suppose that x,yβIFβ. Then there exist a,bβI(M) such that xβ€a and yβ€b and Ξ»aβ(x)βF and Ξ»bβ(y)βF.
Put cβI(M) such that a,bβ€c. Then by Proposition 3.9,
Ξ»cβ(x),Ξ»cβ(y)βF and so Ξ»cβ(x)βΞ»cβ(y)βF.
Since Ξ»cβ(x),Ξ»cβ(y)β€c, Ξ»cβ(x)βΞ»cβ(y)=Ξ»cβ(xβy), hence xβyβIFβ.
Therefore, IFβ is an ideal of M.
β
Proposition 5.4**.**
Let F be a proper filter of an EMV-algebra (M;β¨,β§,β,0).
(i)* For each xβM, the least filter βFβͺ{x}β of M containing
Fβͺ{x} is the set {zβMβ£zβ₯yβxn,Β βnβN,βyβF}.*
(ii)* F is a maximal filter if and only if, for each xβ/F, there are an integer n and an idempotent b with xβ€b such that Ξ»bβ(xn)βF.*
Proof.
The proof of the first part is straightforward.
For the second one, let F be a maximal filter and xβ/F. By (i), there are an integer n and an element cβF such that 0=cβxn. There is an idempotent bβ₯x,c, so that c,x are in the MV-algebra [0,b]. Then cβxn can be calculated in [0,b], so that cβ€Ξ»bβ(xn) and Ξ»bβ(xn)βF.
The converse follows easily from (i).
β
Lemma 5.5**.**
Let F be a proper filter of a proper EMV-algebra (M;β¨,β§,β,0).
(i) Otherwise, aβIFβ implies that there exists bβI(M) such that aβ€b and Ξ»bβ(a)βF and so
Ξ»bβ(a),aβ€b and Ξ»bβ(a),aβF. Thus 0=aβΞ»bβ(a)βF, which is
a contradiction.
(ii) It follows from definition of IFβ.
(iii) Let aβI(M) such that aβ/IFβ. Then by definition, for all bβI(M) with aβ€b, Ξ»bβ(a)β/F.
If aβ/F, then βFβͺ{a}β=M (since F
is maximal) and so there exist nβN and xβF such that
0β₯xβan=xβuβan for some uβI(M) such that x,aβ€u.
Also, ([0,u];β,Ξ»uβ,0,u) is an MV-algebra and a is a Boolean element of it, so an=a and
Ξ»uβ(a) is the greatest element of [0,u] satisfying the equation zβa=0. It follows that
xβ€Ξ»uβ(a) and Ξ»uβ(a)βF, which is a contradiction.
whence Ξ»bβ(a)β€xβJ. Therefore, Ξ»bβ(a)βJ (since J is an ideal).
(v) Let x,yβM. If yβ₯xβFJβ, then there exists aβI(M)βJ such that x<a and Ξ»aβ(x)βJ. Let bβI(M) such that a,y<b. Then Ξ»bβ(y)β€Ξ»bβ(x)=Ξ»aβ(x)βΞ»bβ(a). By the assumption, Ξ»bβ(a)βJ, so Ξ»aβ(x)βΞ»bβ(a)βJ, which implies that Ξ»bβ(y)βJ. Thus yβFJβ and FJβ is an upset.
Moreover, if x,yβFJβ, then there exist a,bβI(M) such that x<a and y<b and Ξ»aβ(x),Ξ»bβ(y)βJ.
Let cβI(M) such that a,b<c. Then by the assumption, Ξ»cβ(a),Ξ»cβ(b)βJ and hence by Proposition 3.9, we have
Ξ»cβ(x)=Ξ»aβ(x)βΞ»cβ(a)βJ and Ξ»cβ(y)=Ξ»bβ(y)βΞ»cβ(b)βJ. It follows that
Ξ»cβ(x)βΞ»cβ(y)βJ. Now, from definition of FJβ, we have \lambda_{c}\big{(}\lambda_{c}(x)\oplus\lambda_{c}(y)\big{)}\in F_{J}.
That is, xβy=xβcββyβFJβ. Therefore, FJβ is a filter of M.
β
Theorem 5.6**.**
Any proper EMV-algebra has at least one maximal ideal. In addition, if F is a maximal filter of M, then
[TABLE]
is a maximal ideal of M.
Proof.
Let (M;β¨,β§,β,0) be a proper EMV-algebra. Then M has a non-zero idempotent a and βa is a proper filter of M.
By Zornβs lemma, we can easily see that, S, the set of filters of M not containing [math], has at least one maximal element which is clearly a maximal filter of M, F say. Set
A proper ideal I of an EMV-algebra (M;β¨,β§,β,0) is called prime if, for each x,yβM,
xβ§yβM implies that xβM or yβM. We denote by P(M) the set of prime ideals of M.
We note that (i) in the next statement was already proved in Proposition 3.23, here we proved it in a different way using e.g. the Riesz Decomposition Property.
Every prime ideal of a proper EMV-algebra (M;β¨,β§,β,0) is contained in a unique maximal ideal of M.
Proof.
Let I be a prime ideal of an EMV-algebra M. Put aβMβI.
For each bβI(M) with a<b, we have
Ξ»bβ(a)β§a=0βI and so Ξ»bβ(a)βI. Hence, I satisfies condition (5.4) and so by Lemma 5.5(v),
FIβ is a filter of M. It can be easily seen that X={FβFil(M)β£FIββF,Β 0β/F} has a maximal element, say H,
(note that by definition [math] does not belong to FIβ) which is clearly a maximal filter of M.
Now, by Lemma 5.5(iii) and the proof of Theorem 5.6, IHβ is a maximal ideal of M containing I.
(i)* Let P be a prime ideal of an EMV-algebra and let I be a proper ideal of M containing P. Then I is a prime ideal of M.*
(ii)* For each prime ideal J of M, the set S(J)={IβIdeal(M)β£JβIξ =M} is a linearly ordered set of prime ideals with respect to the set theoretical inclusion with a top element.*
In the proof of Theorem 5.6 we showed that if F is a maximal filter of a proper EMV-algebra (M;β¨,β§,β,0), then
IFβ is a maximal ideal of M. Now, let I be a maximal ideal of M. By Lemma 5.5(v), FIβ is a filter of M and so FIβ is contained in a maximal filter H.
(i)
Since FIββH, then from definition it follows that IFIβββIHβ.
Let xβI. Put aβMβI such that x<a. Then Ξ»aβ(x)βFIβ and so aβFIβ. By definition,
x=Ξ»aβ(Ξ»aβ(x))βIFIββ. Therefore, IβIFIββ.
From (i), (ii) and (iii) it follows that IβIIFβββIHβξM and so I=IHβ. Therefore, any maximal
ideal I of M is of the form IHβ for some maximal filter H of M.
Theorem 5.12**.**
Let (M;β¨,β§,β,0) be an EMV-algebra and let I be a proper ideal of M, aβMβI. Then there exists an ideal P of M which is maximal with respect to the property IβP and aβMβP. In addition, P is prime.
Let bβI(M) such that x,y,z1β,z2ββ€b. Since ([0,b];β,Ξ»bβ,0,b) is an MV-algebra, then
by [GeIo, Prop 1.17(i)], we have
[TABLE]
which is a contradiction. Therefore, P is prime.
β
Corollary 5.13**.**
Every proper ideal of an EMV-algebra M can be embedded into a maximal ideal of M.
Proof.
Let I be a proper ideal of M. By Theorem 5.12, there is a prime ideal P of M containing I. Applying Proposition 5.9, we have the assertion in question.
β
Note that, if P is a prime ideal of an EMV-algebra (M;β¨,β§,β,0) and x,yβM, then there exists aβI(M) such that x,yβ€a.
Since ([0,a];β,Ξ»aβ,0,a) is an MV-algebra, then by [CDM, Prop 1.1.7],
[TABLE]
which implies that Ξ»aβ(Ξ»aβ(x)βy)βP or Ξ»aβ(Ξ»aβ(y)βx)βP.
Theorem 5.14**.**
Let (M;β¨,β§,β,0) be a proper EMV-algebra. Then the radical \mboxRad(M) of M, the intersection of all maximal ideals of M, is the set
[TABLE]
Proof.
Let xβMββ{Iβ£IβMaxI(M)}. Then there exists a maximal ideal I such that xβ/I.
By Remark 5.11, there exists a maximal filter H such that I=IHβ and so xβ/IHβ.
[TABLE]
If for all nβN, n.xβ€Ξ»aβ(x), then Ξ»aβ(n.x)β₯x and so Ξ»aβ(x)nβ₯x (note that
([0,a];β,Ξ»aβ,0,a) is an MV-algebra). It follows that xβIHβ=I which is a contradiction. Hence
[TABLE]
Now, let xβMβRad(M). Then for all aβI(M) with xβ€a, there exists nβN that n.xβ°Ξ»aβ(x). It follows that
Ξ»aβ(Ξ»aβ(n.x)βΞ»aβ(x))>0. By Theorem 5.12, there is a prime ideal P of M such that
Ξ»aβ(Ξ»aβ(n.x)βΞ»aβ(x))β/P and so Ξ»aβ(xβn.x)βP and aβ/P (otherwise, from
Ξ»aβ(Ξ»aβ(n.x)βΞ»aβ(x))β€a we have Ξ»aβ(Ξ»aβ(n.x)βΞ»aβ(x))βP).
By Proposition 5.9, there is a maximal ideal J of M containing P.
We claim that the maximal filter J induced from Proposition 5.9 does not contain a.
Recall that J=FHβ, where H is a maximal filter of M containing FPβ. Check
[TABLE]
So our claim is true. From Ξ»aβ((n+1).x)βJ it follows that (n+1).xβ/J and so xβ/J. Hence
xβ/β{Iβ£IβMaxI(M)}. Therefore, β{Iβ£IβMaxI(M)}βRad(M).
β
Remark 5.15**.**
Let (B;β¨,β§) be a generalized Boolean algebra that is not a Boolean algebra and (M;β,β²,0,1) be an MV-algebra. In Example 3.2(3),
we showed that MΓB is an EMV-algebra. By [CoDa, Thm. 2.2], there exists a Boolean algebra B such that
B is a maximal ideal of B. Clearly, MΓB is an MV-algebra containing MΓB.
It is straightforward to prove that MΓB is a maximal ideal of MΓB. Therefore, any EMV-algebra of
the form MΓB, where M is an MV-algebra and B is a generalized Boolean algebra is a maximal ideal of an MV-algebra.
Proposition 5.16**.**
Let (M;β¨,β§,β,0) be an EMV-algebra. If there exists {aiββ£iβN}βI(M) such that {a1ββ¨a2ββ¨β―β¨anββ£nβN} is a full subset of I(M) and aiββ§ajβ=0 for all distinct elements i,jβN, then the EMV-algebra M can be embedded into an MV-algebra.
Proof.
Let (M;β¨,β§,β,0) be an EMV-algebra with the mentioned properties. For each nβN, vnβ:=a1ββ¨β―β¨nβ is an idempotent element of M and so [0,vnβ] is an MV-algebra. Define a map f:MββnβNβ[0,vnβ] by f(x)=(xβ§vnβ)nβNβ for all xβM. Clearly, f is a one-to-one map which preserves β¨, β§ and [math]. Now, we show that f preserves β. Let n be an arbitrary positive integer. We will show that Οnββf:Mβ[0,vnβ] is a homomorphism of EMV-algebras, where Οnβ is the n-th canonical projection map. Put x,yβM. Then there exists aβI(M) such that x,y,vnββ[0,a]. Since ([0,a];β,Ξ»aβ,0,a) is an MV-algebra, then from
[TABLE]
it follows that f preserves β. From definition of the unary operation β² in the MV-algebra βnβNβ[0,vnβ] and Remark 3.10, it can be easily seen that f is an EMV-algebra homomorphism. Therefore, M can be embedded into an MV-algebra.
β
Theorem 5.17**.**
Let (M;β¨,β§,β,0) be an EMV-algebra. Then (Ideal(M);β) is a complete Brouwer lattice. Consequently, EMV is a congruence distributive variety.
Proof.
Indeed, we have:
(i) Clearly, for each family {Jiββ£iβT}βIdeal(M), we have
[TABLE]
(ii) Let I be an ideal of M. Then for each xβIβ(βiβTβJiβ) by Proposition 3.20,
there exist nβN and ci1ββ,β¦,cinββββiβTβJiβ such that xβ€ci1ββββ―βcinββ.
Put bβM such that xβ€b. Since ([0,b];β,Ξ»bβ,0,b) is an MV-algebra, then by [GeIo, Prop 1.17],
[TABLE]
Hence, β distributes over arbitrary β.
(iii) For each ideal I of M, from (i) and (ii) it follows that
max{JβIdeal(M)β£IβJ={0}} exists and it is denoted by Iβ₯. In addition, Iβ₯={xβMβ£xβ§y=0\mboxforallyβI}.
(iv) Since (Ideal(M);β) is a Brouwer lattice, it is distributive see e.g. [Bly, p. 151]. Due to Theorem 3.2, the lattice of congruences on M, (Con(M);β) is also a Brouwer lattice. Therefore, EMV is a congruence distributive variety.
β
Now we show that every subdirectly irreducible EMV-algebra is linearly ordered similarly as does every subdirectly irreducible MV-algebra, see [CDM, Thm 1.3.3].
Proposition 5.19**.**
Every subdirectly irreducible EMV-algebra M is linearly ordered.
Proof.
If M={0}, the statement is clear.
Let Mξ ={0} be a subdirectly irreducible EMV-algebra. Due to Theorem 3.16, this means that M has the least non-trivial ideal I. Let a>0 be any idempotent of M. Then [0,a] is an ideal of M as well as of the MV-algebra [0,a] which yields Iβ[0,a] for each aβI(M). It is clear that I is also the least ideal of every MV-algebra [0,a]. Therefore, every [0,a] is a linearly ordered MV-algebra. Let x,yβM. There is an idempotent a of M such that x,yβ€a. Then xβ€y or yβ€x as was claimed.
β
In Theorem 3.18, we showed that any simple EMV-algebra is an MV-algebra and by Theorem 3.26, we proved that
for each maximal ideal I of an EMV-algebra (M;β¨,β§,β,0), M/I is an MV-algebra. Now, we want to generalize this result.
Theorem 5.20**.**
Any EMV-algebra can be embedded into an MV-algebra.
(2) By Theorem 3.11, the class of EMV-algebras is a variety. Therefore, due to the Birkhoff Subdirect Representation Theorem, see [BuSa, Thm 8.6], M is a subdirect product of subdirectly irreducible EMV-algebras which are in view of Proposition 5.19 linearly ordered EMV-algebras. By Theorem 3.25, every linearly ordered EMV-algebra is an MV-algebra which gives the result.
β
From [CoDa, Thm. 2.2], Theorem 4.15 and Corollary 4.16 we conclude that every proper generalized Boolean algebra, every proper EMV-clan and every proper semisimple EMV-algebra can be embedded into an MV-algebra as a maximal ideal of the MV-algebra. In the following, we present a basic result saying that this is true for each proper EMV-algebra.
Theorem 5.21**.**
[Basic Representation Theorem]*
Every EMV-algebra M is either an MV-algebra or M can be embedded into an MV-algebra N as a maximal ideal of N.*
Proof.
If M possesses a top element, then M is an MV-algebra, see Example 3.2(4). If M has no top element, M is proper and according to Theorem 5.20, there is an MV-algebra N such that M can be embedded into N as an EMV-subalgebra of the EMV-algebra N. Let 1 be the top element of N and without loss of generality, we can assume that M is an EMV-subalgebra of N. Let N0β(M) be the least EMV-subalgebra of N containing both M and the element 1. In what follows, we will use ideas of the proof of Theorem 4.15 to describe N0β(M).
For each xβN, let xβ=Ξ»1β(x).
Set
[TABLE]
We assert N0β(M)=N0β.
Clearly N0β contains M and 1. Let x,yβN0β. We have three cases: (i) x=x0β,y=y0ββM. Then xβ¨y,xβ§y,xβyβN0β. (ii) x=x0ββ, y=y0ββ for some x0β,y0ββM. Then xβ¨y=x0βββ¨y0ββ=(x0ββ§y0β)β, xβ§y=(x0ββ¨y0β)β and xβy=xββyβ=(x0ββy0β)ββN0β. (iii) x=x0β and y=y0ββ for some x0β,y0ββM. Then
[TABLE]
where b is an idempotent of M such that x0β,y0ββ€b; for the last equality we use equality (5.1) of Lemma 5.1. Using again Lemma 5.1, we have y0ββΞ»bβ(x0ββ§y0β)βM so that xβyβN0β.
where a is an idempotent of M such that x,yβ€a. So that xβ§yβN0β. Using xβ¨y=(xββ§yβ)β, we have, xβ¨yβN0β.
We have just proved that N0β is an EMV-algebra containing M and 1, so that N0β is an MV-algebra contained in N0β(M). Therefore, N0β=N0β(M) and N0β contains M properly.
Now we prove that M is a maximal ideal of N0β. Since M is a proper EMV-algebra, M is a proper subset of N0β. To show that M is an ideal it is sufficient assume yβ€xβM. If y=y0ββ, this is impossible while 1β/M. Therefore, M is a proper ideal of N0β=N0β(M). Now let yβN0ββM, then y=y0ββ for some y0ββM. Then yβ=y0ββM showing M is a maximal ideal of the MV-algebra N0β.
β
It is important to note that the converse to Theorem 5.21, i.e. whether a maximal ideal of an MV-algebra is an EMV-algebra, is not true, in general. Indeed, if we take the Chang MV-algebra N=Ξ(ZΓβZ,(0,1)), where ZΓβZ denotes the lexicographic product of the group of natural numbers Z with itself, then the set I={(0,n)β£nβ₯0} is a unique maximal ideal of N, but I is only a qEMV-algebra but not an EMV-algebra because I has only one idempotent, namely 0=(0,0). However, if M is an MV-algebra and I is a maximal ideal of I having enough idempotent elements, i.e., for each xβI, there is an idempotent element a of M belonging to I such that xβ€a, then I is an EMV-algebra. It is well known that if a is a Boolean element of M, then [0,a]βI and ([0,a];βaβ,β²a,0,a) is an MV-algebra, where xβaβy=(x+y)β§a=xβy, xβ²a=aβx=Ξ»aβ(x), x,yβ[0,a]. Then due to Theorem 5.21, we have that the set IβͺIβ², where Iβ²={xβ²β£xβI}, is the least MV-subalgebra of M containing I and 1.
More about MV-algebras for which a proper EMV-algebra can be embedded as their maximal ideal will be done at the end of this section, see Theorem 6.4.
Theorem 5.20 allows us to show that the lattice of all subvarieties of the variety EMV of EMV-algebras is countably infinite similarly as in the case of the lattice of subvarieties of the variety MV of MV-algebras.
Theorem 5.22**.**
The lattice of subvarieties of the variety EMV of EMV-algebras is countably infinite.
Proof.
According to Komori [Kom], the lattice of subvarieties of the variety MV of MV-algebras is countably infinite. Di Nola and Lettieri presented in [DiLe] an equational base of any subvariety of the variety MV which consists of finitely many MV-equations using only β and β. Hence, let V be any subvariety of MV-algebras with a finite equational base {fiβ(x1β,β¦,xnβ)=giβ(y1β,β¦,ymβ)β£i=1,β¦,n}, where fiβ,giβ are finite MV-terms using only β and β. Let E(V) be the subvariety of EMV-algebras satisfying equations fiβ(x1β,β¦,xnβ)=giβ(y1β,β¦,ymβ) for i=1,β¦,n.
Now let M be any EMV-algebra. It generates the subvariety Var(M) of EMV-algebras. According to Theorem 5.20, there is an MV-algebra N such that M can be embedded into N. The MV-algebra N generates the subvariety V(N) of MV-algebras, hence, M belongs to the variety E(V(N)) which proves Var(M)βE(V(N)).
On the other hand, by the proof of Theorem 5.20, we know that N can be chosen in such a way that N is the direct product of the family {M/Iβ£IβP(M)}. Clearly, M/I as a homomorphic image of M belongs to Var(M) for all IβP(M), and so the direct product βIβP(M)βM/I also belongs to Var(M), which implies that NβVar(M). Since any MV-algebra is an EMV-algebra, then V(N)βVar(N)βVar(M), where Var(N) is the variety of EMV-algebras generated by N. Then E(V(N))βVar(M) and finally, E(V(N))=Var(M).
Now let {MΞ±ββ£Ξ±βA} be any system of EMV-algebras. For every MΞ±β, there is an MV-algebra NΞ±β such that MΞ±β can be embedded into NΞ±β. Then Var(MΞ±β)=E(V(NΞ±β)) for each Ξ±βA which entails Var({MΞ±ββ£Ξ±βA})=E(V({NΞ±ββ£Ξ±βA})). Hence, the cardinality of the set of subvarieties of EMV is β΅0β.
β
Corollary 5.23**.**
For every subvariety VEβ of the variety EMV, there is a subvariety V of MV-algebras such that VEβ=E(V), and the equational base from [DiLe] for V is also an equational base for VEβ.
Proof.
The statement follows directly from the proof of Theorem 5.22 and [DiLe].
β
For example, (i) the subvariety satisfying the equation x=0 is a singleton containing the one-element EMV algebra {0}. (ii) The equation xβx=x defines the subvariety of generalized Boolean algebras, which is contained in any non-trivial subvariety of EMV-algebras. Indeed, if B is the variety of Boolean algebras, or equivalently, B is the subvariety MV-algebras that satisfy equation xβx=x, then E(B) is the subvariety of generalized Boolean algebras. Then BβV for any non-trivial variety V of MV-algebras, and (B)βE(V).
(iii) The equation x=x determines the whole variety EMV.
6 Categorical Equivalencies
In what follows, we present a categorical equivalence of the category of proper EMV-algebras with the special category of MV-algebras N with a fixed maximal ideal I having enough idempotents and N=IβͺIβ².
Let PEMV be the category of proper EMV-algebra whose objects are proper EMV-algebras and morphisms are homomorphisms of EMV-algebras. Now let PMV be the category whose objects are couples (N,I), where N is an MV-algebra and I is a fixed maximal ideal of N having enough idempotent elements such that N=IβͺIβ². If (N1β,I1β) and (N2β,I2β) are two objects of PMV, then a morphism in PMV from (N1β,I1β) into (N2β,I2β) is a homomorphism of MV-algebras Ο:N1ββN2β such that Ο(I1β)βI2β.
If G is an arbitrary Abelian β-group, then the MV-algebra N=Ξ(GΓβZ,(0,1)) (perfect MV-algebras, see [CDM, Sec 7.4]) has a unique maximal ideal I={(g,0)β£gβG+} and for it we have IβͺIβ²=N. However, I is not an EMV algebra because (0,0) is a unique idempotent of I.
We note that if Ο:(N1β,I1β)β(N2β,I2β) is a morphisms, then (Ο(N1β),Ο(I1β)) is an object of PMV, and it is easy to verify that PEMV and PMV are indeed categories. In addition, we underline that PEMV is not a variety, since due to Theorem 3.26, if I is a maximal ideal of a proper EMV-algebra M, then M/I is an MV-algebra and thus M/I does not
belong to PEMV.
Define a mapping Ξ¦:PMVβPEMV as follows: For any object (N,I)βPMV, let
[TABLE]
and if (N1β,I1β) and (N2β,I2β) are objects of PMV and
Ο:(N1β,I1β)β(N2β,I2β) is a morphism, then
[TABLE]
Proposition 6.1**.**
Ξ¦* is a well-defined functor that is faithful and full from
the category PMV into the category PEMV.*
Proof.
First, we show that Ξ¦ is a well-defined functor. In other words, we
have to establish that if Ο:(N1β,I1β)β(N2β,I2β) is a morphism of proper EMV-algebras, then Ξ¦(Ο) is a morphism in PEMV. Indeed, the mapping Ξ¦(Ο) is in fact an EMV-homomorphism from the EMV-algebra I1β into the EMV-algebra I2β.
Let Ο1β and Ο2β be two morphisms from (N1β,I1β) into (N2β,I2β) such that Ξ¦(Ο1β)=Ξ¦(Ο2β). Then Ο1β(x)=Ο2β(x) for each xβI1β. If xβN1ββI1β, then there is an element x0ββI1β such that x=x0β²β. Then Ο1β(x)=Ο1β(x0β²β)=(Ο1β(x0β))β²=(Ο2β(x0β))β²=Ο2β(x) which entails Ο1β=Ο2β, i.e. Ξ¦ is a faithful functor.
To prove that Ξ¦ is a full functor, let h:I1ββI2β be a morphism from PMV, i.e. h is a homomorphism of EMV-algebras. By Theorem 5.21, there are MV-algebras N1β and N2β such that I1β and I2β can be embedded into N1β and N2β, respectively, as their maximal ideals. Without loss of generality, we can assume that Iiβ is a subalgebra of Niβ for i=1,2. We assert that there is a morphism Ο:(N1β,I1β)β(N2β,I2β) such that Ξ¦(Ο)=h. In other words h can be extended to an MV-homomorphism Ο from N1β into N2β for some objects (N1β,I1β) and (N2β,I2β) from PMV. By (5.6), N1β=N0β(I1β). So let xβN1ββI1β. There is a unique element x0ββI1β such that x=x0β²β. Then we set Ο(x)=h(x0β)β². Clearly Ο(1)=1, Ο(x)=h(x) if xβI1β, and Ο(xβ²)=(Ο(x))β², xβN1β. Now let x,yβN1β. There are three cases: (1) x,yβI1β, then clearly Ο(xβy)=Ο(x)βΟ(y). (2) x=x0β²β and y=y0β²β for some x0β,y0ββI1β. Then Ο(xβy)=Ο(x0β²ββy0β²β)=(Ο(x0ββy0β))β²=(Ο(x0β)βΟ(y0β))β²=(Ο(x)β²βΟ(y)β²)β²=Ο(x)βΟ(y). (3) x=x0β²β and y=y0β for some x0β,y0ββI1β. There is an idempotent aβI1β such that x,yβ€a. Applying (5.1) of Lemma 5.1, we get
[TABLE]
Therefore, Ο is a homomorphism of MV-algebras which is an extension of h. Whence, Ξ¦(Ο)=h and Ξ¦ is a full functor.
β
Proposition 6.2**.**
Let M be a proper EMV-algebra and hiβ:MβNiβ be an embedding of M into an MV-algebra Niβ for i=1,2. Then N0iβ:=N0β(hiβ(Miβ)) are isomorphic MV-algebras and (Ni0β,hiβ(M))βPMV for i=1,2.
Proof.
Let hiβ:MβNiβ be an embedding for i=1,2. By (5.6) of Theorem 5.21, Ni0β=N0β(hiβ(M)) for i=1,2. Let us define Ο:N10ββN20β such that Ο(x)=h2β(x0β) if x=h1β(x0β) for x0ββM1β and Ο(x)=(h2β(x0β))β² if x=h1β(x0β)β² for x0ββM1β. Then, similarly as in the proof of the Proposition 6.1 that Ξ¦ is a full functor, we can prove that Ο is a homomorphism of MV-algebras. In addition, Ο is a bijection, so that it is an isomorphism. Clearly, (Ni0β,hiβ(M))βPMV for i=1,2.
β
Let A and B be two categories and let f:AβB be a functor. Suppose that g,h are functors from B to A such that gβf=idAβ and fβh=idBβ; then g is a left-adjoint of f and h is a right-adjoint of f.
Proposition 6.3**.**
The functor Ξ¦ from the category PMV into the category PEMV has a left-adjoint.
Proof.
We claim, for a proper EMV-algebra M, there is a universal arrow ((N,I),f) i.e., (N,I) is an object in PMV and f is a morphism from M into Ξ¦(N,I)=I such that if (Nβ²,Iβ²) is an object from PEMV and fβ² is a morphism from M into Ξ¦(Nβ²,Iβ²), then there exists a unique morphism fβ:(N,I)β(Nβ²,Iβ²) such that
Ξ¦(fβ)βf=fβ².
Indeed, by Theorem 5.21 and Proposition 6.2, there is a unique (up to isomorphism of MV-algebras) MV-algebra N and an injective EMV-homomorphism f:MβN such that f(N) is a maximal ideal of N. We assert that ((N,I),f) is universal arrow for M. Let (Nβ²,Iβ²) be an object from PEMV and let fβ² be a morphism from M into Ξ¦(Nβ²,Iβ²). We can define a mapping fβ:NβNβ² such that fβ(f(x)):=fβ²(x) if xβM and if yβNβf(M), there is y0ββM such that y=(f(y0β))β², and we set fβ(y)=(fβ²(y0β))β². Then fβ:NβNβ² is a unique MV-homomorphism such that Ξ¦(fβ)βf=fβ².
Define a mapping Ξ¨:PEMVβPMV by Ξ¨(M):=(N,I) whenever ((N,I),f) is a universal arrow for M and if fβ²:MβMβ² is an EMV-homomorphism, there is a unique morphism fβ:(N,I)β(Nβ²,Iβ²), where Ξ¦(Nβ²,Iβ²)=Mβ², then we set Ξ¨(fβ²):=fβ. Using Theorem 5.21, we have that Ξ¨ is a left-adjoint functor of the functor Ξ¦.
β
Theorem 6.4**.**
The functor Ξ¦ defines a categorical equivalence of the category PMV and the category of proper EMV-algebras PEMV.
In addition, if h:Ξ¦(N,I)βΞ¦(Nβ²,Iβ²) is a morphism of proper EMV-algebras, then there is a unique homomorphism Ο:(N,I)β(Nβ²,Iβ²) of MV-algebras such that we have h=Ξ¨(Ο), and
(i)
if h is surjective, so is Ο;
2. (ii)
if h is injective, so is Ο.
Proof.
According to [MaL, Thm IV.4.1 (i),(iii)], since Ξ¨ is faithful and full, it is necessary to show that, for any proper EMV-algebra M there is an object (N,I) in PMV such that Ξ¦(N,I) is isomorphic to M. To show that it is sufficient to take any universal
arrow ((N,I),f) of M.
β
Let I0β be a maximal ideal of N=Ξ(G,u) and
let I be a unique maximal β-ideal of (G,u) generated by I0β. We define Iu={nuβyβ£nβ₯1,yβI,0β€y<nu}. Then I0ββͺI0β²β=N if and only if G+=(I+)βͺIu.
Let I0ββͺI0β²β=Ξ(G,u) and choose xβG+. Then x=x1β+β―+xmβ+(uβy1β)+β―+(uβynβ), where xiβ,yjββI0β, so that x=x0β+nuβy0β, where x0β=x1β+β―+xmββI+ and y0β=y1β+β―+ynββI+. If n=0, then x=x0ββI+. If n>0, then xβ§u=(x1β+β―+xmβ+(uβy1β)+β―+(uβynβ))β§u=x1βββ―βxmββ(uβy1β)ββ―β(uβynβ)βΞ(G,u). Then m=0, x0β and x=nuβy0ββIu.
Let f1β and f2β be two morphisms from (G1β,u1β,I1β) into (G2β,u2β,I2β) such that Ξ(f1β)=Ξ(f2β). Then f1β(x)=f2β(x) for each xβΞ(G1β,u1β). Since fiβ for i=1,2 is a homomorphism of unital β-groups, it is easy to see that f1β(x)=f2β(x) for each xβG1β and f1β=f2β.
Now we introduce the following notions. On every MV-algebra N we can define a partial addition + such that x+y is defined iff xβy=0, and in such a case, x+y:=xβy; if N=Ξ(G,u), then the partial addition coincides with the group addition related to [0,u]. We say that a couple (G,f) is a universal group for an MV-algebra N if (i) f is a mapping from M into a po-group G which preserves partial addition + on N such that G=G+βG+, f(M) generates G+ as a semigroup, (ii) for any group K and any +-preserving mapping h:NβK, there is a group homomorphism Ο:GβK such that h=Οβf. Due to [Dvu2, Thm 5.3] if Nβ Ξ(G,u), then (G,f) is a universal group for N, where f is an isomorphism f:NβΞ(G,u).
Proposition 6.7**.**
The functor ΞIβ from the category PUALG into the category PMV has a left-adjoint.
Define a mapping ΞIβ:PMVβPUALG by ΞIβ(N,I0β)=(G,u,I) if ((G,u,I),f) is a universal arrow for (N,I0β) and I is a maximal β-ideal of G generated by f(I0β). If fβ² is a morphism from (N,I0β) into (Nβ²,I0β²β), there is a unique morphism fβ:(G,u,I)β(Gβ²,uβ²,Iβ²), where Nβ²β Ξ(Gβ²,uβ²) and Iβ² is a maximal β-ideal of Gβ² generated by fβ²(I0β²β), then ΞIβ(fβ²):=fβ. Therefore, ΞIβ is a left-adjoint of ΞIβ.
β
Theorem 6.8**.**
The functor ΞIβ defines a categorical equivalence of the category PUALG and the category PMV.
Proof.
The statement follows from [MaL, Thm IV.4.1(i),(iii)] and Propositions 6.6β6.7.
β
Corollary 6.9**.**
The categories PUALG, PMV and PEMV are mutually categorically equivalent.
We have introduced the notion of an EMV-algebra, Definition 3.1, which generalizes the notion of an MV-algebra and of a generalized Boolean algebra. We have exhibited its basic properties and notions as ideals, congruences, filters, and their mutual relationship, Theorem 3.16. Nevertheless an EMV-algebra M has not necessarily a top element, M has a maximal ideal, Theorem 5.6. We have defined an EMV-clan as an EMV-algebra of fuzzy sets. We have shown that every EMV-algebra is semisimple iff it is isomorphic to some EMV-clan of fuzzy sets, Theorem 4.11. A state-morphism is any EMV-homomorphism from M into the MV-algebra of the real interval [0,1] which attains the value 1. State-morphisms are in a one-to-one relationship with maximal ideals of M, Theorem 4.2.
We have shown that every EMV-algebra can be embedded into an MV-algebra, Theorem 5.20. Theorem 5.21 characterizes any EMV-algebra saying that either it is an MV-algebra or it can be embedded into an MV-algebra as its maximal ideal.
The class of EMV-algebras forms a variety, Theorem 3.11. Using the equational base of any subvariety of the variety of MV-algebras, [DiLe], we describe a functional base of any subvariety of the variety EMV of EMV-algebras, Corollary 5.23, and the cardinality of all subvarieties of the variety EMV is β΅0β, Theorem 5.22. Finally, we presented mutually categorical equivalencies of the category of proper EMV-algebras, a special category of MV-algebras N with a fixed maximal ideal I having enough idempotents, and a special categories of Abelian unital β-groups, Theorem 6.4, Theorem 6.8 and Corollary 6.9.
With the present paper we have opened a new and interesting window into the realm of unbounded generalizations of MV-algebras and generalized Boolean algebra, and we hope to continue in this research, for example with a variant of the LoomisβSikorski theorem for Ο-complete EMV-algebras.
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