aa r X i v : . [ m a t h . R A ] J un PSEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT
ANATOLIJ DVUREˇCENSKIJ , Mathematical Institute, Slovak Academy of Sciences,ˇStef´anikova 49, SK-814 73 Bratislava, Slovakia Depart. Algebra Geom., Palack´y University17. listopadu 12, CZ-771 46 Olomouc, Czech RepublicE-mail: [email protected]
Abstract.
We study algebraic conditions when a pseudo MV-algebra is aninterval in the lexicographic product of an Abelian unital ℓ -group and an ℓ -group that is not necessary Abelian. We introduce ( H, u )-perfect pseudo MV-algebras and strong (
H, u )-perfect pseudo MV-algebras, the latter ones willhave a representation by a lexicographic product. Fixing a unital ℓ -group( H, u ), the category of strong (
H, u )-perfect pseudo MV-algebras is categori-cally equivalent to the category of ℓ -groups. Introduction
MV-algebras were introduced by Chang [Cha] as the algebraic counterpart of Lukasiewicz infinite-valued calculus and during the last 56 years MV-algebras en-tered deeply in many areas of mathematics and logics. More than 10 years ago, anon-commutative generalization of MV-algebras has been independently appeared.These new algebras are said to be pseudo MV-algebras in [GeIo] or a generalizedMV-algebras in [Rac]. For them author [Dvu2] generalized a famous Mundici’srepresentation theorem, see e.g. [CDM, Cor 7.1.8], showing that every pseudo MV-algebra is always an interval in a unital ℓ -group not necessarily Abelian. Suchalgebras have the operation ⊕ as a truncated sum and they have two negations.We note that the equality of these two negations does not necessarily imply thata pseudo MV-algebra is an MV-algebra. According to Komori’s theorem [Kom],[CDM, Thm 8.4.4], the variety lattice of MV-algebras is countably, whereas the oneof pseudo MV-algebras is uncountable, cf. [Jak, DvHo]. Therefore, the structure ofpseudo MV-algebras is much richer than the one of MV-algebras. In [DvHo] it wasshown that the class of pseudo MV-algebras where each maximal ideal is normalis a variety. This variety is also very rich and within this variety many importantproperties of MV-algebras remain.In [DDT], perfect pseudo MV-algebras were studied. They are characterizedas those that every element of a perfect pseudo MV-algebra is either an infinites-imal or a co-infinitesimal. In [DDT] we have shown that the category of perfect Keywords: Pseudo MV-algebra, symmetric pseudo MV-algebra, ℓ -group, strong unit, lexico-graphic product, ideal, retractive ideal, ( H, u )-perfect pseudo MV-algebra, lexicographic pseudoMV-algebra, strong (
H, u )-perfect pseudo MV-algebraAMS classification: 06D35, 03G12This work was supported by the Slovak Research and Development Agency under contractAPVV-0178-11, grant VEGA No. 2/0059/12 SAV, and CZ.1.07/2.3.00/20.0051. pseudo MV-algebras is equivalent to the variety of ℓ -groups, and every such analgebra M is in the form of an interval in the lexicographic product Z −→× G , i.e. M ∼ = Γ( Z −→× G, (1 , n -perfect pseudo MV-algebras were studied in[Dvu3]. They can be characterized as those pseudo MV-algebras that have ( n + 1)-comparable slices, and their representation is again in the form of an interval in thelexicographic product n Z −→× G , i.e. every strong n -perfect pseudo MV-algebra M is of the form Γ( n Z −→× G, (1 , G is any ℓ -group. In the paper [Dvu4], wehave studied so-called ( H , H ,
1) is a unital ℓ -subgroup of the unital ℓ -group of reals ( R , H −→× G, (1 , H −→× G, ( u, H, u ) is an Abelian unital ℓ -group and G is an Abelian ℓ -group. The main aim of the present paper is to generalize suchlexicographic MV-algebras also for the case of pseudo MV-algebras. Therefore, weintroduce so-called ( H, u )-perfect and strong (
H, u )-perfect pseudo MV-algebras,where (
H, u ) is an Abelian unital ℓ -group. We show that strong ( H, u )-perfectpseudo MV-algebras are always of the form Γ( H −→× G, ( u, G is an ℓ -groupnot necessarily Abelian. This category will be always categorically equivalent withthe variety of ℓ -groups. Therefore, we generalize many interesting results that wereknown only in the realm of MV-algebras, see [DiLe2, CiTo, DFL].The paper is organized as follows. Section 2 gathers necessary properties ofpseudo MV-algebras. Section 3 presents a definition of ( H, u )-perfect pseudo MV-algebras as those which can be decomposed into a system of comparable slicesindexed by the elements of the interval [0 , u ] H = { h ∈ H : 0 ≤ h ≤ u ], where( H, u ) is an Abelian unital ℓ -group. Section 4 defines strong ( H, u )-perfect pseudoMV-algebras and we show their representation by Γ( H −→× G, ( u, H, u )-perfect pseudoMV-algebras will be established in Section 8. Finally, in Section 9 we describeweak (
H, u )-perfect pseudo MV-algebras as those that they can be represented inthe form Γ( H −→× G, ( u, g )), where g is an arbitrary element (not necessarily g = 0)of an ℓ -group G . 2. Pseudo MV-algebras
According to [GeIo], a pseudo MV-algebras or a GMV-algebra by [Rac] is analgebra ( M ; ⊕ , − , ∼ , ,
1) of type (2 , , , ,
0) such that the following axioms holdfor all x, y, z ∈ M with an additional binary operation ⊙ defined via y ⊙ x = ( x − ⊕ y − ) ∼ (A1) x ⊕ ( y ⊕ z ) = ( x ⊕ y ) ⊕ z ;(A2) x ⊕ ⊕ x = x ;(A3) x ⊕ ⊕ x = 1;(A4) 1 ∼ = 0; 1 − = 0;(A5) ( x − ⊕ y − ) ∼ = ( x ∼ ⊕ y ∼ ) − ; SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 3 (A6) x ⊕ ( x ∼ ⊙ y ) = y ⊕ ( y ∼ ⊙ x ) = ( x ⊙ y − ) ⊕ y = ( y ⊙ x − ) ⊕ x ; (A7) x ⊙ ( x − ⊕ y ) = ( x ⊕ y ∼ ) ⊙ y ;(A8) ( x − ) ∼ = x. Any pseudo MV-algebra is a distributive lattice where (A6) and (A7) define thejoint x ∨ y and the meet x ∧ y of x, y , respectively.We note that a po-group (= partially ordered group) is a group ( G ; + ,
0) (writtenadditively) endowed with a partial order ≤ such that if a ≤ b, a, b ∈ G, then x + a + y ≤ x + b + y for all x, y ∈ G. We denote by G + = { g ∈ G : g ≥ } the positive cone of G. If, in addition, G is a lattice under ≤ , we call it an ℓ -group(= lattice ordered group). An element u ∈ G + is said to be a strong unit (=order unit) if G = S n [ − nu, nu ], and the couple ( G, u ) with a fixed strong unit u is said to be a unital po-group or a unital ℓ -group , respectively. The commutativecenter of a group H is the set C ( H ) = { h ∈ H : h + h ′ = h ′ + h, ∀ h ′ ∈ H } .Finally, two unital ℓ -groups ( G, u ) and (
H, v ) is isomorphic if there is an ℓ -groupisomorphism φ : G → H such that φ ( u ) = v . In a similar way an isomorphismand a homomorphism of unital po-groups are defined. For more information onpo-groups and ℓ -groups and for unexplained notions about them, see [Fuc, Gla].By R and Z we denote the groups of reals and natural numbers, respectively.Between pseudo MV-algebras and unital ℓ -groups there is a very close connection:If u is a strong unit of a (not necessarily Abelian) ℓ -group G ,Γ( G, u ) := [0 , u ]and x ⊕ y := ( x + y ) ∧ u,x − := u − x,x ∼ := − x + u,x ⊙ y := ( x − u + y ) ∨ , then (Γ( G, u ); ⊕ , − , ∼ , , u ) is a pseudo MV-algebra [GeIo].A pseudo MV-algebra M is an MV-algebra if x ⊕ y = y ⊕ x for all x, y ∈ M. Wedenote by P s MV and MV the variety of pseudo MV-algebras and MV-algebras,respectively.The basic representation theorem for pseudo MV-algebras is the following gen-eralization [Dvu2] of the Mundici’s famous result: Theorem 2.1.
For any pseudo MV-algebra ( M ; ⊕ , − , ∼ , , , there exists a unique ( up to isomorphism ) unital ℓ -group ( G, u ) such that ( M ; ⊕ , − , ∼ , , is isomorphicto (Γ( G, u ); ⊕ , − , ∼ , , u ) . The functor Γ defines a categorical equivalence of thecategory of pseudo MV-algebras with the category of unital ℓ -groups. We note that the class of pseudo MV-algebras is a variety whereas the class ofunital ℓ -groups is not a variety because it is not closed under infinite products.Due to this result, if M = Γ( G, u ) for some unital ℓ -group ( G, u ), then M islinearly ordered iff G is a linearly ordered ℓ -group, see [Dvu1, Thm 5.3].Besides a total operation ⊕ , we can define a partial operation + on any pseudoMV-algebra M in such a way that x + y is defined iff x ⊙ y = 0 and then we set x + y := x ⊕ y. (2 . ⊙ has a higher binding priority than ⊕ . ANATOLIJ DVUREˇCENSKIJ
In other words, x + y is precisely the group addition x + y if the group sum x + y is defined in M .Let A, B be two subsets of M . We define (i) A B if a ≤ b for all a ∈ A and all b ∈ B , (ii) A ⊕ B = { a ⊕ b : a ∈ A, b ∈ B } , and (iii) A + B = { a + b : if a + b existsin M for a ∈ A, b ∈ B } . We say that A + B is defined in M if a + b exists in M foreach a ∈ A and each b ∈ B . (iv) A − = { a − : a ∈ A } and A ∼ = { a ∼ : a ∈ A } . Using Theorem 2.1, we have if y ≤ x , then x ⊙ y − = x − y and y ∼ ⊙ x = − y + x ,where the subtraction − is in fact the group subtraction in the representing unital ℓ -group.We recall that if H and G are two po-groups, then the lexicographic product H −→× G is the group H × G which is endowed with the lexicographic order: ( h, g ) ≤ ( h , g ) iff h < h or h = h and g ≤ g . The lexicographic product H −→× G isan ℓ -group iff H is linearly ordered group and G is an arbitrary ℓ -group, [Fuc, (d)p. 26]. If u is a strong unit for H , then ( u,
0) is a strong unit for H −→× G , andΓ( H −→× G, ( u, M is symmetric if x − = x ∼ for all x ∈ M .The pseudo MV-algebra Γ( G, u ) is symmetric iff u ∈ C ( G ), hence the variety ofsymmetric pseudo MV-algebras is a proper subvariety of the variety MV . Forexample, Γ( R −→× G, (1 , G is Abelian.An ideal of a pseudo MV-algebra M is any non-empty subset I of M such that(i) a ≤ b ∈ I implies a ∈ I, and (ii) if a, b ∈ I, then a ⊕ b ∈ I. An ideal is said tobe (i) maximal if I = M and it is not a proper subset of another ideal J = M ; wedenote by M ( M ) the set of maximal ideals of M , (ii) prime if x ∧ y ∈ I implies x ∈ I or y ∈ I , and (iii) normal if x ⊕ I = I ⊕ x for any x ∈ M ; let N ( M ) bethe set of normal ideals of M. A pseudo MV-algebra M is local if there is a uniquemaximal ideal and, in addition, this ideal also normal.There is a one-to-one correspondence between normal ideals and congruencesfor pseudo MV-algebras, [GeIo, Thm 3.8]. The quotient pseudo MV-algebra over anormal ideal I, M/I, is defined as the set of all elements of the form x/I := { y ∈ M : x ⊙ y − ⊕ y ⊙ x − ∈ I } , or equivalently, x/I := { y ∈ M : x ∼ ⊙ y ⊕ y ∼ ⊙ x ∈ I } . Let x ∈ M and an integer n ≥ .x := 0 , ⊙ x := x, ( n + 1) .x := ( n.x ) ⊕ x,x := 1 , x := x, x n +1 := x n ⊙ x, x := 0 , x := x, ( n + 1) x := ( nx ) + x, if nx and ( nx ) + x are defined in M. An element x is said to be an infinitesimal if mx exists in M for every integer m ≥ . We denote by Infinit( M ) the set of allinfinitesimals of M. We define (i) the radical of a pseudo MV-algebra M , Rad( M ) , as the setRad( M ) = \ { I : I ∈ M ( M ) } , and (ii) the normal radical of M , Rad n ( M ), viaRad n ( M ) = \ { I : I ∈ N ( M ) ∩ M ( M ) } whenever N ( M ) ∩ M ( M ) = ∅ .By [DDJ, Prop. 4.1, Thm 4.2], it is possible to show thatRad( M ) ⊆ Infinit( M ) ⊆ Rad n ( M ) . SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 5
The notion of a state is an analogue of a probability measure for pseudo MV-algebras. We say that a mapping s from a pseudo MV-algebra M into the realinterval is a state if (i) s ( a + b ) = s ( a ) + s ( b ) whenever a + b is defined in M , and(ii) s (1) = 1. We define the kernel of s as the set Ker( s ) = { a ∈ M : s ( a ) = 0 } .Then Ker( s ) is a normal ideal of M .If M is an MV-algebra, then at least one state on M is defined. Unlike for MV-algebras, there are pseudo MV-algebras that are stateless, [Dvu1] (see also a notejust before Theorem 8.5 below). We note that M has at least one state iff M hasat least one maximal ideal that is also normal. However, every non-trivial linearlyordered pseudo MV-algebra admits a unique state, [Dvu1, Thm 5.5].Let S ( M ) be the set of all states on M ; it is a convex set. A state s is said tobe extremal if from s = λs + (1 − λ ) s , where s , s ∈ S ( M ) and 0 < λ <
1, weconclude s = s = s . Let ∂ e S ( M ) denote the set of extremal states. In addition,in view of [Dvu1], a state s is extremal iff Ker( s ) is a maximal ideal of M iff s ( a ∧ b ) = min { s ( a ) , s ( b ) } . Or equivalently, s is a state morphism , i.e., s is ahomomorphism from M into the MV-algebra Γ( R , I is maximal iff I = Ker( s ) for some extremal state s .We say that a net of states { s α } α converges weakly to a state s if s ( a ) =lim α s α ( a ), a ∈ M . Then S ( M ) and ∂ e S ( M ) are compact Hausdorff topologi-cal spaces, in particular cases both can be empty, and due to the Krein-Mil’manTheorem [Go, Thm 5.17], every state on M is a weak limit of a net of convexcombinations of extremal states.Pseudo MV-algebras can be studied also in the frames of pseudo effect algebrawhich are a non-commutative generalization of effect algebras introduced by [FoBe].According to [DvVe1, DvVe2], a partial algebraic structure ( E ; + , , , where +is a partial binary operation and 0 and 1 are constants, is called a pseudo effectalgebra if, for all a, b, c ∈ E, the following hold:(PE1) a + b and ( a + b ) + c exist if and only if b + c and a + ( b + c ) exist, and inthis case, ( a + b ) + c = a + ( b + c );(PE2) there are exactly one d ∈ E and exactly one e ∈ E such that a + d = e + a =1;(PE3) if a + b exists, there are elements d, e ∈ E such that a + b = d + a = b + e ;(PE4) if a + 1 or 1 + a exists, then a = 0 . If we define a ≤ b if and only if there exists an element c ∈ E such that a + c = b, then ≤ is a partial ordering on E such that 0 ≤ a ≤ a ∈ E. It is possibleto show that a ≤ b if and only if b = a + c = d + a for some c, d ∈ E . We write c = a / b and d = b \ a. Then( b \ a ) + a = a + ( a / b ) = b, and we write a − = 1 \ a and a ∼ = a / a ∈ E. If (
G, u ) is a unital po-group, then (Γ(
G, u ); + , , u ) , where the set Γ( G, u ) := { g ∈ G : 0 ≤ g ≤ u } is endowed with the restriction of the group addition + toΓ( G, u ) and with 0 and u as 0 and 1, is a pseudo effect algebra. Due to [DvVe1,DvVe2], if a pseudo effect algebra satisfies a special type of the Riesz DecompositionProperty, RDP , then every pseudo effect algebra is an interval in some unique(up to isomorphism of unital po-groups) ( G, u ) satisfying also RDP such that M ∼ = Γ( G, u ). ANATOLIJ DVUREˇCENSKIJ
We say that a mapping f from one pseudo effect algebra E onto a second one F is a homomorphism if (i) a, b ∈ E such that a + b is defined in E , then f ( a ) + f ( b )is defined in F and f ( a + b ) = f ( a ) + f ( b ), and (ii) f (1) = 1.We say that a pseudo effect algebra E satisfies RDP property if a + a = b + b implies that there are four elements c , c , c , c ∈ E such that (i) a = c + c ,a = c + c , b = c + c and b = c + c , and (ii) c ∧ c = 0 . In [DvVe2, Thm 8.3, 8.4], it was proved that if ( M ; ⊕ , − , ∼ , ,
1) is a pseudo MV-algebra, then ( M ; + , , , where + is defined by (2.1), is a pseudo effect algebrawith RDP . Conversely, if ( E ; + , ,
1) is a pseudo effect algebra with RDP , then E is a lattice, and by [DvVe2, Thm 8.8], ( E ; ⊕ , − , ∼ , , , where a ⊕ b := ( b − \ ( a ∧ b − )) ∼ , (2 . E has RDP iff E isa lattice and E satisfies RDP , see [DvVe2, Thm 8.8].Finally, we note that an ideal of a pseudo effect algebra E is a non-empty subset I such that (i) if x, y ∈ I and x + y is defined in E , then x + y ∈ I , and (ii) x ≤ y ∈ I implies x ∈ I . An ideal I is normal if a + I := { a + i : i ∈ I if a + i exists in E } = I + a := { j + a : j ∈ I } for any a ∈ E . A maximal ideal is definedin a standard way. If M is a pseudo MV-algebra, then the ideal I of M is also anideal when M is understood as a pseudo effect algebra; this follows from the fact x ⊕ y = ( x ∧ y − ) + y .We note that a mapping from a pseudo effect algebra E into a pseudo effectalgebra F is a homomorphism if (i) a + b ∈ E implies h ( a )+ h ( b ) ∈ F and h ( a + b ) = h ( a ) + h ( b ), and (ii) h (1) = 1. A bijective mapping h : E → F is an isomorphism if both h and h − are homomorphisms of pseudo effect algebras.3. ( H, u ) -Perfect Pseudo MV-algebras Generalizing ideas from [DiLe1, DDT, Dvu3, Dvu4], we introduce the basic no-tions of our paper.If (
H, u ) is a unital ℓ -group, we set [0 , u ] H := { h ∈ H : 0 ≤ h ≤ u } . Definition 3.1.
Let (
H, u ) be a linearly ordered group and let u belong to thecommutative center C ( H ) of H . We say that a pseudo MV-algebra M is ( H, u )- perfect , if there is a system ( M t : t ∈ [0 , u ] H ) of nonempty subsets of M such thatit is an ( H, u )- decomposition of M, i.e. M s ∩ M t = ∅ for s < t, s, t ∈ [0 , u ] H and S t ∈ [0 ,u ] H M t = M , and(a) M s M t for all s < t , s, t ∈ [0 , u ] H ;(b) M − t = M u − t = M ∼ t for each t ∈ [0 , u ] H ;(c) if x ∈ M v and y ∈ M t , then x ⊕ y ∈ M v ⊕ t , where v ⊕ t = min { v + t, u } . We note that (a) can be written equivalently in a stronger way: if s < t and a ∈ M s and b ∈ M t , then a < b . Indeed, by (b) we have a ≤ b . If a = b , then a ∈ M s ∩ M t = ∅ , which is absurd. Hence, a < b .In addition, (i) if ( H, u ) = ( Z ,
1) and M is an MV-algebra, we are speaking on a perfect MV-algebra, [DiLe1], (ii) if ( H , u ) = ( n Z , , a ( n Z , n - perfect , see [Dvu3], (iii) if H is a subgroup of the group ofreal numbers R , such that 1 ∈ H , ( H , H - perfect pseudo MV-algebras. SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 7
For example, let M = Γ( H −→× G, ( u, , (3 . u ∈ C ( H ). We set M = { (0 , g ) : g ∈ G + } , M u := { ( u, − g ) : g ∈ G + } andfor t ∈ [0 , u ] H \ { , u } , we define M t := { ( t, g ) : g ∈ G } . Then ( M t : t ∈ [0 , u ] H ) isan ( H, u )-decomposition of M and M is an ( H, u )-perfect pseudo MV-algebra.As a matter of interest, if O is the zero group, then Γ( O −→× G, (0 , Z −→× O, (1 , H −→× O, ( u, ∼ = Γ( H, u ) and it is semisim-ple (that is, its radical is a singleton) iff H is Archimedean. If G = O = H ,Γ( H −→× G, ( u, Theorem 3.2.
Let M = ( M t : t ∈ [0 , u ] H ) be an ( H, u ) -perfect pseudo MV-algebra. (i) Let a ∈ M v , b ∈ M t . If v + t < u , then a + b is defined in M and a + b ∈ M v + t ; if a + b is defined in M , then v + t ≤ u . If a + b is definedin M and v + t = u , then a + b ∈ M u . (ii) M v + M t is defined in M and M v + M t = M v + t whenever v + t < u . (iii) If a ∈ M v and b ∈ M t , and v + t > u , then a + b is not defined in M. (iv) If a ∈ M v and b ∈ M t , then a ∨ b ∈ M v ∨ t and a ∧ b ∈ M v ∧ t . (v) M admits a state s such that M ⊆ Ker( s ) . (vi) M is a normal ideal of M such that M + M = M and M ⊆ Infinit( M ) . (vii) The quotient pseudo MV-algebra
M/M ∼ = Γ( H, u ) . (viii) Let M = ( M ′ t : t ∈ [0 , u ] H ) be another ( H, u ) -decomposition of M satisfying (a)–(c) of Definition , then M t = M ′ t for each t ∈ [0 , u ] H . (ix) M is a prime ideal of M .Proof. (i) Assume a ∈ M v and b ∈ M t . If v + t < u , then b − ∈ M u − t , so that a ≤ b − ,and a + b is defined in M. Conversely, let a + b be defined, then a ≤ b − ∈ M u − t . If v + t > u , then v > u − t and a > b − ≥ a which is absurd, and this gives v + t ≤ u .Now let v + t = u and a + b be defined in M . By (c) of Definition 3.1, we have a + b ∈ M u .(ii) By (i), we have M v + M t ⊆ M v + t . Suppose z ∈ M v + t . Then, for any x ∈ M v , we have x ≤ z . Hence, y = z − x is defined in M and y ∈ M w for some w ∈ [0 , u ] H . Since z = y + x ∈ M v + t ∩ M v + w , we conclude t = w and M v + t ⊆ M v + M t . (iii) It follows from (i).(iv) Inasmuch as x ∧ y = ( x ⊕ y ∼ ) − y ∼ , we have by (c) of Definition 3.1,( x ⊕ y ∼ ) − y ∼ ∈ M s , where s = (( v + u − t ) ∧ u ) − ( u − t ) = v ∧ t. Using a de Morganlaw and property (d), we have x ∨ y ∈ M v ∨ t . (v) Let s be a unique state on Γ( H, u ) which exists in view of [Dvu1, Thm 5.5].Define a mapping s : M → [0 ,
1] by s ( x ) = s ( t ) if x ∈ M t . It is clear that s is awell-defined mapping. Take a, b ∈ M such that a + b is defined in M . Then thereare unique indices v and t such that a ∈ M v and b ∈ M t . By (i), v + t ≤ u and a + b ∈ M v + t . Therefore, s ( a + b ) = s ( v + t ) = s ( v ) + s ( t ) = s ( a ) + s ( b ) . It isevident that s (1) = 1 and M ⊆ Ker( s ).(vi) It is clear that M is an ideal of M . To prove the normality of M , take x ∈ M v and y ∈ M . Then x ⊕ y = (( x ⊕ y ) − x ) + x ∈ M v which implies by (i)–(ii)( x ⊕ y ) − x ∈ M and x ⊕ M ⊆ M ⊕ x . In the same way we proceed for the secondimplication.Since M + M = M , by (ii) we have M ⊆ Infinit( M ) . ANATOLIJ DVUREˇCENSKIJ (vii) Since by (vi) M is a normal ideal, M/M is a pseudo MV-algebra, too.Using (iv), it is easy to verify that x ∼ M y iff there is an h ∈ [0 , u ] H such that x, y ∈ M h . We define a mapping φ : M/M → Γ( H, u ) by φ ( x ) = h iff x ∈ M h forsome h ∈ [0 , u ] H . The mapping φ is an isomorphism from M/M onto Γ( H, u ).(viii) Let M = ( M ′ t : t ∈ [0 , u ] H ) be another ( H, u )-decomposition of M . Weassert M = M ′ . If not, there are x ∈ M \ M ′ and y ∈ M ′ \ M . By (a), wehave x < y as well as y < x which is absurd. Hence, M = M ′ . By (vi), M isnormal and by (vii), M ∼ = Γ( H, u ) ∼ = M/M ′ . If x ∼ M y , then x, y ∈ M h for some h ∈ [0 , u ] H , as well as x ∼ M ′ y implies x, y ∈ M t ′ and t = t ′ which implies M t = M ′ t for any t ∈ [0 , H .(ix) By (vii), M/M ∼ = Γ( H, u ), so that
M/M is a linear pseudo MV-algebra.Applying [Dvu1, Thm 6.1], we conclude that the normal ideal M is prime. (cid:3) Example 3.3.
We define MV-algebras: M = Γ( Z −→× ( Z −→× Z ) , (1 , (0 , , M =Γ(( Z −→× Z ) −→× Z , ((1 , , , and M = Γ( Z −→× Z −→× Z , (1 , , which are mutually iso-morphic. The first one is ( Z , -perfect, the second one is ( Z −→× Z , (1 , -perfect andof course, the linear unital ℓ -groups ( Z , and ( Z −→× Z , (1 , are not isomorphicwhile the first one is Archimedean and the second one is not Archimedean. Bothpseudo MV-algebras define the corresponding natural ( Z , -decomposition ( M t ) t and ( Z −→× Z , (1 , -decompositions ( M q ) q of M and M , respectively. Then M = { (0 , (0 , m )) : m ≥ } ∪ { (0 , ( n, m )) : n > , m ∈ Z } = Ker( s ) = Infinit( M ) ,where s is a unique state on M ; it is two-valued. But M = { ((0 , , m ) : m ≥ } ⊂ Ker( s ) = Infinit( M ) , where s is a unique state on M , it vanishes only on Infinit( M ) ; it is two-valued. In what follows, we show that the normal ideal M of an ( H, u )-decomposition( M t : t ∈ [0 , u ] H ) of an ( H, u )-perfect pseudo MV-algebra is maximal iff (
H, u ) isisomorphic with ( H , H is a subgroup of the group of reals R with naturalorder, and 1 ∈ H . Theorem 3.4.
Let ( M t : t ∈ [0 , u ] H ) be an ( H, u ) -decomposition of an ( H, u ) -perfect pseudo MV-algebra M . The following statements are equivalent: (i) M is maximal. (ii) ( H, u ) is isomorphic to some ( H , , where H is a subgroup of the group R and ∈ H . (iii) M possesses a unique state s and M = Ker( s ) .Proof. (i) ⇒ (ii). By [Dvu1, Prop 3.4-3.5], M is maximal iff M/M is simple,i.e. it does not contain any non-trivial proper ideal. By (vii) of Theorem 3.2,( M/M , u/M ) ∼ = Γ( H, u ) which means by Theorem 2.1 that (
H, u ) is a linear,Archimedean and Abelian unital ℓ -group, and by H¨older’s theorem, [Bir, ThmXIII.12] or [Fuc, Thm IV.1.1], it is isomorphic to some ( H , H is a subgroupof R and 1 ∈ H .(ii) ⇒ (iii). If ( H, u ) ∼ = ( H , H is a subgroup of R and 1 ∈ H , then M is isomorphic to an ( H , M possesses a uniquestate s . This state has the property s ( M ) = H and Ker( s ) = M .(iii) ⇒ (i). If s is a unique state of M and M = Ker( s ), by [Dvu1], M is anormal and maximal ideal of M . (cid:3) SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 9
Remark 3.5.
We note that in Corollary 7.7 it will be shown that if an (
H, u )-perfect pseudo MV-algebra M is of a stronger form, namely a strong ( H, u )-perfectpseudo MV-algebra introduced in the next section, then M has a unique state.In general, the uniqueness of a state for any ( H, u )-perfect pseudo MV-algebra isunknown.We note that there is uncountably many non-isomorphic unital ℓ -subgroups( H ,
1) of the unital group ( R , H is either cyclic ,i.e. H = n Z for some n ≥ H is dense in R . Therefore, if H = H ( α ) is a subgroup of R generated by α ∈ [0 ,
1] and 1 , then H = n Z for some integer n ≥ α is a rational number. Otherwise, H ( α ) iscountable and dense in R , and M ( α ) := Γ( H ( α ) ,
1) = { m + nα : m, n ∈ Z , ≤ m + nα ≤ } , see [CDM, p. 149]. Therefore, we have uncountably many non-isomorphic ( H , Representation of Strong ( H, u ) -perfect Pseudo MV-algebras In accordance with [Dvu4], we introduce the following notions and generalizeresults from [Dvu4] for strong (
H, u )-perfect pseudo MV-algebras. Our aim is tofind an algebraic characterization of pseudo MV-algebras that can be representedin the form of the lexicographic productΓ( H −→× G, ( u, , where ( H, u ) is a linearly ordered Abelian unital ℓ -group and G is an ℓ -group notnecessarily Abelian. In [Dvu4], we have studied a particular case when ( H, u ) =( H , H is a subgroup of reals.We say that a pseudo MV-algebra M enjoys unique extraction of roots of a, b ∈ M and na, nb exist in M , and na = 1 = nb , then a = b. Every linearlyordered pseudo MV-algebra enjoys due to Theorem 2.1 and [Gla, Lem 2.1.4] uniqueextraction of roots. In addition, every pseudo MV-algebra Γ( H −→× G, ( u, H, u ) is a linearly ordered ℓ -group, enjoys unique extraction of roots of 1 for any n ≥ ℓ -group G . Indeed, let k ( s, g ) = ( u,
0) = k ( t, h ) for some s, t ∈ [0 , u ] H , g, h ∈ G , k ≥ . Then ks = u = kt which yields s = t >
0, and kg = 0 = kh implies g = 0 = h. Let n ≥ a of a pseudo MV-algebra M is said tobe cyclic of order n or simply cyclic if na exists in M and na = 1 . If a is a cyclicelement of order n , then a − = a ∼ , indeed, a − = ( n − a = a ∼ . It is clear that 1is a cyclic element of order 1 . Let M = Γ( G, u ) for some unital ℓ -group ( G, u ) . An element c ∈ M such that(a) nc = u for some integer n ≥ , and (b) c ∈ C ( G ) , where C ( G ) is a commutativecenter of G, is said to be a strong cyclic element of order n . We note that if H is a subgroup of reals and t = 1 /n, then c n is a strong cyclicelement of order n. For example, the pseudo MV-algebra M := Γ( Q −→× G, (1 , , where Q is thegroup of rational numbers, for every integer n ≥ , M has a unique cyclic elementof order n, namely a n = ( n , . The pseudo MV-algebra Γ( n Z , (1 , n ≥ , has the only cyclic element of order n, namely ( n , . If M = Γ( G, u )and G is a representable ℓ -group, G enjoys unique extraction of roots of 1 , therefore, M has at most one cyclic element of order n. In general, a pseudo MV-algebra M can have two different cyclic elements of the same order. But if M has a strong cyclic element of order n, then it has a unique strong cyclic element of order n anda unique cyclic element of order n, [DvKo, Lem 5.2].We say that an ( H, u )-decomposition ( M t : t ∈ [0 , u ] H ) of M has the cyclicproperty if there is a system of elements ( c t ∈ M : t ∈ [0 , u ] H ) such that (i) c t ∈ M t for any t ∈ [0 , u ] H , (ii) if v + t ≤ , v, t ∈ [0 , u ] H , then c v + c t = c v + t , and (iii) c = 1 . Properties: (a) c = 0; indeed, by (ii) we have c + c = c , so that c = 0 . (b) If t = 1 /n, then c n is a cyclic element of order n. Let M = Γ( G, u ) , where ( G, u ) is a unital ℓ -group. An ( H, u )-decomposition( M t : t ∈ [0 , u ] H ) of M has the strong cyclic property if there is a system ofelements ( c t ∈ M : t ∈ [0 , u ] H ), called an ( H, u )- strong cyclic family , such that(i) c t ∈ M t ∩ C ( G ) for each t ∈ [0 , u ] H ;(ii) if v + t ≤ , v, t ∈ [0 , u ] H , then c v + c t = c v + t ;(iii) c = 1 . For example, let M = Γ( H −→× G, ( u, , where ( H, u ) is an Abelian linearlyordered unital ℓ -group and G is an ℓ -group (not necessarily Abelian), and M t = { ( t, g ) : ( t, g ) ∈ M } for t ∈ [0 , u ] H . If we set c t = ( t, , t ∈ [0 , u ] H , then the system( c t : t ∈ [0 , u ] H ) satisfies (i)—(iii) of the strong cyclic property, and ( M t : t ∈ [0 , u ] H ) is an ( H, u )-decomposition of M with the strong cyclic property.Finally, we say that a pseudo MV-algebra M is strong ( H, u )- perfect if there isan ( H, u )-decomposition ( M t : t ∈ [0 , u ] H ) of M with the strong cyclic property.A prototypical example of a strong ( H, u )-perfect pseudo MV-algebra is thefollowing.
Proposition 4.1.
Let G be an ℓ -group and ( H, u ) an Abelian unital ℓ -group. Thenthe pseudo MV-algebra M H,u ( G ) := Γ( H −→× G, ( u, . is a strong ( H, u ) -perfect pseudo MV-algebra with a strong cyclic family (( h,
0) : h ∈ [0 , u ] H ) . Now we present a representation theorem for strong (
H, u )-perfect pseudo MV-algebras by (4.1). The following theorem uses the basic ideas of the particularsituation (
H, u ) = ( H ,
1) which was proved in [Dvu4, Thm 4.3].
Theorem 4.2.
Let M be a strong ( H, u ) -perfect pseudo MV-algebra, where ( H, u ) is an Abelian unital linearly ordered ℓ -group. Then there is a unique (up to isomor-phism) ℓ -group G such that M ∼ = Γ( H −→× G, ( u, . Proof.
Since M is a pseudo MV-algebra, due to [Dvu2, Thm 3.9], there is a uniqueunital (up to isomorphism of unital ℓ -groups) ℓ -group ( K, v ) such that M ∼ = Γ( K, v ) . Without loss of generality we can assume that M = Γ( K, v ). Assume ( M t : t ∈ [0 , u ] H ) is an ( H, u )-decomposition of M with the strong cyclic property and withan ( H, u )-strong cyclic family ( c t ∈ M : t ∈ [0 , u ] H ).By (vi) of Theorem 3.2, M is an associative cancellative semigroup satisfyingconditions of Birkhoff’s Theorem [Bir, Thm XIV.2.1], [Fuc, Thm II.4], which guar-antees that M is a positive cone of a unique (up to isomorphism) directed po-group G . Since M is a lattice, we have that G is an ℓ -group.Take the ( H, u )-strong perfect pseudo MV-algebra M H,u ( G ) defined by (4.1),and define a mapping φ : M → M H,u ( G ) by φ ( x ) := ( t, x − c t ) (4 . SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 11 whenever x ∈ M t for some t ∈ [0 , u ] H , where x − c t denotes the difference taken inthe group K . Claim 1: φ is a well-defined mapping. Indeed, M is in fact the positive cone of an ℓ -group G which is a subgroup of K. Let x ∈ M t . For the element x − c t ∈ K, we define ( x − c t ) + := ( x − c t ) ∨ x ∨ c t ) − c t ∈ M (when we use (iii) of Theorem 3.2) and similarly ( x − c t ) − := − (( x − c t ) ∧
0) = c t − ( x ∧ c t ) ∈ M . This implies that x − c t = ( x − c t ) + − ( x − c t ) − ∈ G. Claim 2: The mapping φ is an injective and surjective homomorphism of pseudoeffect algebras. We have φ (0) = (0 ,
0) and φ (1) = (1 , . Let x ∈ M t . Then x − ∈ M − t , and φ ( x − ) = (1 − t, x − c − t ) = (1 , − ( t, x − c t ) = φ ( x ) − . In an analogous way, φ ( x ∼ ) = φ ( x ) ∼ . Now let x, y ∈ M and let x + y be defined in M. Then x ∈ M t and y ∈ M t . Since x ≤ y − , we have t ≤ − t so that φ ( x ) ≤ φ ( y − ) = φ ( y ) − which means φ ( x ) + φ ( y ) is defined in M H,u ( G ) . Then φ ( x + y ) = ( t + t , x + y − c t + t ) =( t + t , x + y − ( c t + c t )) = ( t , x − c t ) + ( t , y − c t ) = φ ( x ) + φ ( y ) . Assume φ ( x ) ≤ φ ( y ) for some x ∈ M t and y ∈ M v . Then ( t, x − c t ) ≤ ( v, y − c v ) . If t = v, then x − c t ≤ y − c t so that x ≤ y. If i < j, then x ∈ M t and y ∈ M v sothat x < y. Therefore, φ is injective.To prove that φ is surjective, assume two cases: (i) Take g ∈ G + = M . Then φ ( g ) = (0 , g ) . In addition g − ∈ M so that φ ( g − ) = φ ( g ) − = (0 , g ) − = (1 , − (0 , g ) = (1 , − g ) . (ii) Let g ∈ G and t with 0 < t < g = g − g , where g , g ∈ G + = M . Since c t ∈ M t , g + c t exists in M and it belongs to M t , and g ≤ g + c t which yields ( g + c t ) − g = ( g + c t ) \ g ∈ M t . Hence, g + c t = ( g + c t ) \ g ∈ M t which entails φ ( g + c t ) = ( t, g ) . Claim 3: If x ≤ y, then φ ( y \ x ) = φ ( y ) \ φ ( x ) and φ ( x / y ) = φ ( x ) / φ ( y ) . It follows from the fact that φ is a homomorphism of pseudo effect algebras. Claim 4: φ ( x ∧ y ) = φ ( x ) ∧ φ ( y ) and φ ( x ∨ y ) = φ ( x ) ∨ φ ( y ) . We have, φ ( x ) , φ ( y ) ≥ φ ( x ∧ y ) . If φ ( x ) , φ ( y ) ≥ φ ( w ) for some w ∈ M, we have x, y ≥ w and x ∧ y ≥ w. In the same way we deal with ∨ . Claim 5: φ is a homomorphism of pseudo MV-algebras. It is necessary to show that φ ( x ⊕ y ) = φ ( x ) ⊕ φ ( y ) . This follows straightforwardfrom the previous claims and equality (2.2).Consequently, M is isomorphic to M H,u ( G ) as pseudo MV-algebras.If M ∼ = Γ( H −→× G ′ , ( u, G ′ , then ( H −→× G, ( u, H −→× G ′ , ( u, ℓ -groups in view of the categorical equivalence, see [Dvu2,Thm 6.4] or Theorem 2.1; let ψ : Γ( H −→× G, ( u, → Γ( H −→× G ′ , ( u, ψ ( { (0 , g ) : g ∈ G + } ) = { (0 , g ′ ) : g ′ ∈ G ′ + } which proves that G and G ′ areisomorphic ℓ -groups. (cid:3) We say that a pseudo MV-algebra is lexicographic if there are an Abelian linearlyordered unital ℓ -group ( H, u ) and an ℓ -group G (not necessarily Abelian) such that M ∼ = Γ( H −→× G, ( u, M is lexicographic iff M is strong ( H, u )-perfect for some Abelian linear unital ℓ -group ( H, u ). We note that in[DFL], a lexicographic MV-algebra denotes an MV-algebra having a lexicographicideal which will be defined below in Section 7. But by Theorem 7.5, we will concludethat both notions are equivalent for symmetric pseudo MV-algebras from M .It is worthy to note that according to Example 3.3, the pseudo MV-algebra M has two isomorphic lexicographic representations Γ( Z −→× ( Z −→× Z ) , (1 , (0 , Z −→× Z ) −→× Z , ((1 , , H , u ) := ( Z ,
1) and ( H , u ) := ( Z −→× Z , (1 , G := Z −→× Z and G := Z are not isomorphic ℓ -groups.5. Local Pseudo MV-algebras with Retractive Radical
In [DiLe2, Cor 2.4], the authors characterized MV-algebras that can be expressedin the form Γ( H −→× G, (1 , H is a subgroup of R and G is an Abelian ℓ -group. In what follows, we extend this characterization for local symmetric pseudoMV-algebras. This result gives another characterization of strong ( H , M the set of pseudo MV-algebras M such that either every max-imal ideal of M is normal or M is trivial. By [DDT, (6.1)], M is a variety.Let M be a symmetric pseudo MV-algebra. For any x ∈ M , we define the order , in symbols ord( x ), as the least integer n such that n.x = 1 if such n exists,otherwise, ord( x ) = ∞ . It is clear that the set of all elements with infinite order isan ideal. An element x is finite if ord( x ) < ∞ and ord( x − ) < ∞ . Lemma 5.1.
Let M be a pseudo MV-algebra from M and x ∈ M . There exists aproper normal ideal of M containing x if and only if ord( x ) = ∞ .Proof. Let x be any element of M and let I ( x ) be the normal ideal of M generatedby x . Then I ( x ) = { y ∈ M : y ≤ m.x for some m ∈ N } . (5 . (cid:3) Lemma 5.2.
Let M be a symmetric pseudo MV-algebra. If ord( x ⊙ y ) < ∞ , then x ≤ y − .Proof. By the hypothesis, ord( x ⊙ y ) = n for some integer n ≥ . Hence ( y − ⊕ x − ) n =0 . By [GeIo, Prop 1.24(ii)], ( y − ⊕ x − ) ∨ ( x ⊕ y ) = 1 which by [GeIo, Lem 1.32]yields ( y − ⊕ x − ) n ∨ ( x ⊕ y ) n = 1, so that ( x ⊕ y ) n = 1 and x ⊕ y = 1, consequently, x ≤ y − . (cid:3) Lemma 5.3.
Let M ∈ M be a symmetric pseudo MV-algebra. The followingstatements are equivalent: (i) M is local. (ii) For every x ∈ M , ord( x ) < ∞ or ord( x − ) < ∞ .Proof. Let M be local. There exists a unique maximal ideal I that is normal.Assume that for some x ∈ M , we have ord( x ) = ∞ = ord( x − ). By Lemma 5.1, x, x − ∈ I which is absurd.Conversely, let for every x ∈ M , ord( x ) < ∞ or ord( x − ) < ∞ . Let I be amaximal ideal of M and assume that x / ∈ M for some x ∈ M with ord( x ) = ∞ . Since I is by the hypothesis normal, by a characterization of normal and maximalideals, [GeIo, Prop 3.5], there is an integer n ≥ x − ) n ∈ I . By Lemma5.1, ord(( x − ) n ) = ∞ and ord((( x − ) n ) − ) < ∞ , i.e. ord( n.x ) < ∞ , which implies SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 13 ord( x ) < ∞ that is impossible. Hence, every element x with infinite order belongsto I , and so I is a unique maximal ideal of M , in addition I is normal. (cid:3) Lemma 5.4.
Let M ∈ M be a local symmetric pseudo MV-algebra and let I be aunique maximal ideal of M . For all x, y ∈ M such that x/I = y/I , we have x < y or y < x .Proof. By hypothesis, we have that x ⊙ y − / ∈ I or y ⊙ x − / ∈ I . By Lemma 5.3, inthe first case we have ord( x ⊙ y − ) < ∞ which by Lemma 5.2 implies x ≤ y andconsequently x < y . In the second case, we similarly conclude y < x . (cid:3) We introduce the following notion. A normal ideal I of a pseudo MV-algebra M is said to be retractive if the canonical projection π I : M → M/I is retractive, i.e.there is a homomorphism δ I : M/I → M such that π I ◦ δ I = id M/I . If a normalideal I is retractive, then δ I is injective and M/I is isomorphic to a subalgebra of M .For example, if M = Γ( H −→× G, ( u, I = { (0 , g ) : g ∈ G + } , then I is a nor-mal ideal, see Theorem 3.2(vi), and due to M/I ∼ = Γ( H, u ) ∼ = Γ( H −→× { } , ( u, ⊆ Γ( H −→× G, ( u, I is retractive. Lemma 5.5.
Let I be a normal ideal of a symmetric pseudo MV-algebra. Thenthe following are equivalent: (i) x/I = y/I . (ii) x = ( h ⊕ y ) ⊙ k − , where h, k ∈ I .Proof. (i) ⇒ (ii) Assume x/I = y/I . Then the elements k = x − ⊙ y and h = x ⊙ y − belong to I . It is easy to see that x ⊕ k = x ∨ y = h ⊕ y . Since k − = y − ⊕ x ≥ x ,we have x = x ∧ k − = ( x ⊕ k ) ⊙ k − = ( h ⊕ y ) ⊙ k − .(ii) ⇒ (i) Then we have x/I = y/I . (cid:3) Let M be a pseudo MV-algebra, and let Sub( M ) be the set of all subalgebrasof M . Then Sub( M ) is a lattice with respect to set theoretical inclusion with thesmallest element { , } and greatest one M . It is easy to see that if M is symmetricand I is an ideal of M , then the subalgebra h I i of M generated by I is the set h I i = I ∪ I − . We recall a subalgebra S of M is said to be a complement of asubalgebra A of M if S ∩ A = { , } and S ∨ A = M .In the following, we characterize retractive ideals of pseudo MV-algebras in ananalogous way as it was done for MV-algebras in [CiTo, Thm 1.2]. Theorem 5.6.
Let M be a symmetric pseudo MV-algebra and I a normal ideal of M . The following statements are equivalent: (i) I is a retractive ideal. (ii) h I i has a complement.Proof. Let I be a retractive ideal of M and let δ I : M/I → M be an injectivehomomorphism such that π I ◦ δ I = id M/I . We claim that δ I ( M/I ) is a complementof h I i . Clearly δ I ( M/I ) ∩ h I i = { , } . Let x ∈ M , then x/I = δ I ( M/I )( x/I ) /I sothat by Lemma 5.5, we have x = ( h ⊕ δ I ( x/I )) ⊙ k − for some h, k ∈ I that implies x ∈ δ I ( M/I ) ∨ h I i .Conversely, assume that h I i has a complement S ∈ Sub( M ). From S ∩ h I i = { , } we conclude that the canonical projection π I is injective on S . Indeed, iffor x, y ∈ S , we have x/I = y/I , then also x/I = ( x ∨ y ) /I = y/I which yields ( x ∨ y ) ⊙ x − ∈ S ∩ I = { } , ( x ∨ y ) ⊙ y − ∈ S ∩ I = { } . Therefore, x = x ∨ y = y and this implies that the restriction π I ↿ S is injective.From S ∨ h I i = M , we have that for each x ∈ M , there is a term in the languageof pseudo MV-algebras, says p ( a , . . . , a m , b , . . . , b n ), such that x = p M ( x , . . . , x m , y , . . . , y n )for some x , . . . , x m ∈ S and y , . . . , y n ∈ h I i . Then x/I = p M/I ( x /I, . . . , x m /I, y /I, . . . , y n /I ) . Since y i /I ∈ { , } for each i = 1 , . . . , n , there is an n -tuple ( t , . . . , t n ) of elementsfrom { , } such that x/I = p M ( x , . . . , x m , t , . . . , t n ) /I . Since( x , . . . , x m , t , . . . , t n ) ∈ S m + n , we have that p M ( x , . . . , x m , t , . . . , t n ) ∈ S . Therefore, the restriction π I ↿ S isan isomorphism from S onto M/I , and setting δ I = ( π I ↿ S ) − , we see that I isretractive. (cid:3) Theorem 5.7.
Let M be a symmetric pseudo MV-algebra from M . The followingstatements are equivalent: (i) M is local and Rad n ( M ) is retractive. (ii) M is strong ( H , -perfect for some subgroup H of R with ∈ H . (iii) There exists a subgroup H of R with ∈ H and an ℓ -group G such that M ∼ = Γ( H −→× G, ( u, .Proof. (i) ⇒ (ii) Let I be a unique maximal and normal ideal of M and let ( K, v )be a (unique up to isomorphism) unital ℓ -group given by Theorem 2.1, such that M ∼ = Γ( K, v ); without loss of generality we can assume that M = Γ( K, v ). By[Dvu1], there is an extremal state (= state morphism) s : M → [0 ,
1] such that I = Ker( s ). The range of s , s ( M ), is an MV-algebra which corresponds to aunique subgroup H of R such that s ( M ) = Γ( H ,
1) is a subalgebra of Γ( R , I = Rad n ( M ), I is a retractive ideal, and M/I is isomorphic to Γ( H , H ,
1) can be injectively embedded into K and H is isomorphic to a subgroupof K .In addition, let h I i be a subalgebra of M generated by I . Then h I i = I ∪ I − = I ∪ I − , I − = I ∼ , and h I i is a perfect pseudo MV-algebra. By [DDT, Prop 5.2],there is a unique (up to isomorphism) ℓ -group G such that h I i ∼ = Γ( Z −→× G, (1 , M ∼ = Γ( H −→× G, (1 , M t = s − ( { t } ), t ∈ [0 , H . We assert that ( M t : t ∈ [0 , H ) is an ( H , M . It is clear that it is a decomposition: Every M t is non-empty,and M − t = M − t = M ∼ t for each t ∈ [0 , H . In addition, if x ∈ M v and y ∈ M t ,then x ⊕ y ∈ M v ⊕ t , x ∧ y ∈ M v ∧ t and x ∨ y ∈ M v ∨ t . By Lemma 5.4, we have M s M t for all s < t , s, t ∈ [0 , H .Since I = Rad n ( M ) is retractive, there is a unique subalgebra M ′ of M suchthat s ( M ′ ) = s ( M ). For any t ∈ [0 , H , there is a unique element x t ∈ M ′ such that s ( x t ) = t . We assert that the system ( x t : t ∈ [0 , H ) satisfies thefollowing properties (i) c t ∈ M t for each t ∈ [0 , H , (ii) c v + t = c v + c t whenever v + t ≤
1, (iii) c = 1. (iv) c t ∈ C ( K ). Indeed, since s is a homomorphismof pseudo MV-algebras, by the categorical equivalence Theorem 2.1, s can beuniquely extended to a unital ℓ -group homomorphism ˆ s : ( K, v ) → ( H , x is any element of K , then x + c t − x ∈ M because M is symmetric, and hence SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 15 ˆ s ( x + c t − x ) = ˆ s ( x ) + ˆ s ( c t ) − ˆ s ( x ) = s ( c t ) = t which implies x + c t − x = c t so that x + c t = c t + x .In other words, we have proved that ( M t : t ∈ [0 , H ) has the strong cyclicproperty, and consequently, M is strong ( H , M ∼ =Γ( H −→× G ′ , (1 , ℓ -group G ′ . Hence G ′ ∼ = G ,where G was defined above, which proves (ii) ⇒ (iii).The implication (iii) ⇒ (i) is evident by the note that is just before Theorem5.7. (cid:3) We note that if M is a local symmetric pseudo MV-algebra with a retractive idealRad n ( M ), then M is a lexicographic extension of Ker n ( M ) in the sense describedin [HoRa]. Proposition 5.8.
Let ( M α : α ∈ A ) be a system of pseudo MV-algebras and let I α be a non-trivial normal ideal of M α , α ∈ A . Set M = Q α M α and I = Q α I α .Then I is a retractive ideal of M if and only if every I α is a retractive ideal of M α .Proof. The set I = Q α I α is a non-trivial normal ideal of M . Then M/I ∼ = Q α M α /I α and without loss of generality, we can assume that M/I = Q α M α /I α .Assume that every I α is retractive. We denote by π α the canonical projection of M α onto M α /I α and by δ α : M α /I α → M α its right inversion i.e. π α ◦ δ α = id M α /I α .Let π : M → M/I be the canonical projection. If we set δ : M/I → M by δ (( x α /I α ) α ) := (( δ α ( x α /I α )) α ), then we have π ◦ δ = id M/I , so that I is retractive.Conversely, let I be a retractive ideal of M . Let π α : Q α M α be the α -thprojection of M onto M α . We define a mapping δ α : M α /I α → M α by δ α = π α ◦ δ ( α ∈ A ). Then Q α π α ◦ δ α ( x α /I α ) = Q π α ◦ π α ◦ δ ( x α /I α ) which yields π α ◦ δ α = id M α /I α . (cid:3) Corollary 5.9.
Let I be a non-trivial normal ideal of a pseudo MV-algebra M andlet α be a cardinal. Then the power I α is a retractive ideal of the power pseudoMV-algebra M α if and only if I is a retractive ideal of M . Free Product and Local Pseudo MV-algebras
In the present section we show that every local pseudo MV-algebra that is astrong ( H , V be a class of pseudo MV-algebras and let { A t } t ∈ T ⊆ V . According to[DvHo1], we say that a V - coproduct (or simply a coproduct if V is known from thecontext) of this family is a pseudo MV-algebra A ∈ V , together with a family ofhomomorphisms { f t : A t → A } t ∈ T such that(i) S t ∈ T f t ( A t ) generates A ;(ii) if B ∈ V and { g t : A t → B } t ∈ T is a family of homomorphisms, then thereexists a (necessarily) unique homomorphism h : A → B such that g t = f t h for all t ∈ T .Coproducts exist for every variety V of algebra, and are unique. They are des-ignated by F V t ∈ T A t (or A ⊔ V A if T = { , } ). If each of the homomorphisms f t is an embedding, then the coproduct is called the free product . By [DvHo1, Thm 2.3], the free product of any set of non-trivial pseudo MV-algebras exists in the variety of pseudo MV-algebras.Now let M be a symmetric local pseudo MV-algebra from M with a uniquemaximal and normal ideal I = Ker n ( M ) = Ker( M ). Let H be a subgroup of R such that M/I ∼ = Γ( H , ℓ -group G such that Γ( Z −→× G, (1 , ∼ = h I i . Let N = Γ( H −→× G, (1 , I is retractive, then by Theorem 5.7, M ∼ = N , and in thissection, we describe this situation using free product of M/I and h I i . We note thatthis was already established in [DiLe2, Thm 3.1] but only for MV-algebras. Forour generalization, we introduce a weaker form of our free product of M/I and h I i which we will denote M/I ⊔ w h I i in the variety of symmetric pseudo MV-algebrasfrom M and which means that (i) remains and (ii) are changed as follows(i*) if φ : M/I → M/I ⊔ w h I i and φ : h I i → M/I ⊔ w h I i are injecive homo-morphisms, then φ ( M/I ) ∪ φ ( h I i ) generates M/I ⊔ w h I i ,(ii*) if κ : M/I → A and κ : h I i → A , where A is a symmetric pseudo MV-algebra from M , are such homomorphisms that κ ( a ) + κ ( b ) = κ ( b ) + κ ( a ), then there is a unique homomorphism ψ : M/I ⊔ w h I i → A suchthat ψ ◦ φ = κ and ψ ◦ φ = κ .We note that if M is an MV-algebra, then our notion coincides with the originalform of the free product of MV-algebras in the class of MV-algebras. Theorem 6.1.
Let M be a symmetric local pseudo MV-algebra from M , I =Rad n ( I ) and N = Γ( H −→× G, (1 , for some unital ℓ -subgroup ( H , of ( R , andsome ℓ -group G . The following statements are equivalent: (i) M ∼ = N . (ii) The free product
M/I ⊔ w h I i in the variety of symmetric pseudo MV-algebrasfrom M is isomorphic to M .Proof. (i) ⇒ (ii) Let M = Γ( K, v ). By Theorem 5.7, I = Ker n ( M ) is a retractiveideal. Define φ : M/I → Γ( H −→× { } , (1 , ⊂ Γ( H −→× G, (1 , N and φ : h I i → Γ( Z −→× G, (1 , ⊂ Γ( H −→× G, (1 , N as follows: Let s be a unique state on M which is guaranteed by Theorem 3.4. We set M t = s − ( { t } ) for any t ∈ [0 , H .Then φ ( x/I ) := ( t,
0) whenever x ∈ M t . Since h I i = I ∪ I − , we set φ ( x ) = (0 , x )if x ∈ I and φ ( x ) = (1 , x −
1) if x ∈ I − . From (4.2) of the proof of Theorem 4.2we see that φ and φ are injective homomorphisms of pseudo MV-algebras into N . Using again (4.2), we see that φ ( M/I ) ∪ φ ( h I i ) generates N .Now suppose that there is a symmetric pseudo MV-algebra A from M and twomutually commuting homomorphisms κ : M/I → A and κ : h I i → A = Γ( W, w ),i.e. κ ( a ) + κ ( b ) = κ ( b ) + κ ( a ) for all a ∈ M/I and b ∈ h I i . Then κ (1 /I ) = w = κ (1) and w commutes with every κ ( a ) and κ ( b ). Claim 1.
Let a = κ φ − ( h, with < h < , h ∈ H , and ǫ = κ φ − (0 , g ) with g ∈ G + . Then ǫ < a < ǫ − . Indeed, by the assumption, from the form of the element a we conclude that itis finite and ǫ and a commute. Then there is an integer n ≥ n.a = 1.Since ǫ ∈ Rad( A ), we have n.ǫ = nǫ < n.a ≤ na which yields 0 ≤ n ( a − ǫ ), sothat ǫ < a . In a similar way we show ǫ < a − , i.e. ǫ < a < ǫ − . Claim 2.
Let α = κ ◦ φ − : Γ( H −→× { } , (1 , → A and β = κ ◦ φ − : φ − ( h I i ) → A . Passing to the corresponding representing unital ℓ -groups, we will denote by SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 17 ˆ α and ˆ β the corresponding extensions of α and β to ℓ -homomorphisms of unital ℓ -groups into the unital ℓ -group ( G A , w ) such that Γ( G A , w ) = A . Then ˆ α (0 , h ) +ˆ β (0 , g ) ≥ for each h ∈ H + and each g ∈ G . If h = 0, the statement is evident. Let h >
0. Then a := ˆ α (0 , h ) + ˆ β (0 , g ) =ˆ α ( h,
0) + ˆ β (0 , g + ) + ˆ β (0 , g − ), where g + = g ∨ g − = g ∧
0. Then a =ˆ α ( h,
0) + β (0 , g + ) + β (1 , g − ) − β (1 , α ( h,
0) + β (0 , g + ) ≥ β (0 , − g − ) = β (1 , − β (1 , g − ) and the claim is proved.Now we define a mapping ψ : H −→× G → G A by ψ ( h, g ) = ˆ α ( h,
0) + ˆ β (0 , g ) , ( h, g ) ∈ H −→× G. Claim 3. ψ is an ℓ -group homomorphism of unital ℓ -groups. (a) We have ψ (0 ,
0) = 0 and ψ (1 ,
0) = w . Moreover, ψ ( h , g ) + ψ ( h , g ) = ˆ α ( h ,
0) + ˆ β (0 , g ) + ˆ α ( h ,
0) + ˆ β (0 , g )= ˆ α ( h ,
0) + ˆ α ( h ,
0) + ˆ β (0 , g ) + ˆ β (0 , g )= ˆ α ( h + h ,
0) + ˆ β (0 , g + g )= ψ ( h + h , g + g ) . (b) According to Claim 2, we see that ψ ( h, g ) ≥ h, g ) ≥ (0 , ψ preserves ∧ . For x := ( h , g ) ∧ ( h , g ), we have three cases (i) x = ( h , g )if h < h , (ii) x = ( h , g ∧ g ) if h = h , and (iii) x = ( h , g ) if h < h . In case (i), we have ψ ( h , g ) − ψ ( h , g ) = ψ ( h − h , g − g ) ≥ ψ preserves ∧ . In case (ii), we have ψ (( h , g ) ∧ ( h , g )) = ψ ( h , g ∧ g ) = ˆ α ( h ,
0) + ˆ β (0 , g ∧ g )= ˆ α ( h ,
0) + ˆ β (0 , g ) ∧ ˆ β (0 , g )= (ˆ α ( h ,
0) + ˆ β (0 , g )) ∧ (ˆ α ( h ,
0) + ˆ β (0 , g ))= ψ ( h , g ) ∧ ψ ( h , g ) . Case (iii) follows from (i).If we restrict ψ to N , then we have ψ ( h, g ) = ( α ( h, ⊕ β (1 , g + )) ⊙ β (1 , g − ) , ( h, g ) ∈ N. Using that ψ is an ℓ -group homomorphism, we have that if g = g + g , where g ≥ g ≤
0, then ψ ( h, g ) = ( α ( h, ⊕ β (1 , g )) ⊙ β (1 , g ) . Uniqueness of ψ . If ψ ′ is another homomorphism from N into A such that φ i ◦ ψ = κ i for i = 1 ,
2, then ψ ′ (0 ,
0) = ψ (0 , ψ ′ (0 , g ) = ψ ′ ( φ φ − (0 , g )) = φ κ (0 , g ) = ψ (0 , g ), g ∈ G + . ψ ′ ( h,
0) = ψ ′ ( φ φ − ( h, φ κ ( h,
0) = ψ ( h, h ∈ [0 , H .Using all above steps, we have that the free product M/I ⊔ w h I i ∼ = N . Since N ∼ = M , we have established (ii).(ii) ⇒ (i) From the proof of the previous implication we have that the freeproduct of Γ( H ,
1) and Γ( Z −→× G, (1 , N = Γ( H −→× G, (1 , M/I ∼ = Γ( H ,
1) and h I i ∼ = Γ( Z −→× G, (1 , M ∼ = N . (cid:3) Pseudo MV-algebras with Lexicographic Ideals
The following notions were introduced in [DFL] only for MV-algebras, and inthis section, we extend them for symmetric pseudo MV-algebras and generalizesome results from [DFL].We say that a normal ideal I is (i) commutative if x/I ⊕ y/I = y/I ⊕ x/I for all x, y ∈ M , (ii) strict if x/I < y/I implies x < y .For example, (i) if s is a state, then Ker( s ) is a commutative ideal, [Dvu1, Prop4.1(ix)], (ii) every maximal ideal that is normal is commutative, [Dvu1]. If M is alocal symmetric pseudo MV-algebra, Rad n is a strict ideal.Now we extend for pseudo MV-algebras the notion of a lexicographic ideal in-troduced in [DFL] only for MV-algebras. We say that a commutative ideal I of apseudo MV-algebra M , { } 6 = I = M , is lexicographic if(i) I is strict,(ii) I is retractive,(iii) I is prime.We note that a lexicographic ideal for MV-algebras was defined in [DFL] by(i)–(iii) and(iv) y ≤ x ≤ y − for all y ∈ I and all x ∈ M \ h I i , where h I i is the subalgebraof M generated by I .But since I is strict, we have y ∈ I − implies z < y for any z ∈ I . Hence, if z / ∈ I , we have z/I > x/I = 0 /I for all x ∈ I which yields z > x . Therefore, h I i = I ∪ I − and (iv) holds, and consequently, (iv) from [DFL] is superfluous, andfor the definition of a lexicographic ideal of an MV-algebra we need only (i)–(iii).Let LexId( M ) be the set of lexicographic ideals of M . If we take the MV-algebra M from Example 3.3, we see that I = { (0 , m, n ) : m > , n ∈ Z or m = 0 , n ≥ } and I = { (0 , , n ) : n ≥ } are two unique lexicographic ideals of M and I ⊂ I . Proposition 7.1. If I, J ∈ LexId( M ) , then I ⊆ J or J ⊆ I . In addition, ev-ery lexicographic ideal is contained in the radical Rad( M ) of M . If one of thelexicographic ideals is a maximal ideal, then M has a unique maximal ideal of M .Proof. Suppose the converse, that is, there are x ∈ I \ J and y ∈ J \ I . Then x/I < y/I and y/J < x/J which yields x < y and y < x which is absurd.Assume that I is a lexicographic ideal of M . If I = Rad( M ), the statement isevident. If there is an element y ∈ Rad( M ) such that y / ∈ I , then by (ii) x < y forany element x ∈ I , so that I ⊆ Rad( M ).Let I be any lexicographic ideal of M . We have two cases. (a) I is a maximalideal of M . We claim M has a unique maximal ideal. Indeed, for any maximalideal J of M , J = I , there are x ∈ I \ J and y ∈ J \ I which implies x < y sothat x ∈ J which is a contradiction. Hence, I is a unique maximal ideal of M ,then Rad( M ) = I and every lexicographic ideal of M is in Rad( M ). (b) I is nota maximal ideal of M . Let J be an arbitrary maximal ideal of M . There exists y ∈ J \ I which yields y > x for any x ∈ I , so that x ∈ J and I ⊆ J . Hence, again I ⊆ Rad( M ). (cid:3) Remark 7.2.
It is clear that if LexId( M ) = ∅ is finite, then LexId( M ) has thegreatest element. If LexId( M ) is infinite, we do not know whether LexId( M ) hasthe greatest element. And if this element exists, is it a maximal ideal of M ? SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 19
We note that in Theorem 7.9(1), we show that if M is symmetric from M andLexId( M ) = ∅ , then M is local.As an interesting corollary we have the following statement. Corollary 7.3. If LexId( M ) is non-empty and s is a state on M , then s vanisheson each lexicographic ideal of M .Proof. Let I be a lexicographic ideal of M . First let s be an extremal state. ThenKer( s ) is by [Dvu1, Prop 4.3] a maximal ideal. Hence, by Proposition 7.1, we have I ⊆ Ker( M ) ⊆ Ker( s ), so that each extremal state vanishes on I . Therefore, eachconvex combination of extremal states, and by Krein–Mil’man Theorem, each stateon M vanishes on I . (cid:3) A strengthening of the latter corollary for lexicographic pseudo MV-algebras M from M will be done in Corollary 7.7 showing that then M has a unique state.Now we present a prototypical examples of a pseudo MV-algebra with lexico-graphic ideal. Proposition 7.4.
Let ( H, u ) be an Abelian linear unital ℓ -group and let G bean ℓ -group. If we set I = { (0 , g ) : g ∈ G + } , then I is a lexicographic ideal of M = Γ( H −→× G, ( u, .In addition, M is subdirectly irreducible if and only if G is a subdirectly irre-ducible ℓ -group.Proof. It is clear that I is a normal ideal of M as well as it is prime.We have x/I = 0 /I iff x ∈ I . Assume (0 , g ) /I < ( h, g ′ ) /I . Then ( h, g ) / ∈ I thatyields h > , g ) < ( h, g ′ ). Hence, if x/I < y/I , then ( y − x ) /I > /I and y − x > x < y .Since M/I ∼ = Γ( H −→× { } , ( u, ⊆ Γ( H −→× G, ( u, I is retractive.Finally, let y ∈ I and x ∈ M \ h I i . Then h I i = I ∪ I − and x = ( h, g ′ ) for some h with 0 < h < u and g ′ ∈ G . Then y = (0 , g ) and hence, y < x < y − .The statement on subdirect irreducibility follows from the categorical represen-tation of pseudo MV-algebras, Theorem 2.1. (cid:3) Theorem 7.5.
Let M be a symmetric pseudo MV-algebra from M and let I be alexicographic ideal of M . Then there is an Abelian linear unital ℓ -group ( H, u ) andan ℓ -group G such that M ∼ = Γ( H −→× G, ( u, .Proof. Similarly as in the proof of Theorem 5.7, we can assume that M = Γ( K, v )for some unital ℓ -group ( K, v ). Since I is lexicographic, then I is normal and prime,so that M/I is a linear, and since I is also commutative, M/I is an MV-algebra.There is an Abelian linear unital ℓ -group ( H, u ) such that
M/I ∼ = Γ( H, u ).Let π I : M → M/I be the canonical projection. For any t ∈ [0 , u ] H , we set M t := π − I ( { t ) } . We assert that ( M t : t ∈ [0 , u ] H ) is an ( H, u )-decomposition of M . Indeed, (a) let x ∈ M s and y ∈ M t for s < t, s, t ∈ [0 , u ] H . Then π I ( x ) = s < t < π ( y ) and x < y because I is strict. (b) Since π I is a homomorphism, M − t = M u − t = M ∼ t for each t ∈ [0 , u ] H . (c) Let x ∈ M s and y ∈ M t , then π I ( x ⊕ y ) = π I ( x ) ⊕ π I ( y ) = s ⊕ t. In addition, h I i = I ∪ I − = I ∪ I − , I − = I ∼ , and h I i is a perfect pseudo MV-algebra. By [DDT, Prop 5.2], there is a unique (up to isomorphism) ℓ -group G suchthat h I i ∼ = Γ( Z −→× G, (1 , Now we show that ( M t : t ∈ [0 , u ] H ) has the strong cyclic property. Being I also retractive, there is a subalgebra M ′ of M such that M ′ ∼ = M/I and π I ( M ′ ) = π I ( M ). Then M ′ is in fact an MV-algebra. For any t ∈ [0 , u ] H , there is a unique c t ∈ M t such that π I ( c t ) = t. We assert that the system of elements ( c t : t ∈ [0 , u ] H )has the following properties: (i) c t ∈ M t , (ii) if s + t ≤ u , then c s + c t ∈ M and c s + c t = c s + t , (iii) c = 1, and (iv) c t ∈ C ( K ) for each t ∈ [0 , u ] H ; indeed let x ∈ K . Being M symmetric, the element x + c t − x ∈ H belongs also to M . Due tothe categorical equivalence, Theorem 2.1, the homomorphism π I can be uniquelyextended to a homomorphism ˆ π I : ( K, v ) → ( H, u ) of unital ℓ -groups. Hence, π I ( x + c t − x ) = ˆ π I ( x + c t − x ) = ˆ π I ( x ) + ˆ π I ( c t ) − ˆ π I ( x ) = π I ( c t ) = t which implies c t = x + c t − x and x + c t = c t + x .Consequently, M is a strong ( H, u )-perfect pseudo MV-algebra. By Theorem4.2, there is an ℓ -group G ′ such that M ∼ = Γ( H −→× G ′ , ( u, ℓ -groups) of G ′ in Theorem 4.2, we have G ′ ∼ = G and consequently M ∼ = Γ( H −→× G, ( u, (cid:3) According to the latter theorem and Proposition 7.4, we see that our notionof a lexicographic pseudo MV-algebra for symmetric pseudo MV-algebras from M coincides with the notion of one defined for MV-algebras in [DFL] as those havingat least one lexicographic ideal.In the following result we compare the class of local pseudo MV-algebras withthe class of lexicographic pseudo MV-algebras. Theorem 7.6. (1)
The class of lexicographic pseudo MV-algebras from M isstrictly included in the class of symmetric local pseudo MV-algebras. (2) The class of symmetric local pseudo MV-algebras with retractive radical isstrictly included in the class of lexicographic pseudo MV-algebras from M .Proof. (1) Let M be a lexicographic pseudo MV-algebra from M . By Theorem7.5, M is symmetric and it is isomorphic to some M ′ := Γ( H −→× G, ( u, H, u ) is an Abelian unital ℓ -group and G is an ℓ -group. Then the ideal I = { (0 , g ) : g ∈ G + } is by Proposition 7.4 a retractive ideal of M ′ . By Proposition7.1, we have I ⊆ Rad( M ′ ) = Rad n ( M ′ ). Since I is prime, so is Rad n ( M ′ ) whichyields M ′ / Rad n ( M ′ ) is linearly ordered and semisimple. Hence, M ′ / Rad n ( M ′ ) isa simple MV-algebra. Therefore, by [Dvu1, Prop 3.3-3.5], Rad n ( M ) is a maximalideal which yields that M ′ is local and, consequently M is local.To show that the class of lexicographic pseudo MV-algebras from M is strictlyincluded in the class of symmetric local pseudo MV-algebras, we can use an examplefrom the proof of [DFL, Thm 4.7] or the pseudo MV-algebra Γ( Z −→× Z , (2 , M . Using an example from [DFL, Thm 4.7], we conclude that thisinclusion is proper. (cid:3) The latter result entails the following corollary.
Corollary 7.7.
Every lexicographic pseudo MV-algebra from M admits a uniquestate.Proof. If M is a lexicographic pseudo MV-algebra from M , by (i) of Theorem 7.6,we see that M is local, that is, it has a unique maximal ideal and this ideal is SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 21 normal. Due to a one-to-one relation between extremal states and maximal andnormal ideals of M , [Dvu1], we conclude M admits a unique state. (cid:3) The following result gives a new look to Theorem 5.7.
Theorem 7.8.
Let M be a lexicographic symmetric pseudo MV-algebra from M .The following statements are equivalent: (i) Rad n ( M ) is a lexicographic ideal. (ii) M is strongly ( H , -perfect for some unital ℓ -subgroup ( H , of ( R , .Proof. Let M ∼ = Γ( H −→× G, ( u, ℓ -group ( H, u ) and an ℓ -group G and let I be a retractive ideal of M such that M/I ∼ = Γ( H, u ). ByProposition 7.4, I ⊆ Rad n ( M ).(i) ⇒ (ii) If Rad n ( M ) is a retractive ideal, then M/ Rad n ( M ) is a semisimpleMV-algebra that is linearly ordered because Rad n ( M ) is a prime normal ideal.Again applying by [Dvu1, Prop 3.4-3.5], M/ Rad n ( M ) ∼ = Γ( H, u ) and Γ(
H, u ) isisomorphic to some ( H , ⇒ (i) Since M/I ∼ = Γ( H , I is a maximal ideal of M . Hence, I = Rad n ( M ) and I is a lexicographic ideal of M and M ∼ = Γ( H −→× G, (1 , (cid:3) We say that a pseudo MV-algebra M from M is I - representable if I is a lex-icographic ideal of M and M ∼ = Γ( H −→× G, ( u, H, u ) is an Abelianunital ℓ -group such that M/I ∼ = Γ( H, u ) and G is an ℓ -group such that h I i ∼ =Γ( Z −→× G, (1 , H, u ) and G are guaranteed by Theorem 7.5. Theorem 7.9.
The class of lexicographic pseudo MV-algebras from M is closedunder homomorphic images and subalgebras, but it is not closed under direct prod-ucts.Moreover, (1) if N is a homomorphic image of M , then N ∼ = Γ( H −→× G , ( u , ,where ( H , u ) and G are homomorphic images of ( H, u ) and G , respectively. (2) If N is a subalgebra of M , then N ∼ = Γ( H −→× G , ( u , , where ( H , u ) and G are subalgebras of ( H, u ) and G , respectively.Proof. Let I be a lexicographic ideal of M such that M is I -representable.(1) Let f : M → N be a surjective homomorphism. Then N is symmetric andfrom M whilst M is a variety. If we set f ( I ) = { f ( x ) : x ∈ I } , then f ( I ) is a normalideal of N = f ( M ) that is also commutative, prime and strict. We claim that f ( I )is a retractive ideal, too. Let π I : M → M/I be the canonical projection and let δ I : M/I → M be a homomorphism such that π I ◦ δ I = id M/I . Let M = δ I ( M/I )be a subalgebra of M that is isomorphic to M/I . If we define ˆ f : M/I → N/f ( I )by ˆ f ( x/I ) = f ( x ) /f ( I ), then ˆ f is a well-defined homomorphism such that ˆ f ◦ π I = π f ( I ) ◦ f . Set N = f ( M ) and let f M be the restriction of f onto M . We define δ f ( I ) : N/f ( I ) → N via δ f ( I ) ( f ( x ) /f ( I )) := f M ( δ I ( x/I )); then δ f ( I ) is a well-defined homomorphism such that δ f ( I ) ( N/f ( I )) = N and f M ◦ δ I = δ f ( I ) ◦ ˆ f .Hence, π f ( I ) ◦ δ f ( I ) ( f ( x ) /f ( I )) = π f ( I ) ◦ f M ◦ δ I ( x/I )= ˆ f ◦ π I ◦ δ I ( x/I ) = ˆ f ( x/I )= f ( x ) /f ( I ) that proves f ( I ) is a retractive ideal of N .Take the unital representation of pseudo MV-algebras given by Theorem 2.1, andlet N ∼ = Γ( K, v ) and let f : ( H −→× G, ( u, → ( K, v ) be a surjective homomorphismof unital ℓ -groups. Let f ( h ) = f ( h, h ∈ H , and f ( g ) = f (0 , g ), g ∈ G . If weset H := f ( H ), u = f ( u, G := f ( G ). Then N ∼ = Γ( H −→× G , ( u , N be a subalgebra of M . Then N is symmetric and belongs to M . Weset J := N ∩ I . Then J is a normal ideal of N that is also commutative and prime.It is strict, too, because if x ∈ N and x / ∈ J , then x / ∈ I and x > y for any y ∈ J andconsequently, for any y ∈ J . Then N/J can be embedded into
M/I by a mapping i J ( x/J ) := x/I ( x ∈ N ) and if i ( x ) = x , x ∈ N , then π I ◦ i = i J ◦ π J . Let M := δ I ( M/I ) and N := M ∩ N . Then δ I ( N/I ) ∈ N ; indeed, if there is x ∈ N such that δ I ( x/I ) / ∈ N , then π I ◦ δ I ( x/I ) = x/I / ∈ N /I . Define δ J : N/J → N by δ J ( x/J ) = i − J ◦ δ I ( x/I ). Since i − I ◦ π I ( x ) = π J ◦ i − ( x ), x ∈ N , then π J ◦ δ J ( x/J ) = π J ◦ i − ◦ δ I ( x/I )= i − I ◦ π I ◦ δ I ( x/I ) = i − I ( x/I ) = x/J. The rest follows the analogous steps as the end of (1).(3) According to Corollary 7.7, every lexicographic pseudo MV-algebra M admitsa unique state. But the pseudo MV-algebra M × M admits two extremal states,and therefore, M × M is not lexicographic. (cid:3) We note that in case (3) of latter Theorem if I is a lexicographic ideal of M ,then I × I is by Proposition 5.8 a retractive ideal but not lexicographic.8. Categorical Representation of Strong ( H, u ) -perfect PseudoMV-algebras In this section, we establish the categorical equivalence of the category of strong(
H, u )-perfect pseudo MV-algebras with the variety of ℓ -groups. This extends thecategorical representation of strong n -perfect pseudo MV-algebras from [Dvu3] andof H -perfect pseudo MV-algebras from [Dvu4] with the variety of ℓ -groups. In whatfollows, we follow the ideas of [Dvu4, Sec 5] and to be self-contained we repeat themmutatis mutandis.Let SPP s MV H,u be the category of strong (
H, u )-perfect pseudo MV-algebraswhose objects are strong (
H, u )-perfect pseudo MV-algebras and morphisms arehomomorphisms of pseudo MV-algebras. Now let G be the category whose objectsare ℓ -groups and morphisms are homomorphisms of ℓ -groups.Define a mapping M H,u : G → SPP s MV H,u as follows: for G ∈ G , let M H,u ( G ) := Γ( H −→× G, ( u, h : G → G is an ℓ -group homomorphism, then M H,u ( h )( t, g ) = ( t, h ( g )) , ( t, g ) ∈ Γ( H −→× G, ( u, . It is easy to see that M H,u is a functor.
Proposition 8.1. M H,u is a faithful and full functor from the category G of ℓ -groups into the category SPP s MV H,u of strong ( H, u ) -perfect pseudo MV-algebras.Proof. Let h and h be two morphisms from G into G ′ such that M H,u ( h ) = M H,u ( h ). Then (0 , h ( g )) = (0 , h ( g )) for each g ∈ G + , consequently h = h . SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 23
To prove that M H,u is a full functor, suppose that f is a morphism from a strong( H, u )-perfect pseudo MV-algebra Γ( H −→× G, ( u, H −→× G , ( u, . Then f (0 , g ) = (0 , g ′ ) for a unique g ′ ∈ G ′ + . Define a mapping h : G + → G ′ + by h ( g ) = g ′ iff f (0 , g ) = (0 , g ′ ) . Then h ( g + g ) = h ( g )+ h ( g ) if g , g ∈ G + . Assumenow that g ∈ G is arbitrary. Then g = g + − g , where g + = g ∨ g − = − ( g ∧ g = − g − + g + . If g = g − g , where g , g ∈ G + , then g + + g = g − + g and h ( g + ) + h ( g ) = h ( g − ) + h ( g ) which shows that h ( g ) = h ( g ) − h ( g ) is awell-defined extension of h from G + onto G .Let 0 ≤ g ≤ g . Then (0 , g ) ≤ (0 , g ) , which means h is a mapping preservingthe partial order.We have yet to show that h preserves ∧ in G , i.e., h ( a ∧ b ) = h ( a ) ∧ h ( b ) whenever a, b ∈ G. Let a = a + − a − and b = b + − b − , and a = − a − + a + , b = − b − + b + .Since , h (( a + + b − ) ∧ ( a − + b + )) = h ( a + + b − ) ∧ h ( a − + b + ) . Subtracting h ( b − ) fromthe right hand and h ( a − ) from the left hand, we obtain the statement in question.Finally, we have proved that h is a homomorphism of ℓ -groups, and M H,u ( h ) = f as claimed. (cid:3) We note that by a universal group for a pseudo MV-algebra M we mean a pair( G, γ ) consisting of an ℓ -group G and a G -valued measure γ : M → G + (i.e., γ ( a + b ) = γ ( a ) + γ ( b ) whenever a + b is defined in M ) such that the followingconditions hold: (i) γ ( M ) generates G . (ii) If K is a group and φ : M → K is an K -valued measure, then there is a group homomorphism φ ∗ : G → K such that φ = φ ∗ ◦ γ .Due to [Dvu2], every pseudo MV-algebra admits a universal group, which isunique up to isomorphism, and φ ∗ is unique. The universal group for M = Γ( G, u )is (
G, id ) where id is the embedding of M into G .Let A and B be two categories and let f : A → B be a functor. Suppose that g, h be two functors from B to A such that g ◦ f = id A and f ◦ h = id B , then g isa left-adjoint of f and h is a right-adjoint of f. Proposition 8.2.
The functor M H,u from the category G into SPP s MV H,u hasa left-adjoint.Proof.
We show, for a strong (
H, u )-perfect pseudo MV-algebra M with an ( H, u )-decomposition ( M t : t ∈ [0 , u ] H ) and an ( H, u )-strong cyclic family ( c t : t ∈ [0 , u ] H )of elements of M , there is a universal arrow ( G, f ), i.e., G is an object in G and f is a homomorphism from the pseudo MV-algebra M into M H,u ( G ) such that if G ′ is an object from G and f ′ is a homomorphism from M into M H,u ( G ′ ), then thereexists a unique morphism f ∗ : G → G ′ such that M H,u ( f ∗ ) ◦ f = f ′ .By Theorem 4.2, there is a unique (up to isomorphism of ℓ -groups) ℓ -group G such that M ∼ = Γ( H −→× G, ( u, . By [Dvu2, Thm 5.3], ( H −→× G, γ ) is a universalgroup for M, where γ : M → Γ( H −→× G, ( u, γ ( a ) = ( t, a − c t ) , if a ∈ M t . (cid:3) Define a mapping P H,u : SPP s MV H,u → G via P H,u ( M ) := G whenever( H −→× G, f ) is a universal group for M . It is clear that if f is a morphism fromthe pseudo MV-algebra M into another one N , then f can be uniquely extendedto an ℓ -group homomorphism P H,u ( f ) from G into G , where ( H −→× G , f ) is auniversal group for the strong ( H, u )-perfect pseudo MV-algebra N . Proposition 8.3.
The mapping P H,u is a functor from the category
SPP s MV H,u into the category G which is a left-adjoint of the functor M H,u . Proof.
It follows from the properties of the universal group. (cid:3)
Now we present the basic result of this section on a categorical equivalence ofthe category of strong (
H, u )-perfect pseudo MV-algebras and the category of G . Theorem 8.4.
The functor M H,u defines a categorical equivalence of the category G and the category SPP s MV H,u of strong ( H, u ) -perfect pseudo MV-algebras.In addition, suppose that h : M H,u ( G ) → M H,u ( G ′ ) is a homomorphism ofpseudo MV-algebras, then there is a unique homomorphism f : G → G ′ of ℓ -groupssuch that h = M H,u ( f ) , and (i) if h is surjective, so is f ; (ii) if h is injective, so is f .Proof. According to [MaL, Thm IV.4.1], it is necessary to show that, for a strong(
H, u )-perfect pseudo MV-algebra M , there is an object G in G such that M H,u ( G )is isomorphic to M . To show that, we take a universal group ( H −→× G, f ). Then M H,u ( G ) and M are isomorphic. (cid:3) An important kind of ℓ -groups are doubly transitive ℓ -groups; for more details onthem see e.g. [Gla]. Every such an ℓ -group generates the variety of ℓ -groups, [Gla,Lem 10.3.1]. The notion of doubly transitive unital ℓ -group ( G, u ) was introducedand studied in [DvHo], and according to [DvHo, Cor 4.9], the pseudo MV-algebraΓ(
G, u ) generates the variety of pseudo MV-algebras.An example of a doubly transitive permutation ℓ -group is the system of allautomorphisms, Aut( R ), of the real line R , or the next example:Let u ∈ Aut( R ) be the translation tu = t + 1, t ∈ R , andBAut( R ) = { g ∈ Aut( R ) : ∃ n ∈ N , u − n ≤ g ≤ u n } . Then (BAut( R ) , u ) is a doubly transitive unital ℓ -permutation group, and it is agenerator of the variety of pseudo MV-algebras P s MV . In addition, Γ(BAut( R ) , u )is a stateless pseudo MV-algebra.The proof of the following statement is practically the same as that of [Dvu4,Thm 5.6] and therefore, we omit it here. Theorem 8.5.
Let G be a doubly transitive ℓ -group. Then the variety generatedby SPP s MV H,u coincides with the variety generated by M H,u ( G ) . Weak ( H, u ) -perfect Pseudo MV-algebras In this section, we will study another kind of (
H, u )-perfect pseudo MV-algebras,called weak (
H, u )-perfect pseudo MV-algebras. Their prototypical examples arepseudo MV-algebras of the form Γ( H −→× G, (1 , b )), where ( H, u ) is an Abelian unital ℓ -group, G is an ℓ -group and b ∈ G . Such pseudo MV-algebras were studied for thecase ( H, u ) = ( H ,
1) in [Dvu4].Let (
H, u ) be an Abelian unital ℓ -group. We say that a pseudo MV-algebra M ∼ = Γ( K, v ) with an (
H, u )-decomposition ( M t : t ∈ [0 , u ] H ) is weak if there is asystem ( c t : t ∈ [0 , u ] H ) of elements of M such that (i) c = 0 , (ii) c t ∈ C ( K ) ∩ M t , for any t ∈ [0 , u ] H , and (iii) c v + t = c v + c t whenever v + t ≤ u. We notice that in contrast to the strong cyclic property, we do not assume c = 1 . In addition, a weak (
H, u )-perfect pseudo MV-algebra M is strong iff c = 1 . SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 25
Example 9.1.
Let ( H, u ) be an Abelian unital ℓ -group. The pseudo MV-algebra M = Γ( H −→× G, ( u, b )) , where b ∈ G , M t = { ( t, g ) : ( t, g ) ∈ M } , t ∈ [0 , u ] H forman ( H, u ) -decomposition of M , is a weak pseudo MV-algebra setting c t = ( t, , t ∈ [0 , u ] H .Proof. We have to verify that ( M t : t ∈ [0 , u ] H ) is an ( H, u )-decomposition. Toshow that it is enough to verify (b) of Definition 3.1, i.e. M − t = M u − t = M ∼ t foreach t ∈ [0 , u ] H . Let ( t, g ) ∈ M t . Then ( t, g ) − = ( u, b ) − ( t, g ) = ( u − t, b − g ) . If we choose ( t, g ), where g = b + g − b , then ( t, g ) ∼ = − ( t, g ) + ( u, b ) =( − t + u, − g + b ) = ( u − t, b − g ) = ( t, g ) − which yields ( t, g ) − ∈ M ∼ t , that is M − t ⊆ M ∼ t . Dually we show M ∼ t ⊆ M − t . Then M − t = M u − t = M ∼ t . (cid:3) Whereas every strong (
H, u )-perfect pseudo MV-algebra is symmetric, weak onesare not necessarily symmetric.For example, the pseudo MV-algebra Γ( H −→× G, ( u, b )), where b > b / ∈ C ( G )and M t := { ( t, g ) ∈ Γ( H −→× G, ( u, b )) } for each t ∈ [0 , u ] H , is weak ( H, u )-perfectbut neither strong (
H, u )-perfect nor symmetric.We note that M is a unique maximal and normal ideal of M . This ideal isretractive iff M is strongly ( H, u )-perfect. For example, let M = Γ( Z −→× Z , (2 , M is weakly ( Z , Z , M = { (0 , n ) : n ≥ } , M = { (1 , n ) : n ∈ Z } , M = { (2 , n ) : n ≤ } , M/M ∼ = Γ( Z , M . In addition, M is not retractive.We notice that even a pseudo MV-algebra of the form Γ( H −→× G, ( u, b )) with b = 0can be strongly ( H, u )-perfect. Indeed, let M = Γ( Z −→× Z , (2 , M := Γ( Z −→× Z , (2 , θ : M → M defined by θ (0 , n ) = (0 , n ), θ (1 , n ) = (1 , n + 1) and θ (2 , n ) = (2 , n + 2)is an isomorphism in question. In addition, M = { (0 , n ) : n ≥ } is a retractiveideal and a lexicographic ideal of M ; M/M = Γ( Z ,
1) and its isomorphic copy in M is the subalgebra { (0 , , (1 , , (2 , } .The next result is a representation of weak ( H, u )-perfect pseudo MV-algebrasby lexicographic product.
Theorem 9.2.
Let M be a weak H -perfect pseudo MV-algebra which is not strong.Then there is a unique (up to isomorphism) ℓ -group G with an element b ∈ G , b = 0 , such that M ∼ = Γ( H −→× G, (1 , b )) . Proof.
Assume M = Γ( K, v ) for some unital ℓ -group ( H, u ) is a weak pseudo MV-algebra with a (
H, u )-decomposition ( M t : t ∈ [0 , u ] H ). Since by (vi) of Theorem3.2 we have M + M = M , in the same way as in the proof of Theorem 4.2, thereexists an ℓ -group G such that G + = M and G is a subgroup of K .Since M is not strong, then c < u. Set b = 1 − c ∈ M \ { } , and define amapping φ : M → Γ( H −→× G, (1 , b )) as follows φ ( x ) = ( t, x − c t )whenever x ∈ M t ; we note that the subtraction x − c t is defined in the ℓ -group K .Using the same way as that in (4.2), we can show that φ is a well-defined mapping.We have (1) φ (0) = (0 , φ (1) = (1 , − c ) = (1 , b ), (3) φ ( c t ) = ( t, , (4) φ ( x ∼ ) = (1 − t, − x + u − c − t ) = (1 − t, − x + b + c t ) , φ ( x ) ∼ = − φ ( x ) + (1 , b ) = − ( t, x − c t ) + (1 , b ) = (1 − t, − x + b + c t ), and similarly (5) φ ( x − ) = φ ( x ) − . Following ideas of the proof of Theorem 4.2, we can prove that φ is an injectiveand surjective homomorphism of pseudo MV-algebras as was claimed. (cid:3) It is worthy of reminding that Theorem 9.2 is a generalization of Theorem 4.2,because Theorem 4.2 in fact follows from Theorem 9.2 when we have b = 0 . Thishappens if c = 1.Also in an analogous way as in [Dvu4], we establish a categorical equivalenceof the category of weak ( H, u )-perfect pseudo MV-algebras with the category of ℓ -groups G with a fixed element b ∈ G .Let WPP s MV H,u be the category of weak (
H, u )-perfect pseudo MV-algebraswhose objects are weak (
H, u )-perfect pseudo MV-algebras and morphisms are ho-momorphisms of pseudo MV-algebras. Similarly, let L b be the category whoseobjects are couples ( G, b ) , where G is an ℓ -group and b is a fixed element from G ,and morphisms are ℓ -homomorphisms of ℓ -groups preserving fixed elements b .Define a mapping F H,u from the category L b into the category WPP s MV H,u as follows:Given (
G, b ) ∈ L b , we set F H,u ( G, b ) := Γ( H −→× G, ( u, b )) , and if h : ( G, b ) → ( G , b ) , then F H,u ( h )( t, g ) = ( t, h ( g )) , ( t, g ) ∈ Γ( H −→× G, ( u, b )) . It is easy to see that F H,u is a functor.In the same way as the categorical equivalence of strong (
H, u )-perfect pseudoMV-algebras was proved in the previous section, we can prove the following theo-rem.
Theorem 9.3.
The functor F H,u defines a categorical equivalence of the category L b and the category WPP s MV H,u of weak ( H, u ) -perfect pseudo MV-algebras. Finally, we present addition open problems.
Problem 9.4. (1) Find an equational basis for the variety generated by the set
SPP s MV H,u . For example, if (
H, u ) = ( Z ,
1) the basis is 2 .x = (2 .x ) , see [DDT,Rem 5.6], and the case ( H, u ) = ( Z , n ) was described in [Dvu3, Cor 5.8].(2) Find algebraic conditions that entail that a pseudo MV-algebra is of the formΓ( H −→× G, ( u, H, u ) is a unital ℓ -group not necessary Abelian.10. Conclusion
In the paper we have established conditions when a pseudo MV-algebra M is aninterval in some lexicographic product of an Abelian unital ℓ -group ( H, u ) and an ℓ -group G not necessarily Abelian, i.e. M = Γ( H −→× G, ( u, H, u )-perfect pseudo MV-algebras as those pseudo MV-algebrasthat can be split into comparable slices indexed by the elements from the interval[0 , u ] H . For them we have established a representation theorem, Theorem 4.2, andwe have shown that the category of strong ( H, u )-perfect pseudo MV-algebras iscategorically equivalent to the variety of ℓ -groups, Theorem 8.4.We have shown that our aim can be solved also introducing so-called lexico-graphic ideals. We establish their properties and Theorem 7.5 gives also a rep-resentation of a pseudo MV-algebra in the form Γ( H −→× G, ( u, SEUDO MV-ALGEBRAS AND LEXICOGRAPHIC PRODUCT 27
Finally, we have studied and represented weak (
H, u )-perfect pseudo MV-algebrasas those that they have a form Γ( H −→× G, ( u, g )) where g ∈ G is not necessary thezero element, Theorem 9.2.The present study has opened a door into a large class of pseudo MV-algebrasand formulated new open questions, and we hope that it stimulate a new researchon this topic. References [Bir] G. Birkhoff, “Lattice Theory” , Amer. Math. Soc. Coll. Publ., Vol. , Providence, RhodeIsland, 1967.[CDM] R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, “Algebraic Foundations of Many-valuedReasoning” , Kluwer Academic Publ., Dordrecht, 2000.[CiTo] R. Cignoli, A. Torrens, Retractive MV-algebras , Mathware Soft Comput. (1995), 157–165.[Cha] C.C. Chang, Algebraic analysis of many valued logics , Trans. Amer. Math. Soc. (1958), 467–490.[DDJ] A. Di Nola, A. Dvureˇcenskij, J. Jakub´ık, Good and bad infinitesimals, and states onpseudo MV-algebras , Order (2004), 293–314.[DDT] A. Di Nola, A. Dvureˇcenskij, C. Tsinakis, On perfect GMV-algebras , Comm. Algebra (2008), 1221–1249.[DFL] D. Diaconescu, T. Flaminio, I. Leu¸stean, Lexicographic MV-algebras and lexicographicstates , Fuzzy Sets and System uzzy Sets and Systems (2014), 63–85. DOI10.1016/j.fss.2014.02.010[DiLe1] A. Di Nola, A. Lettieri,
Perfect MV-algebras are categorical equivalent to abelian ℓ -groups , Studia Logica (1994), 417–432.[DiLe2] A. Di Nola, A. Lettieri, Coproduct MV-algebras, nonstandard reals and Riesz spaces , J.Algebra (1996), 605–620.[Dvu1] A. Dvureˇcenskij,
States on pseudo MV-algebras , Studia Logica (2001), 301–327.[Dvu2] A. Dvureˇcenskij, Pseudo MV-algebras are intervals in ℓ -groups , J. Austral. Math. Soc. (2002), 427–445.[Dvu3] A. Dvureˇcenskij, On n -perfect GMV-algebras , J. Algebra (2008), 4921–4946.[Dvu4] A. Dvureˇcenskij, H -perfect pseudo MV-algebras and their representations, Math. Slovacahttp://arxiv.org/abs/1304.0743[DvHo] A. Dvureˇcenskij, W.C. Holland,
Top varieties of generalized MV-algebras and unitallattice-ordered groups , Comm. Algebra (2007), 3370–3390.[DvHo1] A. Dvureˇcenskij, W.C. Holland, Free products of unital ℓ -groups and free products ofgeneralized MV-algebras, Algebra Universalis (2009), 19–25.[DvKo] A. Dvureˇcenskij, M. Kolaˇr´ık, Lexicographic product vs Q -perfect and H -perfect pseudoeffect algebras , Soft Computing (2014), 1041–1053. DOI: 10.1007/s00500-014-1228-6[DvVe1] A. Dvureˇcenskij, T. Vetterlein, Pseudoeffect algebras. I. Basic properties , Inter. J. Theor.Phys. (2001), 685–701.[DvVe2] A. Dvureˇcenskij, T. Vetterlein, Pseudoeffect algebras. II. Group representation , Inter. J.Theor. Phys. (2001), 703–726.[FoBe] D.J. Foulis, M.K. Bennett, Effect algebras and unsharp quantum logics , Found. Phys. (1994), 1331–1352.[Fuc] L. Fuchs, “Partially Ordered Algebraic Systems” , Pergamon Press, Oxford-New York,1963.[GeIo] G. Georgescu, A. Iorgulescu, Pseudo-MV algebras , Multiple Val. Logic (2001), 95–135.[Gla] A.M.W. Glass, “Partially Ordered Groups” , World Scientific, Singapore, 1999.[Go] K.R. Goodearl, “Partially Ordered Abelian Groups with Interpolation” , Math. Surveysand Monographs No. 20, Amer. Math. Soc., Providence, Rhode Island, 1986.[HoRa] D. Hort, J. Rach˚unek, Lex ideals of generalized MV-algebras , In: Calude, C. S. (ed.) etal., Combinatorics, computability and logic. Proc. 3rd Inter. Conf., DMTCS ’01. Univ.of Auckland, New Zealand and Univ. of Constanta, Romania, 2001. London: Springer.Discrete Mathematics and Theoretical Computer Science. 2001, pp. 125–136. [Jak] J. Jakub´ık,
On varieties of pseudo MV-algebras , Czechoslovak Math. J. (2003), 1031–1040.[Kom] Y. Komori, Super Lukasiewicz propositional logics , Nagoya Math. J. (1981), 119–133.[MaL] S. Mac Lane, “Categories for the Working Mathematician” , Springer-Verlag, New York,Heidelberg, Berlin, 1971.[Rac] J. Rach˚unek, A non-commutative generalization of MV-algebras , Czechoslovak Math. J.52