ABELIAN l-GROUPS WITH STRONG UNIT AND

ABELIAN `-GROUPS WITH STRONG UNIT AND PERFECT MV-ALGEBRAS L. P. BELLUCE, A. DI NOLA, AND B. GERLA

Abstract. We investigate the class of abelian `-groups with strong unit corresponding to perfect MV-algebras via the Γ functor, showing that this is a universal subclass of the class of all abelian `groups with strong unit and describing the formulas that axiomatize it. We further describe results for classes of abelian `-groups with strong unit corresponding to local MV-algebras with finite rank.

1. Introduction The class of MV-algebras arises as the algebraic counterpart of the infinite-valued L Ã ukasiewicz sentential calculus, just as Boolean algebras arise as the algebraic counterpart of classical propositional logic. Due to the non-idempotency of the MV-algebraic disjunction, unlike Boolean algebras, MV-algebras can contain non-archimedean elements, i.e., elements x such that for any n > 0, nx = x ⊕ . . . ⊕ x (n times) is always strictly greater than (n − 1)x and strictly smaller than 1, where ⊕ is the counterpart of the disjunction in the L Ã ukasiewicz sentential calculus. Each such element is contained in a proper ideal, hence in a maximal ideal. If the intersection of maximal ideals of an MV-algebra A (the radical of A) is equal to {0}, then we say that A is a semisimple MV-algebra. There exist MV-algebras which are not semisimple, the radical of A containing non zero-elements called infinitesimals (that are hence nonarchimedean). Perfect MV-algebras are those MV-algebras generated by their infinitesimal elements or, equivalently, generated by their radical. An important example of a perfect MV-algebra is the subalgebra S of the Lindenbaum algebra L of first order L Ã ukasiewicz logic generated by the classes of formulas which are valid but non-provable. Hence perfect MV-algebras are directly connected with the very important phenomenon of incompleteness in L Ã ukasiewicz first order logic (see [17], [3], [5]). 1

As is well known ([16]), MV-algebras form a category equivalent to the category of abelian lattice ordered groups with strong unit (ù groups, for short). This makes the interest in MV-algebras relevant outside the realm of logic. Let us denote by Γ the functor implementing this equivalence. So, via Γ−1 each perfect MV-algebra is associated with an ù -group. Also it has been proved (see [11], Theorem 3.5) that the category of perfect MV-algebras is equivalent to the category of abelian lattice ordered groups (`-groups, for short). Let us denote by ∆ the functor implementing this equivalence. Hence ∆ maps functorially each `-group into a perfect MV-algebra, without the help of a strong unit. In this paper we define the class of antiarchimedean ù -group as the class obtained by the restriction of Γ−1 to the class of perfect M V algebras (cfr. Theorem 14). This is in line with the work done in [14], where the restriction of Γ−1 to the varieties of MV-algebras generated by simple chains is characterized as the class of uniformly hyperarchimedean lattice-ordered groups. We then generalize results on MV-algebras obtained in [9] and [1] to the class of ù -groups corresponding to perfect MV-algebras. This will display the interplay between the two functors ∆ and Γ. By exploiting the properties of ultrapowers of perfect MV-algebras and antiarchimedean ù -groups, we get a result concerning the completeness of a logic essentially based on infinitesimals, see Section 3. Finally, in Section 4 we extend the relationship between perfect MValgebras and antiarchimedean ù -groups to subclasses of local MValgebras. The main results of the paper can be summarized as follows: • we give a characterization of the positive cone of any antiarchimedean ù -group (see Theorem 14, (iv)), • we describe ultraproducts of antiarchimedean ù -groups, showing that they are antiarchimedean too, • we show that the logic Lp of perfect MV-algebras, as defined in [1], is complete with respect to all perfect MV-chains which are ultrapowers of the perfect MV-chain Γ(Z×lex R, (1, 0)). 2. Some preliminary notions

W V Notation: In what follows we shall use the symbols , , ∼ and ⇒ to denote the connectives of classical first order logic, while symbols ∨, ∧, ¬, → will be used for algebraic operations. 2

Definition 1. A structure A = (A, 0, ¬, ⊕) is an MV-algebra iff A satisfies the following equations, for every x, y ∈ A: 1. (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z); 2. x ⊕ y = y ⊕ x; 3. x ⊕ 0 = x; 4. ¬¬x = x; 5. x ⊕ ¬0 = ¬0; 6. ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x. The class of MV-algebras is hence the class of models of the first order theory with equality over the language LMV = {0, ¬, ⊕} where 0 is a constant symbol, ¬ is a unary function symbol and ⊕ is a binary function symbol, axiomatized by (the universal closure of) equations in Definition 1. It is hence a variety (equational class). For any subclass K of elements from MV, V (K) shall denote the subvariety of MV generated by K. If K has just one element A then we also write V (A) for V (K). On A two new operations ∨ and ∧ are defined as follows: x ∨ y = ¬(¬x ⊕ y) ⊕ y and x ∧ y = ¬(¬x ¯ y) ¯ y. Further we set 1 = ¬0, x ¯ y = ¬(¬x ⊕ ¬y) and x → y = ¬x ⊕ y. The structure (A, ∨, ∧, 0, 1) is a bounded distributive lattice. We shall write x ≤ y iff x ∧ y = x. We say that an MV-algebra A is an MV-chain when, as a lattice, A is linearly ordered. Boolean algebras are just the MV-algebras obeying the additional equation x ¯ x = x. For any MV-algebra A we denote by B(A) = {x ∈ A | x ¯ x = x} the biggest Boolean algebra contained in A. Given a subset S of an MV-algebra A we set ¬S = {¬x | x ∈ S}. We write nx instead of x ⊕ ... ⊕ x (n-times) and xn instead of x ¯ ... ¯ x (n-times). The least integer for which nx = 1 is called the order of x. When such an integer exists, we denote it by ord(x) and say that x has finite order, otherwise we say that x has infinite order and write ord(x) = ∞. An element x ∈ A is idempotent if x ¯ x = x. Example. The unit interval [0, 1] of the abelian group R of real numbers, with operations defined by x ⊕ y = min{1, x + y}, x ¯ y = max{0, x + y − 1}, and ¬x = 1 − x is an MV-chain in which every element has finite order. We shall refer to this MV-algebra as [0, 1]. The subset Sn = {0, 1/n, . . . , (n − 1)/n, 1} of [0, 1] can be equipped with a structure of MV-algebra by the restriction of the operations ⊕, ¯, ¬. An ideal of an MV-algebra A is a non-empty subset I of A which is closed under ⊕ and such that if x ≤ y and y ∈ I then x ∈ I. A prime ideal P of A is an ideal of A such that x ∧ y implies x ∈ P or y ∈ P . 3

An ideal M of A is called maximal if M ⊆ I implies I = A or I = M , where I an ideal of A. The set of all prime ideals of A shall be denoted by Spec(A). For each element x of an MV-algebra A the set id(x) = {y ∈ A | y ≤ nx, for some n > 0} is the ideal of A generated by x. Each proper ideal is contained in a maximal ideal. The notions of MV-isomorphism, quotient, subalgebra, product, etc., are just the particular cases of the corresponding universal algebraic notions. An MV-algebra A is called simple if and only if A is non trivial and {0} is its only proper ideal. Every simple MV-algebra is isomorphic to a subalgebra of [0, 1], (see, e.g., [7], Theorem 3.5.1, p. 70). Every non-zero element of a non trivial MV-algebra A has finite order if and only if A is simple. The intersection of all maximal ideals, the radical of A, will be denoted by Rad(A). An MV-algebra A such that Rad(A) = {0} is called semisimple. Let X be a non-empty set. Then the set B = [0, 1]X of all [0, 1]-valued functions over X, equipped with pointwise operations, is an MV-algebra. Up to isomorphism, subalgebras of B provide the most general possible examples of semisimple MV-algebras, (see, e.g., [1], Theorem 4.9, p. 486). Definition 2. An MV-algebra A is local if A has a unique maximal ideal. The class of all local MV-algebras will be denoted by Local. It is also well known that for each x ∈ A, x is a member of a proper ideal, hence of a maximal ideal, if and only if x has infinite order. It turns out that an MV-algebra A is local if and only if for every x ∈ A, ord(x) < ∞ or ord(¬x) < ∞. Definition 3. Let n be a positive integer. Then we say that a local MV-algebra A has rank n iff A/RadA ∼ = Sn . A local MV-algebra A has finite rank iff A has rank n for some integer n. Definition 4. An MV-algebra A is called perfect if for every nonzero element x ∈ A, ord(x) = ∞ if and only if ord(¬x) < ∞. If P is an ideal of A, then alg(P ) = P ∪ ¬P is a subalgebra of A. The class of perfect MV-algebras will be denoted by Perf. This class turns out to be a universal class. Indeed the following holds. Theorem 5. [1] An MV-algebra A is perfect if and only if it satisfies the first order formula α: ³ ´ ^ _ (1) ∀x (x2 ⊕ x2 = (x ⊕ x)2 ) (x2 = x ⇒ (x = 0 x = 1)) . 4

For all unexplained MV-algebraic notions we refer the reader to [7].

Definition 6. An abelian `-group is a structure G = (G, ∧, ∨, +, −, 0) such that (G, ∧, ∨) is a lattice, (G, +, −, 0) is an abelian group, and if ≤ denotes the partial order on G induced by ∧ and ∨, then, for all a, b, x ∈ G, if a ≤ b, then a + x ≤ b + x. A strong unit u in an abelian `-group is an element u ∈ G such that u ≥ 0 and for every x ∈ G there is a positive integer n such that x ≤ nu. If G is an abelian `-group with strong unit u, we say that G is an abelian ù -group. All the groups considered in this paper are abelian, hence throughout the paper we shall refer to abelian `-groups (or abelian ù -groups) simply as `-groups (or ù -groups). Given x ∈ G we let |x| = x ∨ −x. An ideal of an `-group G is a subgroup J of G such that if x ∈ J and |y| ≤ |x| then y ∈ J. The class of `-groups is hence the class of models of a first order theory with equality over L`g = {0, −, +, ∧, ∨, ≤} where 0 is a constant symbol, − is a unary function symbol and +, ∧, ∨ are binary function symbols and ≤ is a binary relation symbol. Axioms of the theory of `-groups are axioms of abelian groups for −, +, 0 plus axioms of lattices for ∧, ∨, ≤ plus the following: (∀xyz)(x ≤ y ⇒ (x + z) ≤ (y + z)). We remind that the class Ab of (abelian) `-groups is an equational class. On the contrary, the class Abu of ù -groups is not even first order definable. We denote by Ab the category of `-groups. For more details on `-groups we refer the reader to [8]. Example 7. The group Z of integer numbers is an `-group with the usual order relation. For any `-group G, the lexicographic product Z×lex G is an `-group. Further, the element (1, 0) ∈ Z×lex G is a strong unit. If G is an `-group we denote by G+ the set {g ∈ G | g > 0} and by the set G+ ∪ {0}. If G is an `-group and u is an element of G+ , then the set [0, u] = {g ∈ G | 0 ≤ g ≤ u} can be equipped with a structure of MV-algebra by setting ¬x = u − x and x ⊕ y = (x + y) ∧ u. In [16] it is shown that a functor Γ can be defined between the category of ù -groups and the category of MV-algebras and that such categories are indeed equivalent: for any ù -group G with strong unit u, Γ(G, u) is the MV-algebra [0, u]. G+ 0

5

Definition 8. Let Z be the `-group of integer numbers with natural order and denote by Z×lex Z the `-group given by the lexicographic product (the order is hence total). The MV-algebra Γ(Z×lex Z, (n, 0)) is an MV-chain of rank n that will be denoted by Snω ([15], [7]). The MV-chain S1ω is also known as Chang MV-algebra and it is denoted by C ([6]). Let Perf be the full subcategory of the category of MV-algebras whose objects are perfect MV-algebras. Following [11], we can associate each perfect MV-algebra A with an `-group G = ∆(A) by the following construction: Let θ be a relation on Rad(A) × Rad(A) be defined by (x, y)θ(x0 , y 0 ) if and only if x ⊕ y 0 = x0 ⊕ y. The relation θ can be shown to be a congruence and we set [x, y] ⊕ [x0 , y 0 ] = [x ⊕ x0 , y ⊕ y 0 ] and −[x, y] = [y, x], where [x, y] denote the congruence class of (x, y). Then ∆(A) = (Rad(A) × Rad(A)/θ, ≤, +, −, [0, 0]) is an `-group. It can be shown that this construction can be extended to a functor between Perf and Ab, and that the two categories are equivalent. Indeed, a functor G from Ab to Perf can be defined extending the map sending each G ∈ Ab to G(G) = Γ(Z×lex G, (1, 0)) ∈ Perf. Further, ∆ and G form an equivalence among the category of perfect MV-algebras and `-groups. As a consequence of the above mentioned categorical equivalence we have: Theorem 9. An MV-algebra A is perfect if and only if A∼ = Γ(Z× G, (1, 0)) lex

for some `-group G. Definition 10. For any sentence σ in LMV we denote by σ b(v) the formula with only one free variable v over L`g defined as follows: Let t(v1 , ..., vk ) be a term in LMV and v a propositional variable different from v1 , ..., vk . We define b t as follows: - if t = 0 then b 0 is 0, if t = vi then vbi is vi , - if t = ¬t1 then b t is v − b t1 , b - if t = t1 ⊕ t2 then t is (t1 + t2 ) ∧ v. Let ϕ(v1 , ..., vk , v) be a formula in LMV such that all the free and bound variables of ϕ are in {v1 , ..., vk , v} where v is a propositional variable different from v1 , ..., vk . We define ϕ b as follows: b is b t1 = b t2 , - if ϕ is t1 = t2 then ϕ b b is ∼ ψ, - if ϕ is ∼ ψ then ϕ W W V - if ϕ is ψ χ then ϕ b is ψb χ b, and similarly for , ⇒, V b - if ϕ is (∀vi )ψ then ϕ b is (∀vi )((0 ≤ vi vi ≤ v) ⇒ ψ), 6

- if ϕ is (∃vi )ψ then ϕ b is (∃vi )((0 ≤ vi

V

b vi ≤ v) ⇒ ψ).

Thus to any formula ϕ(v1 , ..., vk ) in LMV corresponds a formula ϕ(v b 1 , ..., vk , v) in L`g . As a consequence to any sentence σ in LMV corresponds a formula with only one free variable σ b(v) in L`g (see also the proof of Theorem 1 in [10]). Theorem 11. For any sentence σ over LMV and for any MV-algebra A, A²σ iff (G, u) ² σ ˆ (u) −1 where (G, u) = Γ (A). Proof. The proof easily proceeds by structural induction.

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3. Antiarchimedean ù -groups Definition 12. The class AntArcu of antiarchimedean ù -groups is the subclass of Abu defined by the universal formula α ˆ (u), where α is the formula in Theorem 5, i.e., V α ˆ= (∀x)((x ≥ 0 x ≤ u) ⇒ V ((2(2x ∧ u) − u) ∨ 0) = (2((2x − u) ∨ 0) ∧ u)) W (((2x − u) ∨ 0 = x) ⇒ (x = 0 x = u)))) . It is usual to find examples of groups from the above class in the context of M V -algebras, as the group Z ×lex Z with strong unit (1, 0) which is the K0 of the AF C ∗ -algebra considered in [16], and the group (Z ×lex H2 , (1, 0)), where H2 is the `-group of germs vanishing at (0, 0) of two-variable McNaughton functions. Let Abu denote the category of ù -groups, where maps are `-group homomorphisms preserving strong unit. Definition 13. For (G, u) ∈ Abu let R(G) = {g ∈ G+ : ∀n ∈ N ng < u} and let HG = hR(G)i denote the `-subgroup of G generated by R(G). Theorem 14. Let (G, u) ∈ Abu . Then the following are equivalent: (i) (G, u) ∈ AntArcu (ii) Γ(G, u) ∈ Perf, (iii) (G, u) ∼ = (Z×Slex H, (1, 0)), with H an `-group (iv) G+ = HG+ ∪ m>0 (HG + mu). Proof. (i) ⇔ (ii). By Theorem 11, (G, u) satisfies α ˆ if and only if Γ(G, u) satisfies α. (ii) ⇔ (iii). Let Γ(G, u) ∈ Perf. Then by Theorem 9 there exists an `-group H with Γ(G, u) ∼ = Γ(Z×lex H, (1, 0)) and then (G, u) ∼ = 7

(Z×lex H, (1, 0)). Vice versa, if (G, u) ∼ = (Z×lex H, (1, 0)), then Γ(G, u) ∼ = Γ(Z×lex H, (1, 0)). By Theorem 9, Γ(G, u) is a perfect MV-algebra. (iii) ⇒ (iv). Let (G, u) ∼ = (Z×lex H, (1, 0)). Then all the elements of (G, u) of type (0, g), with g ∈ H + are such that n(0, g) = (0, ng) < (1, 0), for any n ∈ N, hence (0, g) ∈ R(G). Any other element z ∈ G+ \ {(0, g) | g ∈ H + } is of type z = (m, h) = (0, h) + m(1, 0) for every m > 0 and h ∈ H. Hence G+ satisfies (iv). (iv) ⇒ (ii). We denote by (−u, u) the subset {g ∈ G | −u < g < u} = [−u, u] \ {−u, u}. Claim 1: HG ⊆ (−u, u). Indeed, let x ∈ HG . Then −x + u ∈ HG + u, hence −x + u ∈ G+ , and so x ≤ u. Similarly, x + u ∈ HG + u and so x + u ∈ G+ , therefore −u ≤ x hence HG ⊆ [−u, u]. Now if u ∈ HG then also mu ∈ HG for any m > 0, but mu > u hence u ∈ / HG and analogously one can prove that −u ∈ / HG . Claim 2: For any x ∈ HG+ , x + x ∧ u = x + x and x + x − u ∨ 0 = 0. Indeed, since HG is an `-group contained in (−u, u), then x + x ∈ HG+ hence x + x < u and x + x − u < 0. In order to prove (ii) we check condition (1) of Theorem 5 on elements of [0, u]. • If x ∈ [0, u]∩HG , then by Claim 2, x2 ⊕x2 = 0 and (x⊕x)2 = 0. Further trivially x2 = x implies x = 0. • If x ∈ [0, u] \ HG then by hypothesis x = z + mu with z ∈ HG and m > 0. Hence 0 ≤ z +mu ≤ u and so −mu ≤ z ≤ (1−m)u that, together with Claim 1, implies m = 1. So x = z +u. Now, x2 = x + x − u ∨ 0 = 2z + u ∨ 0 = 2z + u since 2z > −u. Then (x2 ⊕ x2 ) = 4z + 2u ∧ u = u since 4z > −u. On the other hand, (x⊕x)2 = ((2z +2u∧u)+(2z +2u∧u)−u)∨0 = u+u−u∨0 = u since 2z + 2u > u. If x ¯ x = x then z + u = 2z + u hence z = 0 and x = u. ¤ From this last proof we can conclude the following. Corollary 15. If G is an antiarchimedean ù -group, then HG+ = R(G) is the radical of the perfect MV-algebra [0, u]. Proposition 16. Any antiarchimedean ù -group is local, its only maximal ideal being HG . Proof. If (G, u) is an antiarchimedean ù -group, then Γ(G, u) is a perfect MV-algebra. Let M be the maximal ideal of Γ(G, u), then H = Γ−1 (M ) is the unique maximal ideal of (G, u) since both Γ, Γ−1 preserve inclusion. By Corollary 15, M = HG+ and since, by [7], 8

H = {x ∈ G | |x| ∧ u ∈ HG+ } then HG+ ⊆ H + . In order to prove that H = HG , let us suppose by contradiction that there is x ∈ H + \ HG+ . Then there exists nx such that nx x > u. For every g ∈ G+ \ HG+ , by Theorem 14, g = h + mu with h ∈ HG and m > 0. Hence g ≤ (m + 1)u ≤ nx (m + 1)x ∈ H which implies g ∈ H since H is an `-ideal of G. Hence G+ ⊆ H + but this is contradiction with the fact that H is a maximal ideal and so H = HG . ¤ Proposition 17. In every ù -group (G, u), the subgroup AG ≤ G such that [ + A+ = H ∪ (HG + mu) G G m>0

is the greatest antiarchimedean ù -subgroup of G. Proof. The claim follows by Theorem 14 and noticing that if B is a antiarchimedean ù -subgroup of G then HB ⊆ HG . ¤ We say that a perfect MV-algebra A is radical divisible if and only if G = ∆(A) is a divisible `-group. Hence A is radical divisible iff A∼ = Γ(Z×lex D, (1, 0)) with D a divisible `-group. Proposition 18. Every non trivial perfect MV-chain embeds into a radical divisible perfect MV-chain. Proof. Let A be a perfect MV-chain such that A = Γ(G, u) with G = (Z×lex H, (1, 0)), u = (1, 0) and H a totally ordered `-group. Hence H embeds into a divisible totally ordered group D, say by the embedding α : H ,→ D. Then let: β : (z, h) ∈ Z×lex H ,→ (z, α(h)) ∈ Z×lex D . It is easy to check that β is an embedding of the unital `-group (Z×lex H, (1, 0)) into (Z×lex D, (1, 0)). Hence Γ(β) is an MV-embedding of the perfect MV-chain A into the perfect MV-chain Γ(Z×lex D, (1, 0)), the latter being radical divisible. ¤ Let AntArcu be the full subcategory of Abu having as objects the antiarchimedean ù -groups. Theorem 19. The categories Perf of perfect MV-algebras, AntArcu and the category Ab are mutually equivalent. Proof. Let ∆Z be the functor from Ab to AntArcu defined as follows: - each object H of Ab is mapped to the `-group ∆Z (H) = (Z×lex H, (1, 0)); by Theorem 14 and Definition 12 we get that ∆Z (H) is an antiarchimedean ù -group. 9

- each map f : H → H 0 of `-groups is mapped to ∆Z (f ) : (n, h) ∈ Z×lex H 7→ (n, f (h)) ∈ Z×lex H 0 . We are going to show that ∆Z is faithful and full. Indeed, let β be any homomorphism in homAntArcu (∆Z (G), ∆Z (G0 )), then β(n, x) can be written as (β1 (n), β2 (x)), where β1 and β2 are `-group homomorphisms, since operations are defined componentwise. But β(1, 0G ) = (1, 0G0 ), so β(n, x) = (n, β2 (x)). Hence for every such β there exists a homomorphism β2 : G → G0 such that β = ∆Z (β2 ) hence the map α ∈ homAb (G, G0 ) 7→ ∆Z (α) ∈ homAntArcu (∆Z (G), ∆Z (G0 )) is surjective. On the other hand, if f, g ∈ homAb (G, G0 ) and ∆Z (f ) = ∆Z (g) then for every x ∈ G, (n, f (x)) = (n, g(x)), hence for every x ∈ G, f (x) = g(x), hence f = g. So ∆Z is faithful and full. By Theorem 14, ∆Z is also dense, since every antiarchimedean ù -group G is isomorphic to some Z×lex H, so the categories Ab and AntArcu are equivalent. The theorem is completely proved observing that the categories Perf and Ab are equivalent (see [11]). ¤ 3.1. Ultraproducts of antiarchimedean ù -groups. Let I be and index set and (Ai )i∈I be a family of perfect MV-algebras. Let A = Q i∈I Ai be the usual product in the category MV of MV-algebras. The category Perf admits products too: the product of (Ai )i∈I in Perf is the algebra A0 whose elements are sequences hai i0i∈I such that ord(ai ) = ord(aj ) for all i, j ∈ I. Let F be a non-principal ultrafilter in 2I . In the category MV we have the usual ultraproduct A/F which consists of equivalence classes of sequences: [hai ii∈I ] = [hbi ii∈I ]

iff

{i | ai = bi } ∈ F.

Since perfect MV-algebras are first order definable (Theorem 5), we get that A/F is a perfect MV-algebra. On the other hand, we can consider the ultraproduct in the category Perf by taking A0 /F as the set of equivalence classes [hai i0i∈I ] of elements hai i0i∈I ∈ A0 . Proposition 20. The MV-algebras A/F and A0 /F are isomorphic perfect MV-algebras. Proof. Since A0 is a subalgebra of A then A0 /F is a subalgebra of A/F . We can consider the inclusion map: σ : [hai i0i∈I ] ∈ A0 /F ,→ [hai i0i∈I ] ∈ A/F that is a monomorphism. In order to prove that σ is surjective let [hai ii∈I ] ∈ A/F . 10

If ord([hai ii∈I ]) = ∞, then it is straightforward to prove that {i | ord(ai ) = ∞} ∈ F . Let ui = ai if ord(ai ) = ∞ and let ui = 0 if ord(ai ) < ∞. Then hui ii∈I ∈ A0 since ord(ui ) = ∞ for every i ∈ I, so [hui ii∈I ] ∈ A0 /F . Since {i | ui = ai } = {i | ord(ai ) = ∞} ∈ F then σ([hui ii∈I ]) = [hui ii∈I ] = [hai ii∈I ]. Similarly, if ord([hai ii∈I ]) < ∞ we let ui = ai if ord(ai ) < ∞ and ui = 1 otherwise. Again hui ii∈I ∈ A0 and σ([hui ii∈I ]) = [hai ii∈I ]. Therefore σ is an isomorphism. ¤ Theorem 21. Let ∗ (Gi ) denote the ultraproduct of the `-groups (Gi )i∈I with respect to a non principal ultrafilter F of 2I and ∗ (Γ(Z×lex G, (1, 0i )) denote the ultraproduct of the perfect MV-algebras Γ(Z×lex Gi , (1, 0i )) with respect to F . Then Γ(Z× ∗ (Gi ), (1, 0)) ∼ = ∗ (Γ(Z× Gi , (1, 0)). lex

lex

Proof. Notice that, as already remarked, the ultraproduct ∗

(Γ(Z×lex Gi , (1, 0))

is a perfect MV-algebra and the MV-algebra Γ(Z×lex ∗ (Gi ), (1, 0)) is perfect too. Let Ψ be a mapping: Ψ : Γ(Z×lex ∗ (Gi ), (1, 0)) → ∗ (Γ(Z×lex Gi , (1, 0)) defined as follows: Ψ((0, [hgi i]i∈I )) = [h(0, gi )i]i∈I with gi ∈ G+ i for every i ∈ I, and Ψ((1, [hgi i]i∈I )) = [h(1, gi )i]i∈I with gi ∈ G− i for every i ∈ I. It is a matter of direct verification to see that Ψ is an MV-isomorphism. ¤ Following the above argument, we can describe an ultraproduct of antiarchimedean ù -groups. Since the category AntArcu of antiarchimedean ù -groups is equivalent with the category Ab of `-groups and with the category Perf of Q perfect MV-algebras, then AntArcu has arbitrary products. Let Ab Gi be the direct product of `-groups Gi in the category Ab. Here we give a direct description of the product, in the category AntArcu , of a family {(Z×lex Gi , (1, 0i ))}i∈I , 11

where each Gi is an `-group with zero element 0i . Let G=

Ab Y

(Z×lex Gi , (1, 0i ))

i∈I

where the unit is w = h(1, 0i )ii∈I . Let B be the subset of G defined as follows: g ∈ B if and only if g = h(z, gi )ii∈I with z ∈ Z and gi ∈ Gi for every i ∈ I. Then we have: Proposition 22. Using the above notation we have: • (B, h(1, 0i )ii∈I ) is an ù -subgroup of G. QAb • (B, h(1, 0i )ii∈I ) ∼ = (Z×lex i∈I Gi , (1, h0i ii∈I )) hence (B, h(1, 0i )ii∈I ) is antiarchimedean. • (B, h(1, 0i )ii∈I ) is the product of (Z×lex Gi , (1, 0i )) with i ∈ I in the category AntArcu . Proof. By direct verification and by Theorem 14.

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Now let F be a non principal ultrafilter of I. We can hence write B=

AA Y

(Z×lex Gi , (1, 0i ))

i∈I

as the product in the category AntArcu of the family (Z×lex Gi , (1, 0i ))i∈I of antiarchimedean ù -groups. Then with the above notation we have Proposition 23. The ultraproduct B/F is an antiarchimedean ù group. Proof. Indeed, by Lemma 19 AA Y

(Z×lex Gi ) ∼ = Z×lex

i∈I

and then

ÃAA Y

Ab Y

Gi

i∈I

!

Ã

(Z×lex Gi ) /F ∼ =

i∈I

Z×lex

Ab Y

! Gi /F.

i∈I

Q G )/F is a strong Notice that the element [(1, h0i ii∈I )] ∈ (Z×lex Ab QAA i∈I i unit and [h(1, 0i )ii∈I ] is a strong unit of ( i∈I (Z×lex Gi ))/F . Hence we get Ab Y B/F ∼ Gi , (1, h0i ii∈I ))/F, = (Z×lex 12

and, by Theorem 21, Γ(B/F ) ∼ = Γ((Z×lex

Ab Y

Gi , (1, h0i ii∈I ))/F ) ∼ = ∗ (Γ(Z×lex Gi , (1, 0i ))

the latter being a perfect MV-algebra. Then Γ(B/F, [h(1, 0i )ii∈I ]) is a perfect MV-algebra and (B/F, [h(1, 0i )ii∈I ]) an antiarchimedean ù group. ¤ Theorem 24. Every perfect MV-chain can be embedded into an ultrapower of the perfect MV-chain Γ(Z×lex R, (1, 0)), which is divisibleradical. Proof. If A is a perfect MV-chain, then ∆(A) is a totally ordered group, hence ∆(A) can be embedded in some ultrapower ∗ R of R. Since G preserves embeddings, then G(∆(A)) embeds in G(∗ R) and since G and ∆ form an equivalence among the category of perfect MV-algebras and `-groups, we have an embedding of A into G(∗ R) = Γ(Z×lex ∗ R, (1, 0)). The claim follows by Theorem 21. ¤ 3.2. The logic of perfect MV-algebras. In [1], perfect MV-algebras have been considered as the algebraic counterpart of a many-valued logic dealing with infinitesimals. We will show, see Proposition 23, such a logic is complete with respect to ultrapowers of a kind of standard perfect MV-chain, actually the MV-chain Γ(Z×lex R, (1, 0)). In a more general setting in [13] the notion of strong non-standard completeness with respect to non-standard models of the unit real interval is introduced. Here we would like to show how the role of perfect MV-algebras dwells, beyond the `-group theoretical side, in the logical side. Following the work of [4], we introduced the logic Lp of perfect MValgebras in the following way: the language of Lp contains symbols of variables V ar = {v0 , v1 , . . .}, logical symbols →, ¬; predicate symbols R0 , R1 , . . .; a quantifier symbol ∃; improper symbols (, ); and a function d : N → N, N = {0, 1, 2, ...}. The set W F F of well-formed formulas of Lp is defined as usual. Let A be an MV-algebra and X a nonempty set. An {A, X} − model is a system hA, X, (Fn )n∈N i such that for each n ∈ N there is a function Fn : X d(n) → A. Given an {A, X}−model hA, X, (Fn )n∈N i, an assignment is a function f : V ar → X. If f is an assignment, v ∈ V ar, x ∈ X, then fvx is the assignment defined by fvx (vi ) = f (vi ) if vi 6= v and fvx (vi ) = x, vi = v.P If S ⊆ APwe define S as the least upper bound of S in A if it exists, and S = A, otherwise. Given M = hA, X, (Fn )n∈N i we assign 13

values to each α ∈ W F F . We define, therefore a function V al(α, M, f ) : W F F → A ∪ {A}, where f is an M-assignment inductively defined by the following conditions: (1) V al(Rn (vi1 , vi2 , ..., vid(n) ), M, f ) = Fn (f (vi1 ), f (vi2 ), ..., f (vid(n) )); (2) For α ∈ W F F , ( ¬V al(α, M, f ) if V al(α, M, f ) ∈ A V al(¬α, M, f ) = A, otherwise; (3) For α, β ∈ W F F , then V al(α → β, M, f ) = ¬V al(α, M, f ) ⊕ V al(β, M, f ), provided both V al(α, M, f ), V al(β, M, f ) ∈ A and V al(α → β, M, fP ) = A, otherwise; (4) V al((∃v)α, M, f ) = x∈X V al(α, M, fvx ). Call an M-assignment an interpretation if V al(α, M, f ) ∈ A for all α ∈ WFF. The axioms of Lp are the axioms of first order L Ã ukasiewicz logic plus 2 2 the axiom schema, 2α ↔ (2α) . Such logic has been shown in [1] to be complete with respect to linearly ordered perfect MV-algebras. Then, by Theorem 24, we can refine that result as follows: Proposition 25. The logic Lp defined in [1], is complete with respect to all ultrapowers of Γ(Z×lex R, (1, 0)), i.e., to all perfect MV-chains of type ∗ Γ(Z×lex R, (1, 0)). 4. Beyond perfect MV-algebras and Antiarchimedean lu -groups The class of perfect MV-algebras is the intersection of the class of local MV-algebras with the variety generated by Chang MV-algebra. In [9] a classification has been given of the intersection of the universal class of local MV-algebras with any subvariety of MV-algebras, obtaining universal classes of MV-algebras that can be seen as generalizations of perfect MV-algebras. It is worthwile to observe that applying again the functor Γ−1 to such classes we get classes of ù -groups that are ready to be seen as generalization of antiarchimedean ù -groups. Indeed, the class of antiarchimedean ù -groups can be described as −1 Γ (Loc ∩ V (C)), where Loc is the class of local MV-algebras. We recall the following results from [9]: 14

Theorem 26. The class of all local MV-algebras in V (Snω ) coincides with the universal class generated by local MV-algebras of the form Γ(Z×lex G, (n, 0)) and it is characterized by equations of V (Snω ) plus the formula τn : ³ _¡ ^ ¡ ¢_³ ¢´´ (∀x) (2x = 1) x2 = 0 (n + 1) x = 1 xn+1 = 0 . For example, the class of local MV-algebras in the variety V (S2ω ) are characterized by a finite set Eq(V (S2ω )) of MV-equations of one variable (see [12]) plus ³ ´´ _¡ ^ ¢_³ (∀x) (2x = 1) x2 = 0 3x = 1 x3 = 0 . Let A be a local MV-algebra with maximal ideal M . By Proposition 16, we have that (G, u) = Γ−1 (A) is a local `-group, i.e. an `-group having only one maximal ideal. Proposition 27. An `-group (G, u) is in Γ−1 (Loc ∩ V (Snω )) if and only if G is local and it satisfies the formula (2) ³ ´´ ^ _ _³ (n + 1)x ≥ u (n + 1)x ≤ nu . (∀x) (2x ≥ u) (2x ≤ u) Proof. Assume (G, u) ∈ Γ−1 (Loc ∩ V (Snω )) and denote by A the local MV-algebra Γ(G, u) ∈ Loc ∩ V (Snω ). Hence A is local of rank d, with d divisor of n, and G is local. By Theorem 8.2 in [9] and Theorem 10, we get that (G, u) satisfies the formula (2). Vice versa, if (G, u) is local and satisfies (2), then Γ(G, u) is local and by Theorem 10 and Theorem 18 it is a member of V (Snω ). Hence Γ(G, u) ∈ Loc ∩ V (Snω ), that is (G, u) ∈ Γ−1 (Loc ∩ V (Snω )). ¤ Among all local MV-algebras of rank n, there are the radical retractive MV-algebras, i.e., those MV-algebras A having A/Rad(A) as subalgebra (up to isomorphisms). In the following proposition we recall some results of [9]. Proposition 28. A local MV-algebra A of rank n is radical retractive if and only if it is isomorphic to Γ(Z×lex G, (n, 0)), where G is an `group. Further, every local M V -algebra of rank n can be embedded into a local MV-algebra of rank n which is radical retractive. This result motivates a focus on the class of all local M V -algebras of rank n that are radical retractive. By applying the functor Γ we get Proposition 29. Every ù -group of type (Z×lex G, (n, g)) can be embedded into (Z×lex G, (n, 0)). 15

As a result we get an axiomatization of the class of all ù -group of type (Z ×lex G, (n, 0)). Let ϕ, βn and γn be the following sentences: _ _ ϕ : (∀x)((x ≤ ¬x) (¬x ≤ x) (d(x, ¬x))2 = 0)) βn : (∀x)((2x = 1)

_

(x2 = 0)

_

((n + 1)x = 1)

^

(xn+1 = 0)))

γn : (∃z)((n − 1)z = ¬z) Theorem 30. Let A be an M V -algebra. The the following are equivalent: (i) A is local of rank n and radical retractive; (ii) ϕ, βn and γn are valid in A. Proof. Assume (i). Then by Proposition 3.7 in [9], ϕ is true in A. Since A has rank n, it can be shown that A satisfies βn as it was proved in the proof of Theorem 8.2 in [9]. Also A is radical retractive, hence, up to isomorphism, Sn is a subalgebra of A, and then for z = n1 we get that γn is true for A. Vice versa, assume (ii). Hence ϕ holds in A, and then, again by Proposition 3.7 in [9], A is local. Since A satisfies βn , by Theorem 8.2 of [9], A has rank n hence A/RadA ∼ = Sn . Now, from γn we know that there is an element z ∈ A such that (n − 1)z = ¬z, then A contains a copy of Sn as a subalgebra, (see Lemma 2.2 in [18]). Hence A/RadA is a subalgebra of A and A is radical retractive. ¤ Theorem 31. An ù -group (G, u) is isomorphic to (Z ×lex H, (n, 0)) cn and γbn , where b for some `-group H if and only if (G, u) satisfies ϕ, b β is the translation in Definition 10. Proof. Indeed by Proposition 28 and Theorem 30 an MV-algebra A is local of rank n and radical retractive if and only if A∼ = Γ(Z ×lex G, (n, 0)), if and only if A satisfies ϕ, βn and γn . Hence the claim follows by Theorem 11 by applying Γ functor. ¤ 5. Conclusions Using Mundici’s functor Γ and the characterization of perfect MValgebras as a universal class of MV-algebras, we defined a universal class of ù -groups, the antiarchimedean ù -groups. It turns out that 16

the antiarchimedean ù -groups form a category equivalent to the category of perfect MV-algebras and equivalent to the category of `-groups, too. Thus we got a way to have a faithful translation of all nice properties of perfect MV-algebras, (see [7]), to antiarchimedean ù -groups. Analogous results are obtained moving from perfect MV-algebras to local MV-algebras of a given finite rank, see Section 4. In Section 3, using the ultraproduct of perfect MV-algebras we showed the completeness of the logic Lp with respect to ultrapowers of a standard perfect MV-chain, actually the perfect MV-chain Γ(Z×lex R, (1, 0)). References [1] L.P. Belluce, A. Di Nola, B. Gerla, Perfect MV-algebras and their Logic, Applied categorical structures, 15 (2007), 135–151. [2] L. P. Belluce, Semisimple Algebras of Infinite-valued Logic , Canadian J. Math., 38(1986), 1356-1379. [3] L. P. Belluce, The Going Up and Going Down Theorems in MV-algebras and Abelian `-groups, J. of Math. Anal. and Appl., 241(2000), 92-106. [4] L. P. Belluce, C. C. Chang, A weak completeness theorem for infinite valued predicate logic, J. Symbolic Logic, 28(1963), 43-50. [5] L.P. Belluce, A. Di Nola, The MV-algebra of first order L Ã ukasiewicz logic. General algebra and ordered sets. Tatra Mt. Math. Publ. 27 (2003), 7–22. [6] C.C. Chang, Algebraic analysis of many valued logics. Trans. Amer. Math. Soc., 88: 467-490, 1958. [7] R. Cignoli, I. D’Ottaviano, D. Mundici Algebraic Foundations of Many-valued Reasoning , Trends in Logic, Vol. 7, Kluwer, Dordrecht, 2000. [8] M.L. Darnel, Theory of Lattice-ordered Groups Dekker, New York, 1994. [9] A. Di Nola, I. Esposito, B. Gerla, Local MV-algebras in the representation of M V -algebras, Algebra Universalis, 56(2007), 133-164. [10] A. Di Nola. Representation and reticulation by quotients of MV-algebras, Ricerche di Matematica, Vol. XL (1991), 291-297. [11] A. Di Nola, A. Lettieri Perfect MV-algebras are Categorically Equivalent to Abelian `-Groups, Studia Logica, 53(1994), 417-432. [12] A. Di Nola, A. Lettieri Equational Characterization of all Varieties of MValgebras, J. of Algebra, 221(1999), 463-474. [13] T. Flaminio, Strong non-standard completeness for fuzzy logics. Soft Computing, 12:321-333, (2008). [14] A.W. Hager, C.M. Kimber, Uniformly Hyperarchimedean Lattice-Ordered Groups. Order, 24:121-131, (2007). Ã ukasiewicz propositional logics. Nagoya Mathematical [15] Y. Komori, Super L Journal, 84: 119-133, (1981). [16] D. Mundici, Interpretation of AF C ∗ -Algebras in Lukasiewicz Sentential Calculus, J. Funct. Analysis 65, (1986), 15–63. Die Nichaxiomatisierbarkeit des unendlichwertigen [17] B. Scarpellini, Pradikatenkalkulus von Lukasiewicz, Journ. Symb. Log.,27(1962),159-170. [18] A. Torrens, Cyclic elements in M V -algebras and Post Algebras, Math. Logic Quart. 40, (1994), 431-444. 17

Department of Mathematics, British Columbia University, Vancouver, B.C. Canada. E-mail address: [email protected] Department of Mathematics and Informatics, University of Salerno, Via S. Allende, 84081 Baronissi, Italy. E-mail address: [email protected] Department of Informatics and Communications, University of Insubria, Varese, Italy. E-mail address: [email protected]

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