von Neumann regular Hyperrings and applications to Real Reduced Multirings
aa r X i v : . [ m a t h . A C ] J a n von Neumann regular Hyperrings and applications to Real ReducedMultirings ∗ Hugo Rafael de Oliveira Ribeiro † University of S˜ao Paulo, Institute of Mathematics and Statistic (IME), S˜ao Paulo, Brasil
Hugo Luiz Mariano ‡ University of S˜ao Paulo, Institute of Mathematics and Statistic (IME), S˜ao Paulo, Brasil
January 19, 2021
Abstract
A multiring ([
Mar3 ]) is a kind of ring where is allowed the sum of two elements to be anon-empty subset of the structure instead of just one element -and an hyperring is a multiringwith a strong distributive property. Thus a reduced hyperring where the prime spec is a Booleantopological space is called von Neumann regular hyperring (vNH). It is possible to associate toevery such object a structural presheaf in the same way it is made with rings but there are somevNH such that this presheaf is not a sheaf. In this sense, we give a first-order characterizationof vNH with a structural sheaf (geometric vNH or just GvNH) and how to transform a vNH in aGvNH -in fact, this transformation shows that the category
GvNH is a reflexive subcategory of vNH . We also build a von Neumann regular hull for multirings and use this to give applicationsfor algebraic theory of quadratic forms. More precisely, we work with Real Reduced Multiring(RRM, [
Mar3 ]) -also known as Real Semigroup (RS, [
DP1 ])-, a special kind of multirings thatis useful to explore the real structure of rings, and show that a von Neumann hull of a RRM isagain a RRM. This gives a generalization of sheafs arguments present in [
DM4 ]. Keywords— multirings, von Neumann hyperrings, hull, quadratic forms, real semigroupMSC: 1301, 14P10Data sharing not applicable to this article as no datasets were generated or analysed during the currentstudy.
Introduction
In 2006, Marshall presented in [
Mar3 ] the notion of multiring in the setting of quadratic form theory, astructure like a ring with multivaluated sum. This concept enables a unified way to deal with rings, SpecialGroup, Real Semigroup and several other structures from Algebraic Theory of Quadratic Forms ([
RRM ]).Besides that, in [
Mar3 ] is given a simple characterization of RS named Real Reduced Multiring (RRM)-more precisely, in [
RRM ] is described an equivalence between the categories RS , RRM and them are dualto
ARS (Abstract Real Spectra).Also in [
Mar3 ] is proved that for each multiring A can be associated a ”canonical” RRM Q ( A ) -infact, we proved that this RRM is the best in the sense that the natural projection π A : A → Q ( A ) is initial ∗ The first author was supported by Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES) andConselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPQ). † [email protected] (Corresponding Author) ‡ [email protected] Theorem 1 . Q .In [ Mar2 ] (section 8.8), Marshall shows, in the dual language of
ARS , an example of RRM that is notof the form Q ( A ) for any ring A . An open problem in this sense is given a Real Reduced Hyperfield M (equivalently Reduced Special Group or Abstract Order Space) if exist a ring A such that M ∼ = Q ( A ) (or ifexist a field K with M ∼ = Q ( K )). In [ DM3 ] exist a partial result: every RSG is realized as quotient of amultiplicative subset in the ring of continuous real-valued functions over some Boolean space. The questionremains open also for a Real Reduced Hyperring M but as corollary of Theorem 4 .
12 we show that if exist asemi-real ring A with Q ( A ) ∼ = M , then exists a semi-real von Neumann regular ring A ′ such that M ∼ = Q ( A ′ ). Overview of the paper:
We will start section 1 with some definitions and constructions with concerning multirings, emphasizingthe universal property and adaptations concerning the prime spectrum. In section 2 we describe the structuralpresheaf associated to every multiring. In section 3 we present the notion of von Neumann regular hyperring(vNH) and characterize when it is geometric (that is, the structural pre-sheaf is sheaf) (Theorem 3 . Jun ] where is proved that hyperdomains are geometric and in [
DP2 ] is provedthat Real Reduced Multiring (in Real Semigroup language) are also geometric. Furtheremore, we describe ageometric hull for vNH (Corollary 3 .
9) and give a new representation for the Real Reduced Hull of a GvNH(Theorem 3 .
13) using Marshall quotient. We finalize the results with section 4 introducing the von Neumannregular hull of a multiring (Theorem 4 . In this section we give a complete list of construction and related results probably known and very importantto what follows. The proofs are given for the reader convenience. { multiDef } Definition 1.1.
A multiring is a tuple ( A, + , − , · , ,
1) where ( A, · ,
1) is a commutative semigroup, + : A → ℘ ( A ) ∗ := ℘ ( A ) \ {∅} and − : A → A are functions and 0 , ∈ A are constants such that for all a, b, c ∈ R :i) a ∈ b + c ⇒ c ∈ − b + a and b ∈ a + ( − c ).ii) a ∈ b + 0 ⇔ a = b .iii) (Associativity) If x ∈ g + c with g ∈ a + b , then exist h ∈ b + c such that x ∈ a + h .iv) a + b = b + a .v) a · a ∈ b + c , then for all d ∈ A , ad ∈ bd + cd .In some multirings, as the usual rings, the property vi) admits a reciprocal x ∈ bd + cd ⇒ exist a ∈ b + c such that x = ad. The multirings that satisfies this are called hyperrings.Some notions for rings have useful generalization to our context. A multiring A is called multidomain iffor all a, b ∈ A with ab = 0 we have a = 0 or b = 0.We define the set of invertible elements A × = { a ∈ A : exists b ∈ A such that ab = 1 } and the set ofweak-invertible elements A × w = { a ∈ A : exists b , . . . , b n ∈ A such that 1 ∈ ab + · · · + ab n } . Note that A × ⊆ A × w and if A is an hyperring, then A × = A × w . A multiring A is called multifield if 1 = 0 and allnon-zero element is weak-invertible ( A × w = A \ { } ) and A is called hyperfield if 1 = 0 and all non-zeroelement is invertible ( A × = A \ { } ). Remark 1. • Note that if A is a hyperfield, then A is an hyperring because given x ∈ bd + cd , if d = 0,take a = 0; if d = 0, take a = xd − ; in both cases, x = ad and a ∈ b + c . Because of this, we will usethe term hyperfield instead of the term multifield used in [ Mar3 ]. It is possible to define multiring as first-order structure by interpretate the multivalued sum as ternaryrelation: given a multiring ( A, + , − , · , , π ⊆ A such that c ∈ a + b iff( a, b, c ) ∈ π . Thus every axiom is a Horn-geometric sentence. • In the definition of multiring, the multivaluated sum is a priori non-empty but this follows by theaxioms. In fact, given a, b ∈ A , since a ∈ a and 0 ∈ b + ( − b ), associativity implies the existence of h ∈ a + b with a ∈ − b + h . Lemma 1.2.
Let A be a multiring. Given a, b, c ∈ A i) − − ( − a ) = a .ii) a ∈ b + c if, and only if, − a ∈ − b + − c . Proof. i) 0 ∈ ⇒ ∈ − ⇒ − x ∈ x + 0 ⇒ ∈ − x + x ⇒ x ∈ − ( − x ) + 0 ⇒ − ( − x ) = x .ii) a ∈ b + c ⇔ c ∈ − b + a ⇔ − b ∈ c + − a ⇔ − a ∈ − b + − c . Example 1.3. • Given a ring ( A, + , − , · , , ′ : A × A → P ( A ) ∗ as a + ′ b = { a + b } . Then( A, + ′ , − , · , ,
1) is a multiring. • The Krasner hyperfield K = { , } is given by usual multiplication and multivalued sum by 0 + 0 = { } , { } , { , } . • The signal hyperfield = { , , − } is given by usual multiplication and multivalued sum by 0 + x = { x } ∀ x ∈ , 1 + 1 = { } , − −
1) = {− } , −
1) = { , , − } . Definition 1.4.
Let f : A → B be a map between multirings. The function f is a morphism of multiringsif for all a, b, c ∈ A :i) If a ∈ b + A c , then f ( a ) ∈ f ( b ) + B f ( c ).ii) f ( a · A b ) = f ( a ) · B f ( b ).iii) f (0 A ) = 0 B , f (1 A ) = 1 B .iv) f ( − A a ) = − B f ( a ).Given A a multiring and S, T ⊆ A , it is defined S + T = S s ∈ S,t ∈ T s + t and S · T = { st : s ∈ S, t ∈ T } .Given x , . . . , x n ∈ A , we define by induction x + · · · + x n = { x } + ( x + · · · + x n ). A set α ⊆ A is anideal if α + α ⊆ α and Aα ⊆ α and α is a proper ideal if 1 / ∈ α . An ideal α is prime if it is a proper idealand given a, b ∈ A such that ab ∈ α then a ∈ α or b ∈ α and α is maximal if it is maximal between allproper ideals. An equivalent way to define prime ideals is through morphism to the Krasner hyperfield: if f : A → K is a morphism, then f − (0) is a prime ideal and given p ⊆ A prime ideal, the characteristic map χ p : A → K given by χ p ( x ) = 0 iff x ∈ p is a multiring morphism. The set of all prime ideals are denoted byspec( A ). A set S ⊆ A is multiplicative if 1 ∈ S and S · S ⊆ S . Given a subset X ⊆ A , the ideal generatedby X is S { t x + · · · + t n x n : n ≥ , t i ∈ A, x i ∈ X for all i = 1 , . . . , n } . Note that if X = { x , . . . , x n } isfinite, in general I = S { t x + · · · + t n x n : t i ∈ A for all i = 1 , . . . , n } is not the ideal generated by X (it ispossible not closed by addition) but if A is an hyperring this is true. { PIT } Theorem 1.5.
Let A be a multiring.i) Every maximal ideal is a prime ideal.ii) (Prime Ideal Theorem) Let I ⊆ A an ideal and S ⊆ A a multiplicative set such that I ∩ S = ∅ . Thenexists a prime ideal p such that I ⊆ p and p ∩ S = ∅ . In particular, A = 0 if, and only if, spec ( A ) = ∅ .iii) Given a ∈ A , define D ( a ) = { P ∈ spec ( A ) : a / ∈ P } . Then D ( a ) ∩ D ( b ) = D ( ab ) and the family { D ( a ) : a ∈ A } is a basis for a spectral topology in spec ( A ) . roof. i) Let m ⊆ A be a maximal ideal and a, b ∈ A such that ab ∈ m . If a / ∈ m , consider the ideal I generated by m ∪ { a } I = [ { m + t a + · · · + t n a : n ≥ , m ∈ m , t i ∈ A for all i = 1 , . . . , n } . Since m is maximal ideal and a ∈ I \ m , 1 ∈ I . So exist m ∈ m and t , . . . , t n ∈ A such that1 ∈ m + t a + · · · + t n a . Then b ∈ bm + t ab + · · · t n ab ⊆ m .ii) Let X = { J ⊆ A : J is an ideal with I ⊆ J, J ∩ S = ∅} . With the order given by inclusion, ( X, ⊆ )is a non-empty partial order ( I ∈ X ). It is straightforward to conclude by Zorn’s lemma that exists p ∈ X maximal. If p is not prime, exists a, b ∈ A such that ab ∈ p and a, b / ∈ p . Consider the ideal J = p + ( a ) = S { x + at + · · · + at n : x ∈ p, t , . . . , t n ∈ A } . Since a ∈ J \ p and p ⊆ J , by p maximalitywe have J ∩ S = ∅ . Then exists s ∈ S, x ∈ p and t , . . . , t n ∈ A such that s ∈ x + at + · · · + at n .Using the ideal p + ( b ), also exists s ∈ S, y ∈ p and l , . . . , l k ∈ A such that s ∈ y + bl + · · · + bl n .Then s s ∈ xy + xbl + · · · + xbl n + yat + · · · yat n + P i,j abt i l j ⊆ S ∩ p , an absurd. Thus p is prime.iii) The proof is analogous to the ring case and can be found in Proposition 2.3 of [ Mar3 ]. { remarkOpen } Remark 2. • If f : A → B is a morphism between multirings, for each p ∈ spec( B ), f − ( p ) ∈ spec( A )is a prime ideal. This induces a map f ∗ = f − : spec( B ) → spec( A ) such that for each a ∈ A ,( f ∗ ) − ( D A ( a )) = D B ( f ( a )) and f ∗ ( D B ( f ( a ))) = D A ( a ) ∩ Im( f ∗ ). In particular, f ∗ is a spectral mapand if f is surjective, f ∗ : spec( B ) → Im( f ∗ ) ⊆ spec( A ) is open. • Let A be a multiring and p ∈ spec( A ). Then { p } = { q ∈ spec( A ) : p ⊆ q } . In fact, given q ∈ spec( A )with p ⊆ q and x ∈ A with q ∈ D ( x ), then p ∈ D ( x ) and thus q ∈ { p } . Reciprocally, if q ∈ { p } , given x / ∈ q , that is, q ∈ D ( x ), we have p ∈ D ( x ). Thus p ⊆ q . In particular, p ∈ spec( A ) is maximal if, andonly if, { p } ⊆ spec( A ) is closed.In what follows, A is a multiring, S ⊆ A is a multiplicative set and I ⊆ A is an ideal. We sumarize somebasic constructions with multirings. Quocient by ideal.
Given a, b ∈ A , a and b are equivalent module I (notation a ∼ I b ) if a − b ∩ I = ∅ .Let A/I the set of all equivalence classes of ∼ I . The multivaluated sum is given by a ∈ b + c if, and only if,exists a ′ , b ′ , c ′ ∈ A with a ′ ∼ I a , b ′ ∼ I b and c ′ ∼ I c such that a ′ ∈ b ′ + c ′ . This relation can be described by a ∈ b + c if, and only if, exist i ∈ I such that a ∈ b + c + i . The product is a · b = ab and − a = − a . Theconstants are induced by those of A . Then A/I is a multiring and exist a canonical projection π I : A → A/I which is a morphism of multirings with π I ( a ) = 0 if, and only if, a ∈ I . { idealq } Proposition 1.6.
Let A be a multiring and I an ideal.i) Let a, b, c ∈ A . Then π ( a ) ∈ π ( b ) + π ( c ) if, and only if, exist a ′ ∈ A such that a ′ ∼ I a and a ′ ∈ b + c . { } ii) If A is hyperring, then A/I is hyperring.iii) I is prime if, and only if, A/I is a multidomain. { idealqPrime } iv) The ideal I is maximal if, and only if, A/I is a multifield. In particular, if A is an hyperring, I ismaximal if, and only if, A/I is a hyperfield. { idealqMax } v) Given a multiring morphism f : A → B with f ( I ) = { } , then exists an unique morphism f : A/I → B such that f = f ◦ π .vi) The induced spectral map π − : spec( A/I ) → spec( A ) determines an homeomorphism between spec( A/I )and { P ∈ spec( A ) : I ⊆ P } . Given
X, Y spectral topological spaces, a function f : X → Y is spectral if pre-image of compact open is compactopen. In particular, spectral maps are continuous and a continous map between Boolean topological spaces is spectral. ii) Let f : A → B a multiring morphism and J ⊆ B an ideal. Given I ⊆ f − ( J ) ideal, then exists a uniquemorphism f I,J : A/I → B/J such that
A BA/I B/Jff
I,J is a commutative diagram.
Proof. i) If π ( a ) ∈ π ( b ) + π ( c ), we know that exist i ∈ I such that a ∈ b + c + i . Then − b ∈ c + ( − a + i ).So exist − a ′ ∈ − a + i such that − b ∈ c − a ′ and then a ′ ∈ b + c and a ′ ∼ I a .ii) Assume that A is an hyperring and let a, b, c ∈ A . Take π ( x ) ∈ π ( ca ) + π ( cb ). Then by i ) exists x ′ ∈ A with x ′ ∼ I x and x ′ ∈ ca + cb . Since A is an hyperring, exist y ∈ a + b such that x ′ = cy . Then π ( y ) ∈ π ( a ) + π ( b ) and π ( x ) = π ( x ′ ) = π ( cy ).iii) Analogous to the ring case.iv) Assume that I is maximal and let a / ∈ I ( a = 0 in A/I ). Noting that the ideal generated by I ∪ { a } is S { i + x a + · · · + x n a : n ≥ , i ∈ I, x , . . . , x n ∈ A } , by I maximality exists i ∈ I and x , . . . , x n ∈ A suchthat 1 ∈ i + x a + · · · + x n a . Thus 1 ∈ x a + · · · + x n a in A/I . The reciprocal follows by observing thatif the ideal I has the desired property, then for every a / ∈ I , the ideal generate by I ∪ { a } is improper.v) Let f : A → B such that f ( I ) = { } . Note that • If π ( a ) = π ( b ), exist i ∈ I such that a ∈ b + i . Then f ( a ) ∈ f ( b ) + 0 = { f ( b ) } . • If π ( a ) ∈ π ( b ) + π ( c ), then exist a ′ ∈ A such that π ( a ) = π ( a ′ ) and a ′ ∈ b + c . Then f ( a ) = f ( a ′ ) ∈ f ( b ) + f ( c ).Thus, we can define f ( π ( a )) = f ( a ), which by previous observations is a multiring morphism. Theuniqueness of f is trivial.vi) The bijection between spec( A/I ) and { p ∈ spec( A ) : I ⊆ A } follows by the universal property of π usingthe equivalent characterization of prime ideals as morphisms to Krasner hyperfield. By Remark 2, π − is an homeomorphism with the image.vii) Let f : A → B a multiring morphism, J ⊆ B ideal and I ⊆ f − ( J ). Consider the map π J : B → B/J .Since π J ◦ f ( I ) = 0, by the universal property for π I : A → A/I , exist unique multiring morphism f I,J : A/I → B/J such that f I,J ◦ π I = π J ◦ f . Example 1.7. • The signal function sgn : R → is an example of non-injective morphism that satisfiessgn( x ) = 0 ⇔ x = 0. On the other hand, if f : A → B is surjective morphism satisfying the propertyin 1 . , i ), that is, for x, y, z ∈ A ,if f ( x ) ∈ f ( y ) + f ( z ) , then exists x ′ ∈ A such that f ( x ′ ) = f ( x ) and x ′ ∈ y + z, then the induced morphism f : A/I → B is an isomorphism. • A multiring D satisfies a strong cancellative property if for all a, b, c ∈ D with ab = ac and a = 0,then a = b . There are multidomains D that not satisfies strong cancellative property. For example, let R be a RRM (see, for instance, Definition 1 .
19) that has a prime ideal p that is not maximal. Then R/p is a multidomain (1 . , iii )) and satisfies x = x for all x ∈ R/p . Thus if
R/p would satisfy thestrong cancellative property, then
R/P would be a hyperfield and by 1 . , iv ) p ∈ spec( R ) should be amaximal ideal, a contradiction. emark 3. • The map f I,J is usually denoted only by f J when I = f − ( J ). Localization.
The elements of S − A are of the form a/s with a ∈ A and s ∈ S and a/s = b/t if, andonly if, exist u ∈ S such that atu = bsu . The sum is defined by a/s ∈ b/t + c/u if, and only if, exist v ∈ S such that atuv ∈ bsuv + cstv . The product is a/s · b/t := ab/st . The unit element is 1 / /
1. All those are well-defined and make S − A into a multiring. The canonical map ρ S : A → S − A givenby ρ S ( a ) = a/ ρ S ( S ) ⊆ ( S − A ) × . { localization } Proposition 1.8.
Let A be a multiring and S a multiplicative set.i) If A is hyperring, then S − A is hyperring.ii) S − A = 0 if, and only if, 0 ∈ S .iii) Given a multiring morphism f : A → B with f ( S ) ⊆ B × , exist unique morphism f : S − A → B suchthat f = f ◦ ρ S . In particular, ρ S is an epimorphism.iv) The induced spectral map ρ − : spec( S − A ) → spec( A ) determines an homeomorphism between spec( S − A )and { P ∈ spec( A ) : P ∩ S = ∅} .v) Let f : A → B a multiring morphism, T ⊆ B a multiplicative set and S ⊆ f − ( T ) another multiplicativeset. Then exists an unique morphism f S,T : S − A → T − B such that A BS − A T − Bff
S,T is a commutative diagram.
Proof. i) Let x, a, b, c ∈ A and s, u, t, w ∈ S such that x/s ∈ c/w ( a/u ) + c/w ( b/t ) = ca/wu + cb/wt . Thenexist p ∈ S such that xputw ∈ capstw + cbpsuw . Since A is hyperring, exist d ∈ at + bu such that xputw = d ( scwp ). Then d/tu ∈ a/u + b/t and x/s = d/tu · c/w . So S − A is hyperring.ii) If 0 ∈ S , it is obvious that S − A = 0. Reciprocally, if S − A = 0, then 1 = 0 in S − A , that is, exists s ∈ S such that s = s · s · f : A → B with f ( S ) ⊆ B × . Note that Claim. If a/s ∈ b/t + c/u , then f ( a ) f ( s ) − ∈ f ( b ) f ( t ) − + f ( c ) f ( u ) − . In particular, if a/s = b/t , then f ( a ) f ( s ) − = f ( b ) f ( t ) − . Proof.
Take p ∈ S such that aptu ∈ bpsu + cpst . Since f is a morphism, f ( a ) f ( p ) f ( t ) f ( u ) ∈ f ( b ) f ( p ) f ( s ) f ( u ) + f ( c ) f ( p ) f ( s ) f ( t ). But f ( p ) , f ( s ) , f ( t ) , f ( u ) ∈ B × and so f ( a ) f ( s ) − ∈ f ( b ) f ( t ) − + f ( c ) f ( u ) − .Thus we can define f : S − A → B by f ( a/s ) = f ( a ) f ( s ) − and then by above claim f is a multiringmorphism. The uniqueness of f is trivial.iv) The bijection between spec( S − A ) and { p ∈ spec( A ) : p ∩ S = ∅} follows by the universal property of ρ using the equivalent characterization of prime ideals as morphisms to Krasner hyperfield. By Remark2, ρ − is an homeomorphism with the image.v) Follows directly by the universal property of A → S − A . emark 4. • If A is a multidomain, we can define the fraction field of A by f f ( A ) := ( A \ − A . Notethat f f ( A ) is a hyperfield and the canonical map ρ : A → f f ( A ) is not necessarily injective (Example2.5 in [ Mar3 ]). Given a prime ideal p ∈ spec( A ), we define K A ( p ) := f f ( A/p ) and A p = ( A \ p ) − A .Let ρ : A → A p the canonical morphism. Note that, by Proposition 1 .
8, iv), the multiring A p has anunique maximal ideal given by A p ρ ( p ) = { xs : x ∈ p, s ∈ A \ p } and it is denoted by pA p . • The morphism f S,T is usually denoted only by f T when S = f − ( T ). { kAp } Proposition 1.9.
Let A a multiring and p ∈ spec( A ) a prime ideal. Then exists an unique map A p /pA p → K A ( p ) such that AA p /pA p K A ( p )is a commutative diagram. Furtheremore, it is an isomorphism. Proof.
Consider the compositions i : A → A/p → K A ( p ) , i : A → A p → A p /pA p . Note that, by proposi-tions 1 . .
8, they satisfies the following universal property: given a map f : A → B such that f ( p ) = 0and f ( A \ p ) ⊆ B × , then exists uniques f : K A ( p ) → B, f : A p /pA p → B such that f = f ◦ i = f ◦ i .These universal properties assure that exists unique f with i = i ◦ f and f should be an isomorphism. Marshall Quotient.
Given a, b ∈ A , a, b are said Marshall equivalent (denoted by a ∼ S b ) if exists s, t ∈ S such that as = bt . The set of all induced equivalence classes is denoted by A/ m S . The multivaluatedsum is defined by a ∈ b + c if, and only if, exists a ′ , b ′ , c ′ ∈ A such that a ′ ∼ S a , b ′ ∼ S b , c ′ ∼ S c and a ′ ∈ b ′ + c ′ (note that a ∈ b + c if, and only if, exists s, t, u ∈ S such that as ∈ bt + cu ). The productis defined by a · b = ab and the constants are those induced by A . Then A/ m S is a multiring and exist acanonical map π S : A → A/ m S is a surjective morphism and π S ( S ) = { } . Definition 1.10.
Let A a multiring and S ⊆ A multiplicative subset. If for all xs ∈ S with s ∈ S we have x ∈ S , then S is called cancellative. Define also S = { x ∈ A : xs ∈ S for some s ∈ S } . Note that S iscancellative if, and only if, S = S . { marshallQ } Proposition 1.11.
Let A be a multiring and S a multiplicative set.i) S is cancellative multiplicative set and given T ⊆ A multiplicative, S, T induces the same equivalencerelation in A if, and only if, S = T . In particular, A/ m S = A/ m S .ii) If A is hyperring, A/ m S is hyperring.iii) Given a multiring morphism f : A → B with f ( S ) = { } , exists an unique morphism f : A/ m S → B such that f = f ◦ ρ .iv) The induced spectral map π − : spec( A/ m S ) → spec( A ) determines an homeomorphism between spec( A/ m S )and { P ∈ spec( A ) : P ∩ S = ∅} .v) Let f : A → B , T ⊆ B a multiplicative set and S ⊆ f − ( T ) another multiplicative set. Then existunique morphism f S,T : A/ m S → B/ m T such that A BA/ m S B/ m T f f S,T is a commutative diagram. roof. i) First note that S is multiplicative and cancellative.Since S ⊆ S , 1 ∈ S . Given x, y ∈ S , exists s, t ∈ S such that xs, yt ∈ S . Then xyst ∈ S and so xy ∈ S .Also if xs ∈ S with s ∈ S , then exists s , s ∈ S such that xss , ss ∈ S . Therefore x ( ss ) s ∈ S andso x ∈ S .Now assume that S, T satisfies ∼ S = ∼ T . Since S = { x ∈ A : x ∼ S } , we have S = T . Reciprocally, if S = T , given x, y ∈ A with x ∼ S y , exists s, s ′ ∈ S such that xs = ys ′ . By hypothesis exists t, t ′ ∈ T with st, st ′ ∈ T . Then xs ( tt ′ ) = xs ′ ( tt ′ ) and x ∼ T y . Change the roles of S and T , we see that ∼ S = ∼ T .ii) Let x, a, b, c ∈ A such that π ( x ) ∈ π ( ca ) + π ( cb ). Then exists s, t, u ∈ S such that xs ∈ cat + cbu . Since A is an hyperring, exist d ∈ at + bu such that xs = cd . Then π ( x ) = π ( cd ) and π ( d ) ∈ π ( a ) + π ( b ). Thus A/ m S is hyperring.iii) Let f : A → B a multiring morphism such that f ( S ) ⊆ { } . The note that • If π ( a ) ∈ π ( b ) + π ( c ), then f ( a ) ∈ f ( b ) + f ( c ). In particular, if π ( a ) = π ( b ), then f ( a ) = f ( b ).Thus, we just need to define f ( π ( a )) = f ( a ): f : A/ m S → B is a well-defined multiring morphism. Theuniqueness is trivial.iv) The bijection π − : spec( A/ m S ) → { P ∈ spec( A ) : P ∩ S = ∅} is a direct consequence of the universalproperty of π : A → A/ m S and the characterization of prime ideals as morphisms to Krasner hyperfield.By Remark 2, π − is an homeomorphism with the image.v) Follows directly by the universal property of A → A/ m S . Remark 5.
The morphism f S,T is usually denoted only by f T when S = f − ( T ). Proposition 1.12.
Let A a multiring. Let S a multiplicative set and π : A → A/ m S . Given q ∈ spec( A/ m S ),let p = π − ( q ) ∈ spec( A ). Consider the canonical map g : A → K A ( p ) and let S p = g ( S ). Then exist uniquemorphism K A/ m S ( q ) → K A ( p ) / m S p such that AK A/ m S ( q ) K A ( p ) / m S p is a commutative diagram. Furtheremore, it is an isomorphism. Proof.
The strategy is the same as in Proposition 1 .
9. Consider the compositions j : A → A/ m S → K A/ m S ( q ) , j : A → K A ( p ) → K A ( p ) / m S p . They satisfies the following universal property: given a map f : A → B such that f ( S ) = 1 , f ( p ) = 0 and f ( A \ p ) ⊆ B × , then exists uniques f : K A/ m S ( q ) → B, f : K A ( p ) / m S p → B such that f = f ◦ j = f ◦ j . Inductive limits.
Let L be a language and consider the category L-mod of all L -models. Given a right-directed set (or simply directed) h I, ≤i , an I-inductive system of L -models is a functor M : h I, ≤i → L-mod (if i ≤ j , we denote the morphism M ( i ≤ j ) by µ i,j ). A colimit of such system is denoted by lim −→ M orlim −→ M i .A dual cone over the inductive system M is a tuple h A, { µ i : i ∈ I }i such that A is a L -strucutre and µ i : M i → A is a L -morphism with µ i ◦ µ j,i = µ j for every j ≤ i . For the convenience of the reader, wesummarize the relations between this notions: { filtColim } Proposition 1.13.
Let M = h I, ≤i → L − mod be a inductive system of L -structures. Then the colimitexists in L -mod and a dual cone over M , h M, { µ i : i ∈ I }i , is isomorphic to lim −→ M if, and only if,a) M = S i ∈ I µ i ( M i ). ) If ϕ ( v , . . . , v n ) is an atomic L -formula and s , . . . , s n ∈ M i satisfies M (cid:15) ϕ ( µ i ( s ) , . . . , µ i ( s n )), thenexists k ∈ I with k ≥ i such that M k (cid:15) ϕ ( µ i,k ( s ) , . . . , µ i,k ( s n )) . In particular, given i, j ∈ I and a ∈ M i , b ∈ M j , we have µ i ( a ) = µ j ( b ) if, and only if, exists k ≥ i, j such that µ i,k ( a ) = µ j,k ( b ).Furthermore, let ϕ ( v ) be a disjunction of geometric formulas and s ∈ M i . Let S ϕ = { k ∈ I : k ≥ i and M k | = ϕ ( µ i,k ( s )) } . If S ϕ is cofinal in I , then M | = ϕ ( µ i ( s )). Proof.
The result can be found in [
Mir1 ] or [
DM4 ]. Remark 6.
Let L multi = { π, · , − , , } be the language of multirings where π is ternary relation, · binaryfunction, − unary function and 0 , π : a ∈ b + c if, and only if, π ( a, b, c )). Let M : h I, ≤i → Multi be a inductive system of multirings (for each i ∈ I , denote M ( i ) = M i ).Then lim −→ M is a multiring because all axioms of 1 . A be a multiring. A prime cone of A is just a subset P ⊆ A such that A ⊆ P , P + P ⊆ P, P · P ⊂ P, P ∪ − P = A and supp( P ) := P ∩ − P (support of P ) is a prime ideal (when A is an hyperfield, P = { } ,the unique prime ideal). An order of A is just a morphism σ : A → A ) and is called real spectra. These data are equivalent in the following sense: Proposition 1.14.
Let A be a multiring. Then the following data are in bijective correspondence:i) P C = { P ⊆ A : P is a prime cone } .ii) P Q = { ( p, P ) : where p ∈ spec( A ) and P is a prime cone of K A ( p ) } .iii) sper( A ) = { σ : A → σ is a morphism } . Proof.
The bijections are described in terms of sper( A ). i ) ⇔ iii ): Given a prime cone P ⊆ A , consider the morphism σ : A → σ ( a ) = a ∈ P \ − P a ∈ P ∩ − P − a ∈ − P \ P . (1)Reciprocally, if σ ∈ sper( A ), then σ − ( { , } ) ⊆ A is a prime cone. ii ) ⇔ iii ): Given a prime ideal p ∈ spec( A ) and a prime cone P ⊆ K A ( p ), consider the morphism σ : A → σ ( a ) = a / ∈ p and a ∈ P a ∈ p − a / ∈ p and − a ∈ P . (2)Reciprocally, if σ : A → σ − (0) and the image of σ − ( { , } ) by the canonicalcomposition A → A/p → K A ( p ) gives a prime cone in K A ( p ). a formula ϕ ( t ) is geometric if it is a negation of an atomic formula or of the form ∀ v ( ψ ( v, t ) → ∃ wψ ( w, v, t )),where ψ , ψ are positive and quantifier-free. et A be a multiring. Given a , . . . , a n ∈ A , define U ( a , . . . , a n ) = { σ ∈ sper( R ) : σ ( a i ) = 1 for all i =1 , . . . , n } . The multiring A is called semi-real if − / ∈ P A and an ideal α ⊆ A is real if given a , . . . , a n ∈ A such that a + · · · + a n ∩ A = ∅ , then a i ∈ A for all i = 1 , . . . , n . { basicReal } Proposition 1.15.
Let A be a multiring.i) sper( A ) = ∅ if, and only if, A is semi-real.ii) The set sper( A ) endowed with the topology generated by the sets U ( a , . . . , a n ), a i ∈ A , is a spectralspace.iii) Let p a prime ideal of A . Then the following are equivalent:a) p is real.b) The multfield K A ( p ) is semi-real.c) p is the support of some order. Proof.
The proofs are in Propositions 5.1, 6.1 and Corollary 5.3 of [
Mar3 ]. Definition 1.16.
Let A be a multiring. • A subset T ⊆ A is called a preorder if A ⊆ T, T · T ⊆ T and T + T ⊆ T and it is proper if − / ∈ T . • If T is a proper preorder of A , the pair ( A, T ) is called a p-multiring. • We denote by sper T ( A ) = { σ ∈ sper( A ) : σ ( T ) ⊆ { , }} the orders that preserve T .If ( A, T ) and (
B, P ) are p-multirings, a multiring morphism f : A → B is a morphism of p-multirings if f ( T ) ⊆ P . We denote by pMult the category of all p-multirings and its morphisms.Let A be multiring and T ⊆ A a proper preorder (note that A is necessarily semi-real). Consider thenatural mapˆ: A → sper T ( A ) given by ˆ a ( σ ) = σ ( a ). Let Q T ( A ) = { ˆ a : for all a ∈ A } . This set has a productgiven by ˆ a · ˆ b = ˆ ab and a sum given by ˆ a ∈ ˆ b + ˆ c if exists a ′ , b ′ , c ′ ∈ A with ˆ a ′ = ˆ a, ˆ b ′ = ˆ b, ˆ c ′ = ˆ c and a ′ ∈ b ′ + c ′ .If T = 1 + P A we denote Q T ( A ) just by Q ( A ). { Q(A)Mult } Fact 1.17.
Let (
A, T ) be a p-multiring and π : A → Q T ( A ) the natural projection. Then Q T ( A ) is amultiring and given a, b, c ∈ Q T ( A ), π ( a ) ∈ π ( b ) + π ( c ) if, and only if, for all σ ∈ sper T ( A ) we have σ ( a ) ∈ σ ( b ) + σ ( c ). Proof.
Proposition 7.3 of [
Mar3 ]. { piIso } Fact 1.18.
Let A be a semi-real multiring. Then the map A → Q ( A ) is an isomorphism if, and only if, forall a, b ∈ A i) a = a .ii) a + b a = { a } iii) a + b has an unique element. Proof.
Proposition 7.5 of [
Mar3 ]. { rrmdefinition } Definition 1.19.
A multiring A is called real reduced multiring (RRM) if it is semi-real and satisfies thethree conditions of above theorem. The category of all real reduced multirings with multiring morphism isdenoted by RRM . Remark 7. • If (
A, T ) is a p-multiring, then by Fact 1 .
17 the multring Q T ( A ) is a real reduced multi-ring. • Let f : ( A, T ) → ( B, P ) a p-morphism. Then we have an induced map sper P ( B ) → sper T ( A ) givencomposition with f . sepTheorem } Corollary 1.20.
Let A be a real reduced multiring and a, b, c ∈ A . Then a ∈ b + c if, and only if, for all σ ∈ sper( A ) we have σ ( a ) ∈ σ ( b ) + σ ( c ). In particular, if σ ( a ) = σ ( b ) for all σ ∈ sper( A ), then a = b . Proof.
Direct consequence of Fact 1 .
17 and Fact 1 . Remark 8.
Let A be a multiring that satisfies the conditions of Fact 1 .
18. Given a, b ∈ A we have c ∈ a + b satisfying c = c because since a b + a = { a } and a b + b = { b } , c ∈ ( a + b )( a + b ) ⊆ a + a b + b + a b = a + b and thus c = c . Therefore A semi-real if, and only if, 1 = 0 (if − ∈ P A ,then − x for some x ∈ A and so 0 ∈ x = { } ). Furthermore, if A is RRM, since P A = A = Id( A ),then ( A, Id( A )) is a p-multiring. { hyperfieldRR } Corollary 1.21.
Let F be a hyperfield. Then F is a real reduced hyperfield if, and only if for all a ∈ ˙ F . 1 = 0.. a = 1, whenever a = 0.. 1 + 1 = { } . Proof. If F is a real reduced hyperfield, then it is imediate that F satisfies the above axioms. Reciprocally, F is semi-real and a = a for all a ∈ F . Furthermore, given x ∈ a + b a , if b = 0, then x = a and if b = 0,then x ∈ a + a = { a } . It is also easy to see that a + b has just a unique element 0 or 1. { uniProMRR } Theorem 1.22. i) Let f : ( A, T ) → ( B, P ) a morphism of p-multirings. Then exist unique map Q ( f ) : Q T ( A ) → Q P ( B ) such that ( A, T ) (
B, P ) Q T ( A ) Q P ( B ) fQ ( f ) is a commutative diagram. Furthermore, Q defines a functor from category pMulti to RRM .ii) Let ( A, T ) be a p-multiring and let f : ( A, T ) → ( R, P R ) be a morphism of multirings where R isRRM. Then exist unique f : Q T ( A ) → R morphism such that the diagram A Q T ( A ) R f f is commutative. In other words, the functor Q : pMulti → RRM is left-adjoint to the inclusion functor i : RRM → pMulti given by i ( R ) = ( R, P R ) .Proof. i) Let π ( A,T ) : A → Q T ( A ) and π ( B,P ) : B → Q P ( B ) be the canonical projections. Consider the map Q ( f ) : Q T ( A ) → Q P ( B ) given by Q ( f )( π ( A,T ) ( a )) = π ( B,P ) ( f ( a )) (note that if π ( A,T ) ( a ) = π ( A,T ) ( b ),then in particular for all σ ∈ sper P ( B ) we have σ ◦ f ∈ sper T ( A ) and so σ ( f ( a )) = σ ( f ( b )); thus π ( B,P ) ( f ( a )) = π ( B,P ) ( f ( b ))). So given a, b, x ∈ A • Q ( f )( π ( A,T ) ( a ) π ( A,T ) ( b )) = Q ( f )( π ( A,T ) ( ab )) = π ( B,P ) ( f ( ab )) = π ( B,P ) ( f ( a )) π ( B,P ) ( f ( b )) = Q ( f )( π ( A,T ) ( a )) Q ( f )( π ( A,T ) ( b )). • If π ( A,T ) ( x ) ∈ π ( A,T ) ( a ) + π ( A,T ) ( b ), then for all σ ∈ sper P ( B ) we have σ ◦ f ∈ sper T ( A ) and so σ ( f ( x )) ∈ σ ( f ( a )) + σ ( f ( b )). Therefore, by Fact 1 . π ( B,P ) ( f ( x )) ∈ π ( B,P ) ( f ( a )) + π ( B,P ) ( f ( b )),that is, Q ( f )( π ( A,T ) ( x )) ∈ Q ( f )( π ( A,T ) ( a )) + Q ( f )( π ( A,T ) ( b )), hus Q ( f ) : Q T ( A ) → Q P ( B ) is a multiring morphism -and unique that satisfies the diagram commu-tativity because π ( A,T ) , π ( B,P ) are surjective.ii) Consider the map f : Q T ( A ) → R given by f ( π ( A,T ) ( a )) = f ( a ) -note that if π ( A,T ) ( a ) = π ( A,T ) ( b ), thenin particular for all σ ∈ sper( R ) we have σ ◦ f ∈ sper T ( A ) and so σ ( f ( a )) = σ ( f ( b )); thus f ( a ) = f ( b )because R is RRM). The proof that f is multiring morphism is analogous to the preceding item. Thus f is the unique morphism satisfying the diagram commutativity because π is surjective. { sper(Q(A)) } Corollary 1.23.
Let (
A, T ) be a p-multiring. Then the natural map π ( A,T ) : A → Q T ( A ) induces anhomeomorphism sper( Q T ( A )) → sper T ( A ). Proof.
Direct consequence of Theorem 1 . { isoRS } Lemma 1.24.
Let f : A → B be surjective multiring morphism with A, B
RRM. If sper( f ) : sper( B ) → sper( A ) is surjective, then f is isomorphism and thus sper( f ) is an homeomorphism. Proof.
First we will prove that f is injective. Let x, y ∈ A with f ( x ) = f ( y ). Given σ ∈ sper( A ), by thesurjective of sper( f ), exist τ ∈ sper( B ) such that σ = τ ◦ f . Thus σ ( x ) = τ ( f ( x )) = τ ( f ( y )) = σ ( y ). Since σ ∈ sper( A ) was arbitrary, we have x = y . To conclude that f is an isomorphism, we need to show thatif f ( x ) ∈ f ( y ) + f ( z ) for x, y, z ∈ A , then x ∈ y + z . But using the same argument for injectivity we have σ ( x ) ∈ σ ( y ) + σ ( z ) for all σ ∈ sper( A ) and so x ∈ y + z . { } Theorem 1.25.
Let A be a semi-real multiring and π : A → Q ( A ) the canonical projection.i) The multiring A/ m P A is semi-real and the canonical map A → A/ m P A induces an isomor-phism Q ( A ) ∼ = Q ( A/ m P A ) .ii) If A is hyperfield, then the morphism π induces isomorphisms A/ m P ˙ A ∼ = Q ( A ) ∼ = A/ m P A .Proof. i) Given a morphism σ : A → σ (1 + P A ) = { } . Thus by Proposition 1 .
11, the canonicalmorphism ρ : A → A/ m P A induces a bijection sper( A/ m P A ) ∼ = sper( A ) and then byProposition 1 . A is semi-real if, and only if, A/ P A is semi-real.Furthermore, since ρ is surjective, Q ( ρ ) : Q ( A ) → Q ( B ) is surjective and by Corollary 1 .
23 sper( Q ( ρ )) : sper( Q ( B )) → sper( Q ( A )) is surjective (in fact an homeomorphism). Then by Lemma 1 .
24 the morphism Q ( ρ ) is anisomorphism.ii) By item i ), we have Q ( F ) ∼ = Q ( F/ m P F ). On the other hand, • Since F is semi-real, 1 = 0 in F/ m P F . • If x = 0 in F , then x (1 + x − ) = (1 + x ) and so x = 1 in F/ m P F . • If y ∈ F/ m P F , then exists s , s , s ∈ P F such that ys ∈ s + s . Since1 + P F is closed under sums, we have ys ∈ P F . Thus y = 1 in F/ m P F .Therefore by Corollary 1 .
21, the hyperfield F/ m P F is real reduced. Then follows by Theorem1 .
22 that Q ( F/ m P F ) ∼ = F/ m P F . On the other side, since 1 + P F ⊆ P ˙ F , we have asurjective morphism f : F/ m P F → F/ m P ˙ F . Claim.
Let x, y, z ∈ F such that x ∈ y + z in F/ m P ˙ F . Then x ∈ y + z in F/ m P F . Proof.
By hypothesis, exists s , s , s ∈ P ˙ F such that xs ∈ ys + zs ( ∗ ). Thus it should exist x , x , x ∈ ˙ F such that s i x − i ∈ P F , i = 1 , , s ∈ x + · · · + x n and x = 0, then sx − ∈ P F ). But by item ii ) above exists m i , n i ∈ P F such that m i x − i = n i . Consequently,multiplying ( ∗ ) by n n n x ( s x − ) m n n ∈ y ( s x − ) m n n + z ( s x − ) m n n . Therefore x ∈ y + z in F/ m P F . y the above claim, the morphism f reflects the multivaluated sum. Thus to conclude that f is anisomorfism, we have to prove that f is injective but if f ( x ) = f ( y ), then f ( x ) ∈ f ( y )+ f (0) in F/ m P ˙ F .So by the claim x ∈ y + 0 in F/ m P F , that is, x = y .Thus Q ( F ) ∼ = F/ m P F ∼ = F/ m P ˙ F . Here we present a structural presheaf for multiring. This presheaf generalize the sheaf for rings, the sheaf forhyperdomains in [
Jun ] and the sheaf for real reduced multrings in [
DP2 ] (in the language of real semigroups).Thus rings, hyperdomains and real reduced multirings are geometric (the presheaf is a sheaf). But there aremultirings non-geometric (example 3 .
8) and in the next section it is given a first-order characterization ofgeometric von Neumann hyperrings.Let A be a multiring, α an ideal and a ∈ A . We define √ α = { x ∈ A : exist natural n ≥ x n ∈ α } and S a = { x ∈ A : exists n ≥ a n ∈ ( x ) } (if a = 0, we consider 0 = 1.). We also define A a = { , a, a , . . . } − A . Note that, if ρ : A → A a is the canonical map, then S a = ρ − ( A × w a ). { sheaf2 } Proposition 2.1.
Let A multiring. Let α be an ideal and a ∈ A .i) √ α is an ideal and S a is a multiplicative set.ii) √ α = T α ⊆ p ∈ spec ( A ) p . In particular, T p ∈ spec ( A ) p is the set of nilpotents.iii) S a = T a/ ∈ p p c . In particular, T p ∈ spec ( A ) p c = A × w .iv) Furthermore, the following are equivalent:a) D ( a ) ⊆ D ( b ).b) √ a ⊆ √ b .c) a ∈ √ b .d) S b ⊆ S a .v) If D ( a ) ⊆ D ( b ), then exist unique morphism ρ D ( b ) ,D ( a ) : S − b A → S − a A such that AS − b A S − a A ρ D ( b ) ,D ( a ) is a commutative diagram. In addition, if D ( a ) ⊆ D ( b ) ⊆ D ( c ), then ρ D ( c ) ,D ( a ) = ρ D ( c ) ,D ( b ) ◦ ρ D ( c ) ,D ( b ) and ρ D ( a ) ,D ( a ) = Id S − a A .vi) Exists unique morphism A a → S − a A such that AA a S − a A is a commutative diagram. Proof. i) Given x, y ∈ √ α , choose n ≥ x n , y n ∈ α . Then given z ∈ x − y , we have z n ∈ P ni =0 (cid:0) ni (cid:1) x i ( − y ) n − i ⊆ α . Since for all i = 0 , . . . , n , i ≥ n or 2 n − i ≥ n , we have z ∈ √ α . Then it iseasy to see that √ α is an ideal.Note that 1 ∈ S a and given x, y ∈ S a , exists m, n ≥ t , . . . , t n , l , . . . , k k ∈ A such that a m ∈ xt + · · · + xt n and a n ∈ yl + · · · + yl k . Then a m + n ∈ P i,j xyt i l j l and so xy ∈ S a . i) It is easy to see that √ α ⊆ T α ⊆ p p . Let x / ∈ √ α and consider the multiplicative set S = { , x, x , . . . } .Since α ∩ S = ∅ , by Theorem 1 . p with α ⊆ p and x / ∈ p . Then x / ∈ T α ⊆ p p .iii) It is imediate that S a ⊆ T a/ ∈ p p c . Reciprocally, let x / ∈ S a . Then ( x ) ∩ { , a, a , . . . } = ∅ and by Theorem1 . p such that x ∈ p and a / ∈ p . Thus x / ∈ T a/ ∈ p p c .iv) Let α = ( a ) + · · · + ( a n ). Given any prime ideal p , we have α p if, and only if, exists 1 ≤ i ≤ n suchthat a i / ∈ p . Thus by item ii ) √ α = T α ⊆ p p and p ( a ) = T a ∈ p p . Therefore D ( a ) ⊆ D ( a ) ∪ · · · ∪ D ( a n ) ⇔ for all p ∈ spec( A ) , a / ∈ p → α p ⇔ p ( a ) ⊆ √ α. In particular, we have the equivalence a ) ⇔ b and by iii ) b ) ⇔ d ). We also have b ) ⇒ c ). c ) ⇒ b ): Since a ∈ √ b , exists n ≥ a n ∈ ( b ) ( ∗ ). Let x ∈ √ a . Then exists m ≥ l , . . . , l o ∈ A such that x m ∈ al + · · · + al o . Thus we have x nm ∈ P i + ··· + i o = n (cid:0) ni ,...,i o (cid:1) a n l i · · · l i o o ⊆ ( b )by ( ∗ ).v) If D ( a ) ⊆ D ( b ), we have S b ⊆ S a and so the existence and uniqueness of ρ D ( b ) ,D ( a ) follows by 1 .
8. Theother stated properties follows by the uniqueness just proved.vi) Since { , a, a , . . . } ⊆ S a , the existence and uniqueness of the morphism follows by 1 . A be a multiring and a ∈ A . We say that A has the a -invertible property if A × w a = A × a and A hasthe invertible property if it has the a -invertible property for every a ∈ A . Proposition 2.2.
Let A be a multiring and a ∈ A . Let ρ a : A → A a , ρ S a : A → S − a A be the canonicalmorphisms. Let ρ S a : A a → S − a A the induced morphism. The following are equivalent:i) ρ a ( S a ) ⊆ A × a .ii) A satisfies the a -invertible property.iii) S a = { x ∈ A : exists n ≥ , y ∈ A such that a n = xy } .iv) ρ S a : A a → S − a A is an isomorphism.In particular, if for every a ∈ A , ( a ) = Aa , then A has the invertible property. Proof. i ) ⇒ ii ): Let xa k ∈ A × w a . By definition, exists t a n , . . . , t l a nl ∈ A a such that 1 ∈ xa k t a n + · · · + xa k t a n .Thus exists m, n ′ , . . . , n ′ l such that a m ∈ xt a n ′ + · · · + xt l a n ′ l in A and so x ∈ S a . By hypothesis, ρ ( x ) ∈ A × a and exists ya t ∈ A a with x ya t = 1. Then xa k ∈ A × a . ii ) ⇒ iii ): Let x ∈ S a . Then exists n ≥ a n ∈ ( x ) and so x ∈ A × w a = A × a . By definition, thismeans that exists ya t ∈ A a with x ya t = 1. Then exists m ≥ xya m = a m + t . iii ) ⇒ iv ): Observe that, given x ∈ S a , by hypothesis exists n ≥ y ∈ A with a n = xy and thus ρ a ( x ) ∈ A × a . Thus by the universal property of ρ S a , exists morphism ρ a : S − a A → A a such that ρ a = ρ a ◦ ρ S a .To conclude, note that ( ρ a ◦ ρ S a ) ◦ ρ a = ρ a and ( ρ S a ◦ ρ a ) ◦ ρ S a = ρ S a . Thus, by the uniqueness of universalproperty of ρ a and ρ S a , we have ρ a ◦ ρ S a = Id A a and ρ S a ◦ ρ a = Id S a . iv ) ⇒ i ): Let x ∈ S a . Since ρ S a ( ρ a ( x )) = ρ S a ( x ) ∈ ( S − a A ) × and ρ S a is an isomorphism, ρ a ( x ) ∈ A × a . Example 2.3. • If A is an hyperring, then for every a ∈ A ( a ) = aA ; thus A has the invertible property. • Let A be a real reduced multiring and a, t , . . . , t n ∈ A . Let x ∈ at + · · · + at n . Then given σ ∈ sper( A ),if σ ( a ) = 0, we have σ ( x ) = 0; thus σ ( x ) = σ ( a x ). Since σ ∈ sper( A ) was arbitrary, by 1 .
20 we have x = a x = a ( ax ). Therefore ( a ) = aA and A has the invertible property. Let A be a multiring and consider S = { x ∈ A : 1 ∈ ( x ) } = A × w . Then S × A has the 1-invertibleproperty. In fact, if xs ∈ S × A is a weak-invertible, then 1 ∈ ( xs ). This implies that exists u ∈ S with u ∈ ( x ) in A but since 1 ∈ ( u ) we have 1 ∈ ( x ). Thus x ∈ S and xs ∈ S × A is invertible. Furthermore,the natural map A → S × A is initial between morphism from A to multirings with the 1-invertibleproperty.Let A be a multiring and B A := { D ( a ) } a ∈ A be the set of basic open sets of spec( A ). Consider the relation F A : B A → MultR given by F A ( D ( a )) = S − a A and if D ( a ) ⊆ D ( b ), the morphism F A,D ( b ) ,D ( a ) = ρ D ( b ) ,D ( a ) .By 2 . F A is in fact a functor. When there is no risk of confusion, F A is denoted just by F and for D ( a ) ⊆ D ( b ), the restriction map F D ( b ) ,D ( a ) is denoted by F b,a . Furthermore, given p ∈ spec( A ), we denoteby F p = lim −→ a/ ∈ p F ( D ( a )) the stalk at p . Definition 2.4.
Let A be a multiring. The functor F A is called structural presheaf associated to A .i) F is a monopresheaf if for all cover D ( e ) = S i ∈ I D ( e i ) and all a, b, c ∈ A e with F e,e i ( a ) ∈ F e,e i ( b ) + F e,e i ( c ) in A e i for every i ∈ I, then a ∈ b + c in A e .ii) F is a sheaf if it is a monopresheaf and if for all cover D ( e ) = S i ∈ I D ( e i ) and for x i ∈ A e i for all i ∈ I with F e i ,e i e j ( x i ) = F e j ,e i e j ( x j ) for every i, j ∈ I, then exist (necessarily unique) x ∈ A e such that F e,e i ( x ) = x i for every i ∈ I . Definition 2.5.
Let A be a multiring. If F A is a monopresheaf, A is called a mono-multiring; if F A if asheaf, then A is called geometric. { fiber } Proposition 2.6.
Let A a multiring. Given p ∈ spec( A ), then F p ∼ = A p naturally. Proof.
Given a / ∈ p , since S a ⊆ A \ p , exist unique map i S a : S − a A → A p such that AS − a A A pρ p i Sa is a commutative diagram. Thus, by the uniqueness property, it is easy to see that if D ( a ) ⊆ D ( b ), then A p S − b A S − a A ρ D ( b ) ,D ( a ) i Sb i Sa is a commutative diagram. Thus, the family I = { i S a : S − a A → A p : a / ∈ p } is a cocone. To see it isin fact a limit cocone, consider a family of morphism { h S a : S − a A → B : a / ∈ p } such that if D ( a ) ⊆ D ( b ), h S b = h S a ◦ ρ D ( b ) ,D ( a ) . We have to prove that exist unique morphism h : A p → B such that given a / ∈ p , h S a = h ◦ i S a . Existence:
Define h ′ = h S ◦ ρ S : A → B . To use the universal property of the map ρ p : A → A p , wehave to prove that for all a / ∈ p , h ′ ( a ) ∈ B × . But given a / ∈ p , since S − a A B h ′ ρ Sa h Sa is a commutative diagram and ρ S a ( a ) ∈ S − a A is invertible, we have h ′ ( a ) ∈ B × . Then exist a morphism h : A p → B such that AA p B h ′ ρ p h Then for all a / ∈ p , since i S a ◦ ρ S a = ρ p , we have that h ◦ i S a ◦ ρ S a = h ◦ ρ p = h ′ = h S a ◦ ρ S a . But since ρ S a is an epimorphism, we have h ◦ i S a = h S a . Uniqueness:
Just note that for every x ∈ A p , exists a / ∈ p and y ∈ S − a A such that i S a ( y ) = x . { fibertoOpen } Remark 9.
Let A be a multiring, a, b, c ∈ A and p ∈ spec( A ). Note that if a ∈ b + c in A p , then exists x / ∈ p with ax ∈ bx + cx . Thus a ∈ b + c in A x , with p ∈ D ( x ). Thus given e ∈ A , if a ∈ b + c in A p forall p ∈ D ( e ), then exists e , . . . , e n with D ( e ) = S ni =1 D ( e i ) and a ∈ b + c in A e i for all i = 1 , . . . , n because D ( e ) is compact. { cVn } Proposition 3.1.
Let A be an hyperring. The following are equivalent:i) spec( A ) is a Boolean topological space and √ a ∈ A exist b ∈ A such that a = a b . Proof. i ) ⇒ ii ) Let a ∈ A . Since spec( A ) is a boolean topological space, exists a , . . . , a n ∈ A such that D ( a ) c = D ( a ) ∪ · · · ∪ D ( a n ). In particular, D ( aa i ) = D ( a ) ∩ D ( a i ) = ∅ and so aa i ∈ √ i = 1 , . . . , n . Seeing that spec( A ) = D ( a ) ∪ D ( a ) c = D ( a ) ∪ D ( a ) ∪ · · · ∪ D ( a n ), by Proposition 2 . b, t , . . . , t n ∈ A such that 1 ∈ ab + a t + · · · + a n t n . Then multiplying by a entails a = a b . ii ) ⇒ i ) Let a ∈ A and take b with a = a b . Then, since A is an hyperring, exist l ∈ − ab such that al = 0. It is easy to see that D ( a ) c = D ( l ) and thus spec( A ) is a Boolean topological space. Furthermore,given x ∈ √
0, exists n ≥ x n = 0. Let b ∈ A with x = x b . Since ( xb ) = x b = xb , we have xb = ( xb ) n = x n b n = 0 and so x = ( xb ) b = 0. Definition 3.2.
Let A be an hyperring. If any of the equivalent conditions of 3 . A is called vonNeumann hyperring. The category of all von Neumann hyperring with the usual notion of morphism isdenoted by HV N . { vNremarks } Remark and Notation 1.
Let A be a von Neumann hyperring. • Since all axioms of von Neumann hyperring are geometric sentences, the inductive limit of vNH areagain vNH. • Since spec( A ) is Boolean, by remark 2 every prime ideal of A is maximal. • The basic open sets of spec( A ) are given by idempotents elements. Indeed, given a ∈ A , take b ∈ A with a = a b ; then ab is idempotent and D ( a ) = D ( ab ). For each a ∈ A idempotent, since a = a and A is an hyperring, exist a c ∈ − a such that aa c = 0.First note that if x, y ∈ − a and ax = ay = 0, then xy ∈ y − ya = { y } and xy ∈ x − xa = { x } .So x = xy = y and a c is unique determined. Furtheremore, ( a c ) ∈ a c − aa c = { a c } and thus a c isidempotent. We also have D ( a ) c = D ( a c ). • Given a, b ∈ A idempotent, if D ( a ) = D ( b ), then D ( ab c ) = D ( a ) ∩ D ( b ) c = ∅ = D ( a c b ). Thus, since √ ab c ∈ a (1 − b ) and 0 = a c b ∈ b (1 − a ). Then a = ab = b . • Given a ∈ A , we denote by i ( a ) the (unique) idempotent with D ( a ) = D ( i ( a )) and define a c := i ( a ) c . • The topological space sper( A ) is also Boolean. Given a ∈ A , let x ∈ a − a c . Let σ ∈ sper( A ). If σ ( a ) = 0 = σ ( i ( a )), then σ ( x ) = − σ ( i ( a )) c = −
1; if σ ( a ) = 0, then σ ( a c ) = 0 and σ ( x ) = σ . Thus σ ( x ) ∈ { , − } and σ ( a ) = 1 if, and only if, σ ( x ) = 1. Therefore U ( a ) c = U ( x ) c = U ( − x ). • Let f : A → B be a morphism of von Neumann hyperring. Given a ∈ A , we have f ( i ( a )) = i ( f ( a ))and f ( a c ) = f ( a ) c . In fact, let b ∈ A with a = a b . Then i ( a ) = ab and i ( f ( a )) = f ( a ) f ( b ) because f ( a ) = f ( a ) f ( b ). Thus f ( i ( a )) = i ( f ( a )). On the other hand, since i ( a ) c ∈ − i ( a ) , i ( a ) i ( a ) c = 0, wehave f ( i ( a ) c ) ∈ − f ( i ( a )) , f ( i ( a )) f ( i ( a ) c ) = 0 and so f ( i ( a )) c = f ( i ( a ) c ). Thus f ( a c ) = f ( i ( a ) c ) = f ( i ( a )) c = i ( f ( a )) c = f ( a ) c . • If D ( a ) ⊆ D ( b ), a, b idempotents, then a ∈ p ( b ). Thus exist l such that a = bl . Then ab = bbl = a . Definition 3.3.
Let A be a von Neumann hyperring. A set { e , . . . , e n } ⊆ Id( A ) is partition if for all i = j , e i e j = 0 and it is a partition of unity if it is a partition and 1 ∈ e + · · · + e n . { orth } Lemma 3.4.
Let A be a von Neumann hyperring and B = { e , . . . , e n } a partition.i) If x ∈ e + · · · + e n , then D ( x ) = S ni =1 D ( e i ). In particular, given U ⊆ spec( A ) clopen subset, exists e ∈ Id ( A ) such that U = D ( e ).ii) Assume that B is partition of unity and e i + e ci = { } for all i = 1 , . . . , n . Then e + · · · + e n = { } .iii) If B = { e , . . . , e n } and { f , . . . , f k } are partitions of unity, then { e f , . . . , e f k , . . . , e n f k } is also apartition of unity and e + · · · + e n , f + · · · + f k ⊆ e f + · · · + e f k + e f + · · · + e n f k . Proof. i) If p ∈ D ( x ), then exists i with p ∈ D ( e i ). Reciprocally, let p ∈ D ( e i ). For all j = i we have e j e i = 0 ∈ p and thus e j ∈ p . Then if x ∈ p , then e i ∈ p , an absurd. Thus p ∈ D ( x ).Let U ⊆ spec( A ) clopen. Then exists e , . . . , e n ∈ Id( A ) with U = S ni =1 D ( e i ). Let e ′ = e , e ′ i = e i ( e c · · · e ci − ), i = 2 , . . . , n . Note that D ( e ′ i ) = D ( e i ) ∩ T i − k =1 D ( e k ) c and e ′ i e ′ j = 0 if i = j . Then given x ∈ e ′ + · · · + e ′ n , we have D ( x ) = S ni =1 D ( e ′ i ) = S ni =1 D ( e i ) = U .ii) Given i = j , e i e j = 0 implies e i e cj = e i because e cj ∈ − e j . Then e i e c · · · e ci − = e i for all i = 2 , . . . , n − ∈ e + · · · + e n by e c · · · e cn − entails e n = e c · · · e cn − . Therefore e + e + · · · + e n − + e n = e + e + · · · + e n − e c · · · e cn − + e c · · · e cn − = e + e + · · · + e c · · · e cn − ( e n − + e cn − )= e + e + · · · + e n − e c · · · e cn − + e c · · · e cn − ...= e + e c = { } . iii) Since 1 ∈ e + · · · + e n and 1 ∈ f + · · · + f k , we have 1 = 1 · ∈ e f + · · · + e f k + e f + · · · e n f k and if ( i, j ) = ( i ′ , j ′ ), then ( e i f j )( e i ′ f j ′ ) = 0. Furthermore, the relation 1 ∈ e + · · · + e n implies that f j ∈ e f j + · · · + e n f j and so f + · · · + f k ⊆ e f + · · · + e f k + e f k + · · · + e n f k . By a symmetric sameargument, we conclude e + · · · + e n ⊆ e f + · · · + e f k + e f + · · · + e n f k . et A be a von Neumann hyperring. By the above lemma, B A is the Boolean algebra of clopens sets ofspec( A ). Let F be the structural pre-sheaf of A . Given p ∈ spec( A ), note that pA p ⊆ A p is is the zero idealbecause given x ∈ p , x c / ∈ p and xx c = 0. Then by Proposition 1 . F p ∼ = A p ∼ = A p /pA p ∼ = K A ( p ) isan hyperfield. { geoVon } Theorem 3.5.
Let A be a von Neumann hyperring. The following are equivalent:i) A is a mono-hyperring.ii) If u ∈ A satisfies u = 1 in K A ( p ) for every p ∈ spec ( A ) , then u = 1 em A .iii) For every e ∈ A idempotent, e + e c = { } .If any of the above conditions is valid, A is geometric and a ∈ b + c in A if, and only if, a ∈ b + c in K A ( p ) for every p ∈ spec ( A ) . In particular, a = b if, and only if, for all p ∈ spec ( A ) , a = b in K A ( p ) .Proof. i ) ⇒ ii ): Let u ∈ A with u = 1 in K A ( p ) for all p ∈ spec( A ). By remark 9, exists e , . . . , e n ∈ A withspec( A ) = S ni =1 D ( e i ) and u = 1 in A e i . Since F A is a monopresheaf, it follows u = 1. ii ) ⇒ iii ): Immediate since any u ∈ e + e c is locally equal to 1. iii ) ⇒ i ): Assume that spec( A ) = S ni =1 D ( t i ) and a ∈ b + c in A t i for all i = 1 , . . . , n . Consider the set X = { x ∈ A : ax ∈ bx + cx } . Then AX ⊆ X and I = S { x + · · · + x k : k ≥ , x i ∈ X for all i = 1 , . . . , k } is the ideal generated by X . By hypothesis, I is not contained in any prime ideal. Then 1 ∈ I and soexist x , . . . , x k ∈ X such that 1 ∈ x + · · · + x n . Since exist b i ∈ A such that b i x i is idempotents and D ( x i ) = D ( b i x i ), we can assume that exists e , . . . , e k ∈ A idempotent such that. spec( A ) = S ki =1 D ( e i ).. ae i ∈ be i + ce i for all i = 1 , . . . , k .Furtheremore, change e i by e i e c · · · e ci − , we can assume that e i e j = 0 if i = j . By Lemma 3 .
4, we have e + · · · + e n = { } . Then a ∈ ae + · · · + ae k ⊆ b ( e + · · · + e k ) + c ( e + · · · + e k ) = b + c because A is an hyperring.The general case follows easily by observing that given t ∈ A , spec( A t ) ∼ = D ( t ) ⊆ spec( A ) is a compact set.Now, assuming that A is mono-hyperring, we prove that it is in fact geometric. Let { x i } i ∈ I , { e i } i ∈ I familyof elements of A and e ∈ A with D ( e ) = S i ∈ I D ( e i ) and x i = x j in A e i e j for all i, j . Since D ( e ) is compact,we can assume I = { , . . . , n } finite and e, e i ∈ Id( A ). Furthermore, substituting e i by e i ( e c · · · e ci − ), we canalso assume that e i e j = 0 if i = j . Let x ∈ x e + · · · + x n e n . Then xe i = x i e i and so x = x i in A e i .Let A be an hyperring and S ⊆ A a multiplicative set (1 ∈ S and S · S ⊆ S ). Given a, b ∈ A , define D S ( a, b ) = { x ∈ A : exists u, v, s ∈ S such that xu ∈ av + bs } . Definition 3.6.
Let A be a von Neumann hyperring and S ⊆ A . The set S is called a von Neumannsubgroup if . S is multiplicative. . Given a ∈ A idempotent, if x ∈ D S ( a, a c ), then exist s ∈ S such that xs ∈ S . { quoVN } Theorem 3.7.
Let A be a von Neumann hyperring and S ⊆ A a multiplicative set. Then S is a vonNeumann subgroup if, and only if, A/ m S is a von Neumann geometric hyperring. In particular, S is vonNeumann subgroup if, and only if, S also is. roof. Let π : A → A/ m S be the natural projection. ⇒ : Since A/ m S is von Neumann hyperring, by 3 . x + x c = { } for all x ∈ A/ m S idempotent. Let x ∈ A such that π ( x ) is an idempotent. Then π ( i ( x )) = i ( π ( x )) = π ( x ). Thusgiven π ( z ) ∈ π ( x ) + π ( x ) c = π ( i ( x )) + π ( i ( x ) c ), exists u, v, w ∈ S such that zu ∈ i ( x ) v + i ( x ) c w and so z ∈ D S ( i ( x ) , i ( x ) c ). Since S is a von Neumann subgroup, π ( z ) = 1. Therefore A/ m S is a von Neumanngeometric hyperring. ⇐ : Let a ∈ A idempotent and take x ∈ D S ( a, a c ). Then exists s , s , s ∈ S such that xs ∈ as + a c s .Then π ( x ) ∈ π ( a ) + π ( a ) c but since A/ m S is geometric, π ( x ) = 1 and so exist s ∈ S such that xs ∈ S .The particular conclusion about S follows by Proposition 1 . { nonGeometric } Example 3.8.
The Marshall quotient usually not preserve geometric hyperrings. Let A = R × R and S = { , (2 , , (2 , ) , . . . } . Let x = (2 ,
3) and a = (1 , a = a is idempotent and a c = (0 , x = a (2 , ) + a c (2 ,
3) but there is no s ∈ S such that xs ∈ S and so A/ m S is von Neumannhyperring which is not a mono-hyperring. Remark 10. If A be a von Neumann GH and S ⊆ Id( A ) is a multiplicative set (or more generaly if S isa multiplicative set such that for all a, b ∈ S exist c ∈ S such that ac = bc ), then A/ m S is also a geometrichyperring. If A is a von Neumann hyperring and S is a general multiplicative set it is possible to find thesmaller S ′ multiplicative such that S ⊆ S ′ and A/ m S ′ is geometric. This is the content of the next result.Let A be a von Neumann hyperring. Consider the set S u := [ { e + · · · + e n : { e , . . . , e n } is a partition of unity } . By Lemma 3 .
4, it is a upward-union closed by multiplication. { geoHullvN } Corollary 3.9.
Let A be a von Neumann hyperring. Then A/ m S u is a von Neumann geometric hyperringand given a morphism f : A → B such that B is a mono-multiring, exist unique map f : A/ m S u → B suchthat A A/ m S u B f f is a commutative diagram. Proof.
In order to show that A/ m S is geometric, we prove that S is a von Neumann subgroup. Let a ∈ A idempotent and take x ∈ D S ( a, a c ). Then exists s , s , s ∈ S such that xs ∈ as + a c s . Since S u isan upward-union, exist a partition of unity { e , . . . , e n } such that s , s , s ∈ e + · · · + e n . Then xs ∈ ae + · · · + ae n + a c e + · · · a c e n ⊆ S . Thus S is a von Neumann subgroup.To conclude the desired universal property, we have to prove that for all map f : A → B , B mono-multiring, f ( S ) = { } . Let x ∈ S and { e , . . . , e n } partition of unity with x ∈ e + · · · + e n . Then f ( x ) ∈ f ( e ) + · · · + f ( e n ). Since { f ( e ) , . . . , f ( e n ) } is a partition of unity in B and it is mono-multiring, wehave f ( x ) = 1.Let A be a geometric von Neumann hyperring and a ∈ A . Given x, y ∈ a − a c , then xi ( a ) = a = yi ( a )and xi ( a ) c = − yi ( a ) c . Thus x ∈ xi ( a ) + xi ( a ) c = yi ( a ) + yi ( a ) c = { y } . Therefore, we denote ∇ ( a ) theunique element in a − a c . { nablai } Proposition 3.10.
Let A be a geometric von Neumann hyperring and let a ∈ A .i) ∇ ( a ) ∈ A × and a = i ( a ) ∇ ( a ).ii) If f : A → B is a morphism of GvNH, given a ∈ A , then ∇ ( f ( a )) = f ( ∇ ( a )).iii) Given x, y ∈ A , x = y if, and only if, ∇ ( i ( x )) = ∇ ( i ( y )) and ∇ ( x ) = ∇ ( y ). In particular, if e, f ∈ A areidempotents, then ∇ ( e ) = ∇ ( f ) ⇔ e = f . roof. i) It is enough to observe that given b ∈ A with a = a b , then ab is an idempotent and D ( a ) = D ( ab ).ii) We know that i ( a ) c is defined to be the unique element x such that x ∈ − i ( a ) and xi ( a ) = 0 (remark1). Then taking some b ∈ A with a = a b , we know that i ( a ) = ab and so xi ( a ) = 0 if, and only if, xa = 0.iii) Since ∇ ( a ) ∈ a − a c , given p ∈ D ( a ), we have a c ∈ p and so ∇ ( a ) / ∈ p ; if p ∈ D ( a ) c = D ( a c ) we alsoconclude that ∇ ( a ) / ∈ p . Thus by Proposition 2 . ∇ ( a ) ∈ A × . To see that a = i ( a ) ∇ ( a ), note that since ∇ ( a ) ∈ a − a c , we have ∇ ( a ) i ( a ) ∈ ai ( a ) − a c i ( a ) = { ai ( a ) } = { a } .iv) Since ∇ ( a ) ∈ a − a c , we have f ( ∇ ( x )) ∈ f ( a ) − f ( a c ) = f ( a ) − f ( a ) c and so ∇ ( f ( a )) = f ( ∇ ( a )).v) ⇒ : Immediate. ⇐ : Assume that ∇ ( i ( x )) = ∇ ( i ( y )) and ∇ ( x ) = ∇ ( y ). We only have to prove that if e, f ∈ A areidempotents with ∇ ( e ) = ∇ ( f ), then e = f because this imples in our case that i ( x ) = i ( y ) and so x = i ( x ) ∇ ( x ) = i ( y ) ∇ ( y ) = y . So assume ∇ ( e ) = ∇ ( f ) , e, f ∈ Id( A ). Since A is geometric, it is enoughto show that for all p ∈ spec( A ) e = f in K A ( p ). Fixed a prime ideal p , if e = 1 in K A ( p ), then1 = ∇ ( e ) = ∇ ( f ) and thus f = 1 = e in K A ( p ); if e = 0 in K A ( p ), then − ∇ ( e ) = ∇ ( f ) and thus f = 0 = e in K A ( p ). { GvNHR } Proposition 3.11.
Let A be a multiring. The following are equivalent:i) A is real reduced multiring and for each a ∈ A , exists x ∈ A such that ax = 0 and x ∈ − a .ii) A is real reduced hyperring.iii) A is a geometric von Neumann hyperring and for all a ∈ A , 1 + a = { } . Proof. i ) ⇒ ii ): Let d ∈ ba + ca . We have to prove that exists f ∈ b + c such that d = f a . By hypothesis,exists x ∈ − a such that ax = 0. Let l ∈ b + c and consider f ∈ lx + da .Using the Corollary 1 .
20, the proof is completed if we prove that for all σ ∈ sper( A ), σ ( f ) ∈ σ ( b ) + σ ( c )and σ ( d ) = σ ( f ) σ ( a ). Fix σ ∈ sper( A ). Note that if σ ( a ) = 0, then σ ( x ) ∈ − σ ( a ) = { } and if σ ( a ) = 0,0 = σ ( x ) σ ( a ) = σ ( x ) σ ( a ) = σ ( x ) and σ ( a ) = 1. • σ ( f ) ∈ σ ( b ) + σ ( c ).If σ ( a ) = 0, then σ ( x ) = 1 and σ ( f ) ∈ σ ( lx ) + σ ( da ) = { σ ( l ) } ⊆ σ ( b ) + σ ( c ). If σ ( a ) = 0, then σ ( x ) = 0and thus σ ( f ) ∈ σ ( lx ) + σ ( da ) = { σ ( da ) } ∈ σ ( b ) σ ( a ) + σ ( c ) σ ( a ) = σ ( b ) + σ ( c ). • σ ( d ) = σ ( f a ).If σ ( a ) = 0, then σ ( d ) ∈ σ ( ba ) + σ ( ca ) = { } and thus σ ( d ) = 0 = σ ( f a ). If σ ( a ) = 0, then σ ( x ) = 1and σ ( f a ) ∈ σ ( lxa ) + σ ( da ) = { σ ( d ) } because ax = 0. ii ) ⇒ iii ): Since A is a RRM, then in the language of real semigroup A is a Real Semigroup (see [ RRM ])and, by Theorem III.1.1 in [
DP2 ] A is a geometric. On the other hand, by Proposition 3 . A is an vonNeumann hyperring. iii ) ⇒ i ): First we prove that A is a real reduced multiring. • ∀ a ∈ A , a = a .By hypothesis, we have 1 + ∇ ( a ) − = { } . Thus {∇ ( a ) } = ∇ ( a ) (1 + ∇ ( a ) − ) = 1 + ∇ ( a ) = { } . Thus a = ( i ( a ) ∇ ( a )) = i ( a ) ∇ ( a ) = a . • a + ab = { a } .Since A is an hyperring, by hypothesis a + ab = a (1 + b ) = { a } . If x, y ∈ a + b , then x = y .First note that, since a , b ∈ Id( A ), we have i ( a ) = a . Note also that xa , ya ∈ a + b a = { a } xa c , ya c ∈ b a c = { b a c } . Thus xa = ya , xa c = ya c . Since a + a c = { } , x ∈ x ( a + a c ) = xa + xa c = ya + ya c = y ( a + a c ) = { y } . To complete the proof, note that given a ∈ A , the element a c satisfies aa c = 0 and a c ∈ − i ( a ) = 1 − a . Lemma 3.12.
Let A be von Neumann hyperring and T ⊆ A a proper preorder. Then 1 + T ⊆ A is a vonNeumann subgroup. Proof.
Let a ∈ A be an idempotent and u ∈ a + a c . First note that by Proposition 2 . , iii ) u is invertible.Observing that u ∈ P A ⊆ T by hypothesis and u − = u − u ∈ T , the expression (1 + u − ) u = 1 + u assuresthe existence of t ∈ T such that ut ∈ T .Let x ∈ D T ( a, a c ). Then exists s, u, v ∈ T such that xs ∈ au + va c . Taking u ′ , v ′ ∈ T such that u ∈ u ′ , v ∈ v ′ , we have xs ∈ au + a c v ⊆ ( a + au ′ ) + ( a c + a c v ′ ) = ( a + a c ) + ( au ′ + a c v ′ ) . Then exists t ′ ∈ T such that xst ′ ∈ t ′ + au ′ t ′ + a c v ′ t ′ ∈ T . Thus 1 + T ⊆ A is von Neumannsubgroup. { reprevN } Theorem 3.13.
Let A be von Neumann hyperring and T ⊆ A a proper preorder. Then the natural projection π : A → Q T ( A ) induces an isomorphism Q T ( A ) ∼ = A/ m (1 + T ) .Proof. Let ρ : A → A/ m (1 + T ) be the natural projection. By the preceding Lemma, 1 + T ⊆ A is vonNeumann subgroup and thus by Theorem 3 . A/ m (1 + T ) is GvNH. On the other hand, given x, a ∈ A suchthat ρ ( x ) ∈ ρ ( a ) in A/ m (1 + T ), exists t , t , t ∈ T with xt ∈ t + a t ⊆ T and thus ρ ( x ) = 1.Therefore, by Proposition 3 .
11, the hyperring A/ m (1 + T ) is RRM. To see that Q T ( A ) ∼ = A/ m (1 + T ) wewill prove that ρ : A → A/ m (1 + T ) satisfies the same universal property of π : A → Q T ( A ) (Theorem1 . f : A → R a multiring morphism with R RRM such that f ( T ) ⊆ P R = R = Id ( R ). Then f (1 + T ) ⊆ Id ( R ) = { } and thus by Proposition 1 .
11 exists unique f : A/ m (1 + T ) → R such that f = f ◦ ρ , as desired.As the last result of this section, we analyse some constructions that Q : pvNH → RRM preserve inorder to deduce a logical-preservation result (Theorem 3 . h M i : { µ i,j : i ≤ j in I }i be an inductive system of multrings and µ i : M i → lim −→ M the inclusionsmorphisms. For each i ∈ I , let S i ⊆ M i be a subset. We denote by lim −→ S i := S i ∈ I µ i ( S i ). If each S i ⊆ A i is multiplicative and µ i,j ( S i ) ⊆ S j when i ≤ j in I , then h M i / m S i : { µ i,j : i ≤ j in I }i is also an inductivesystem, where µ i,j : M i / m S i → M j / m S j is the induced map. { marshalQl } Lemma 3.14. i) Let h M i : { µ i,j : i ≤ j in I }i be an inductive system of multrings and S i ⊆ M i multi-plicative set for each i ∈ I with µ i,j ( S i ) ⊆ S j when i ≤ j in I . Then lim −→ ( M i /S i ) ∼ = (lim −→ M i ) / m lim −→ S i .ii) Let { A i } i ∈ I be a family of multirings and S i ⊆ A i a multiplicative set for each i ∈ I . Then Q i ∈ I ( A i / m S i ) ∼ =( Q i ∈ I A i ) / m ( Q i ∈ I S i ). Proof.
The proof can be found in [
RRM ]. et L pMulti = L multi ∪ { T } = { π, · , − , , , T } the language of p-multirings where T is a unary relation.Note that the pre-order axioms are geometric and thus exist inductive limits in pMulti . Consider thecategory pvNH as faithfully subcategory of pMulti of pairs ( A, T ) where A is a von Neumann hyperring.Let { ( A i , T i ) } i ∈ I be a family of p-multirings. Then ( Q i ∈ I A i , Q i ∈ I T i ) is the product of the family { ( A i , T i ) } i ∈ I in the category pMulti . Since arbitrary product of von Neumann hyperring is again vNHand pvNH is faithfully subcategory of pMulti , if the original family { ( A i , T i ) i ∈ I } is in pvNH , then( Q i ∈ I A i , Q i ∈ I T i ) is the product in pvNH .Another useful construction in pMulti is inductive limit. Let h ( A i , T i ) : { µ i,j : i ≤ j in I }i be an in-ductive system of p-multirings. Then the lim −→ i ∈ I ( A i , T i ) exists in pMulti because the first-order axioms ofp-multrings are geometric. Furtheremore, we have lim −→ ( A i , T i ) ∼ = (lim −→ A i , lim −→ T i ). As in product case, thesame considerations abou inductive limits in pMulti holds in pvNH .From now on we prove that the functor Q : pvNH → RRM preserves elementar equivalence (twostructures are elementar equivalent if they satisfies the same first-order sentences). We need the classi-cal Keisler-Shelah Theorem that gives an algebraic characterization of elementar equivalence in terms ofultrapowers.
Fact 3.15 (Keisler-Shelah Theorem’s) . Let L be a first-order language and A, B two models of L . Then A ≡ B if, and only if, exists set I and ultrafilter U over I such that Q i ∈ I A/U ∼ = Q i ∈ I B/U . Proof.
The proof can be found in [ CK ]. { ultrap } Remark 11.
Let L be a first-order language and { A i } i ∈ I a family of L -models. If U is an ultrafilter over I ,then the ultraproduct Q i ∈ I A i /U is an inductive limit of products. More precisely, define F : U → L− modelsby F ( A ) = Q i ∈ A A i and if A ⊆ B in U , F B,A : F ( B ) → F ( A ) is the natural projection. Then F is acontravariant functor over a left-directed set and lim −→ F ∼ = Q i ∈ I A i /U (see more details in Propostion 1 .
13 or[ CK ]). { theEleE } Theorem 3.16.
Let ( A, T ) , ( B, P ) be von Neumann p-hyperrings. If ( A, T ) ≡ L pMulti ( B, P ) , then Q T ( A ) ≡ L multi Q P ( B ) .Proof. In order to prove that Q : pvNH → RRM preserves elementar equivalence, by Keisler-Shelah Theo-rem and Remark 11 we have to shows that Q preserves inductive limit and arbitrary products. Inductive Limits Let h ( A i , T i ) : { µ i,j : i ≤ j in I }i be an inductive system of pvNH. Note that 1 + lim −→ T i = lim −→ (1 + T i ).Using the Theorem 3 .
13 and Lemma 3 .
14, we have Q (lim −→ ( A i , T i )) ∼ = Q (lim −→ A i , lim −→ T i ) ∼ = lim −→ A i / m (1 + lim −→ T i )= lim −→ A i / m lim −→ (1 + T i ) ∼ = lim −→ A i / m (1 + T i ) ∼ = lim −→ Q ( A i , T i ) . Products:
Let { ( A i , T i ) } i ∈ I be a family of pvNH. Then by Theorem 3 .
13 and Lemma 3 . Q ( Y i ∈ I ( A i , T i )) ∼ = Q ( Y i ∈ I A i , Y i ∈ I T i ) ∼ = ( Y i ∈ I A i ) / m (1 + Y i ∈ I T i )= ( Y i ∈ I A i ) / m Y i ∈ I (1 + T i ) ∼ = Y i ∈ I A i / m (1 + T i ) ∼ = Y i ∈ I Q ( A i , T i ) . Another way to prove this is to observe that the functor Q : pMulti → RRM is left-adjoint to the inclusionfunctor
RRM → pMulti by Theorem 1 .
22 and thus preserve inductive limits. et A be a von Neumann hyperring. For each x ∈ P A , consider the number P ( x ) = min { n ∈ N : exists x , . . . , x n ∈ A such that x ∈ x + . . . + x n } . Also consider P ( A ) = sup { n ( x ) : x ∈ X A } called the Pythagorean number of A . It is possible to prove that P ( A ) = sup { P ( K A ( q )) : q ∈ spec( A ) } and P ( A ) < ∞ if, and only if, P ( K A ( q )) < ∞ for every q ∈ spec( A ). Corollary 3.17.
Let
A, B semi-real von Neumann hyperrings and assume P ( A ) , P ( B ) < ∞ . If A ≡ L multi B ,then Q ( A ) ≡ L multi Q ( B ). Proof.
Since P ( A ) < ∞ , the preorder the set P A is definible by finitary first-order formuluas in thelanguage L multi (the same true for B ). Thus if A ≡ L multi B , then ( A, P A ) ≡ L pMulti ( B, P B ) and byTheorem 3 .
16 we have Q ( A ) ≡ L multi Q ( B ). In this section, it is built a geometric von Neumann hyperring hull for every multiring. The intention is togeneralize the von Neumann hull for rings ([ AM ]). Definition 4.1.
Let A be a multiring. A function f : spec( A ) → F p ∈ spec ( A ) K A ( p ) is called a constructiblesection if for all p ∈ spec( A ), f ( p ) ∈ K A ( p ) and given p ∈ spec( A ), exists clopen U ⊆ (spec( A )) const and x, y ∈ A such that • p ∈ U ⊆ D ( y ). • For all q ∈ U , f ( q ) = xy ∈ K A ( q ).The set of all constructible sections are denoted by V ( A ) and v A : A → V ( A ) is the natural map givenby v A ( a )( p ) = a ∈ K A ( p ) for every p ∈ spec( A ) and a ∈ A . Remark 12.
Let A be a multiring and f ∈ V ( A ) a constructible section. Note that since (spec( A )) cons iscompact, exists U , . . . , U n ⊆ (spec( A )) const clopens and x i , y i ∈ A , i = 1 , . . . , n , such that. spec( A ) = S ni =1 U i , U i ⊆ D ( y i ).. For all q ∈ U i , f ( q ) = x i y i ∈ K A ( q ).More than that, if we substitute U i by U ′ i = U i ∩ U c ∩ · · · ∩ U ci − , we can assume that the family { U , . . . , U n } is disjoint. { intHull } Proposition 4.2.
Let A is a geometric von Neumann hyperring, then v A : A → V ( A ) is a bijection. Proof.
Since A is geometric, it follows by Theorem 3 . v A is injective. On the other hand, given f ∈ V ( A ), exists U , . . . , U n ⊆ spec( A ) clopens and x i , y i ∈ A such that. spec( A ) = F ni =1 U i and U i ⊆ D ( y i ).. For all p ∈ U i , f ( p ) = x i y i ∈ K A ( p ).Because A is von Neumann, exists e i ∈ A idempotent such that U i = D ( e i ). Consider the element a ∈ e x ∇ ( y ) − + · · · + e n x n ∇ ( y n ) − . Then for all p ∈ U i , in K A ( p ) we have v A ( a )( p ) = x i ∇ ( y i ) − = x i y i = f ( p ).Thus v A ( a ) = f . he preceding Proposition suggest that V ( A ) can be a candidate do von Neumann hull. Consider in V ( A )the product punctually defined and the multivaluated sum defined in a similar manner: given f, g, h ∈ V ( A ) f ∈ g + h ⇔ for every p ∈ spec( A ) f ( p ) ∈ g ( p ) + h ( p ) in K A ( p )First we need to prove that V ( A ) is a multiring and in the sequence that it is von Neumann hyperring. { lemma1 } Lemma 4.3.
Let A be a multiring and x i , y i ∈ A , i = 1 , ,
3. Given p ∈ D ( y y y ) such that x y ∈ x y + x y in K A ( p ), exist a clopen U ⊆ D ( y y y ) with p ∈ U such that for all q ∈ U , x y ∈ x y + x y in K A ( q ). Proof. If x y ∈ x y + x y in K A ( p ) = ff( A/p ), by definition exists s / ∈ p and t ∈ p such that x ( y y ) s ∈ x ( y y ) s + x ( y y ) s + t in A. Let U = D ( y y y ) ∩ D ( s ) ∩ D ( t ) c . Then U ⊆ (spec( A )) const is clopen with p ∈ U and by definition forall q ∈ U we have x y ∈ x y + x y in K A ( q ). Theorem 4.4.
Let A be a multiring. Then V ( A ) is a multiring and v A : A → V ( A ) a multiring morphism.Proof. It is not difficult to see that V ( A ) satisfies all multiring axiom except the associative property. Solet f, g, h, z, t ∈ V ( A ) with z ∈ f + t, t ∈ g + h. We should find t ′ ∈ f + g such that z ∈ t ′ + h . For each p ∈ spec( A ), since z ( p ) ∈ f ( p ) + t ( p ) , t ( p ) ∈ g ( p ) + h ( p ) in K A ( p ), exist x, y ∈ A such that p ∈ D ( y ) and z ( p ) ∈ xy + h ( p ) , xy ∈ f ( p ) + g ( p ) in K A ( p ) . By the Lemma 4 .
3, exist clopen U ⊆ (spec( A )) const with p ∈ U such that for all q ∈ U , z ( q ) ∈ xy + h ( q ) , xy ∈ f ( q ) + g ( q ) in K A ( q ). Since (spec( A )) const is Boolean, exists U , . . . , U n clopens and x i , y i ∈ A , i = 1 , . . . , n such that. spec( A ) = S ni =1 U i , with U i ⊆ D ( y i ).. For each p ∈ U i , z ( p ) ∈ x i y i + h ( p ) , x i y i ∈ f ( p ) + g ( p ) in K A ( p ).Substituing U i by U i ∩ U c ∩ · · · ∩ U ci − , we can assume that U i ∩ U j = ∅ for i = j . Define the function t ′ : spec( A ) → F p ∈ spec ( A ) K A ( p ) given byIf p ∈ U i , then t ′ ( p ) = x i y i . Then t ′ ∈ V ( A ) and z ∈ t ′ + h , with t ′ ∈ f + g .Given f ∈ V ( A ), consider the functions i ( f ) , ∇ ( f ) , f c : spec( A ) → F p ∈ spec ( A ) K A ( p ) given by • i ( f )( p ) = ( f ( p ) = 00 if f ( p ) = 0. ∇ ( f )( p ) = ( f ( p ) if f ( p ) = 0 − f ( p ) = 0. • f c ( p ) = ( f ( p ) = 01 if f ( p ) = 0. { lemma3 } Lemma 4.5.
Let A be a multiring and f, g ∈ V ( A ). Theni) i ( f ) , ∇ ( f ) , f c ∈ V ( A ) and i ( f ) = i ( f ) , ( f c ) = f c , ∇ ( f ) ∈ V ( A ) × .ii) f c = i ( f ) c .iii) f f c = 0 and f c ∈ − i ( f ).iv) If f = f , then f g + f c g = { g } .v) f = ∇ ( f ) i ( f ). Proof.
Straightforward.
Theorem 4.6.
Let A be a multiring. Then V ( A ) is a geometric von Neumann hyperring.Proof. First we prove that V ( A ) is an hyperring. Let x ∈ f g + f h . We need to find y ∈ g + h such that x = f y . Let l ∈ g + h and take y ∈ f c l + ∇ ( f ) − x. Note that since f c x ∈ f c f g + f c f h = { } , we have by Lemma 4 . f c x ∈ x − xi ( f ) and so x = xi ( f ). Then yf = i ( f ) ∇ ( f ) − xf = i ( f ) x = x . On the other hand, we have f c l ∈ f c g + f c h and ∇ ( f ) − ∈ ∇ ( f ) − f g + ∇ ( f ) − f h = i ( f ) g + i ( f ) h . Then y ∈ ( f c g + f c h ) + ( i ( f ) g + i ( f ) h ) = ( f c g + i ( f ) g ) + ( f c h + i ( f ) h ) = g + h. Therefore, V ( A ) is an hyperring. Lastly, V ( A ) is a geometric von Neumann hyperring because given f, g ∈ V ( A ) , g = g , we have by Lemma 4 .
5. ( i ( f ) ∇ ( f ) − ) f = ( i ( f ) ∇ ( f ) − )( i ( f ) ∇ ( f )) = i ( f ) ∇ ( f ) = f .. g + g c = { } { remarktopAlg } Remark 13.
Since { D ( a ) ∩ D ( b ) c ∩ · · · ∩ D ( b n ) c : n ≥ , a, b , . . . , b n ∈ A } is base of (spec( A )) const , given f ∈ V ( A ) and p ∈ spec( A ), exists a, b , . . . , b n , x, y ∈ A such thati) p ∈ D ( a ) ∩ D ( b ) c ∩ · · · ∩ D ( b n ) c ⊆ D ( y ) . ii) For all q ∈ D ( a ) ∩ D ( b ) c ∩ · · · ∩ D ( b n ) c , f ( q ) = xy in K A ( q ).Furthermore assuming that D ( a ) ∩ D ( b ) c ∩ · · · ∩ D ( b n ) c ⊆ D ( y ), the item ii ) is equivalent to v A ( y ) f i ( v A ( a )) v A ( b ) c · · · v A ( b n ) c = v A ( x ) i ( v A ( a )) v A ( b ) c · · · v A ( b n ) c . univ(f) } Lemma 4.7.
Let f : A → B a multiring morphism. Let g, h : V ( A ) → V ( B ) two morphism such that A BV ( A ) V ( B ) fv A v B gh is a commutative diagram. Then g = h . Proof.
Let t ∈ V ( A ) and p ∈ spec( B ). We need to prove that g ( t )( p ) = h ( t )( p ). Let q = f − ( p ). Since t isa section, exists a, b , . . . , b n , x, y ∈ A such that. q ∈ D ( a ) ∩ D ( b ) ∩ · · · ∩ D ( b n ) c ⊆ D ( y ).. For all o ∈ D ( a ) ∩ D ( b ) c ∩ · · · D ( b n ) c , t ( o ) = xy ∈ K A ( o ).The last item implies that v A ( y ) ti ( v A ( a )) v A ( b ) c · · · v A ( b n ) c = v A ( x ) i ( v A ( a )) v A ( b ) c · · · v A ( b n ) c . Notethat, since g is a morphism, g ( i ( v A ( a ))) = i ( g ( v A ( a ))) = i ( v B ( f ( a ))) and g ( v A ( b i ) c ) = g ( v A ( b i )) c = v B ( f ( b i )) c for i = 1 , . . . , n . Thus v B ( f ( y )) g ( t ) i ( v B ( f ( a ))) v B ( f ( b )) c · · · v B ( f ( b n )) c = g ( v A ( y )) g ( t ) g ( i ( v A ( a ))) g ( v A ( b ) c ) · · · g ( v A ( b n ) c )= g ( v A ( y ) ti ( v A ( a )) v A ( b ) c · · · v A ( b n ) c )= g ( v A ( x ) i ( v A ( a )) v A ( b ) c · · · v A ( b n ) c )= v B ( f ( x )) i ( v B ( f ( a ))) v B ( f ( b )) c · · · v B ( f ( b n )) c . By the remark 13, the above equality is equivalent to. For all o ∈ D ( f ( a )) ∩ D ( f ( b )) c ∩ · · · ∩ D ( f ( b n )) c , g ( t )( o ) = f ( x ) f ( y ) ∈ K B ( o ).In particular, since p ∈ D ( f ( a )) ∩ D ( f ( b )) c , we have g ( t )( p ) = f ( x ) f ( y ) . Applying the same argument for h ,we conclude that h ( t )( p ) = f ( x ) f ( y ) = g ( t )( p ). Since p ∈ spec( A ) was arbitrary, g ( t ) = h ( t ). Thus g = h . { uniPGvNH } Theorem 4.8.
Let
A, B be multirings.i) If f : A → B is a multiring morphism, then exist unique morphism V ( f ) : V ( A ) → V ( B ) such that A BV ( A ) V ( B ) fV ( f ) is a commutative diagram. Furthermore, V : Multi → GvNH is a functor.ii) Given a morphism f : A → B , where B is a GvNH, exist unique morphism f : V ( A ) → B such that A V ( A ) B v A f f is a commutative diagram. In other words, the functor V : Multi → GvNH is a left-adjoint functor tothe inclusion functor
GvNH → Multi . roof. i) Given morphism f : A → B , consider the induced map spc( f ) : spec( B ) → spec( A ) and given p ∈ spec( B ), the morphism K A (spc( f )( p )) → K B ( p ) (also denoted by f ). Given a ∈ V ( A ) function a : spec( A ) → F q ∈ spec ( A ) K A ( q ), consider f ( a ) : spec( B ) → F p ∈ spec ( B ) K B ( p ) defined by f ( a )( p ) = f ( a (spc( f )( p ))). Note that f ( a ) ∈ V ( B ), that is, it is a section. In fact, given p ∈ spec( B ), existsclopen U ⊆ spec( A ) and x, y ∈ A with. spc( f )( p ) ∈ U and U ⊆ D ( y ).. For all q ∈ U , a ( q ) = xy ∈ K A ( q ).Then V = spc( f ) − ( U ) ⊆ spec( B ) is a clopen and. p ∈ V and V ⊆ D ( f ( y )).. For all q ∈ V , f ( a )( q ) = f ( x ) f ( y ) ∈ K B ( q ).Thus we have a function f : V ( A ) → V ( B ). It is easy to see that f (1) = 1 , f ( −
1) = −
1. Furthermore,given a, b ∈ V ( A ), we have for p ∈ spec( B ) f ( ab )( p ) = f (( ab )(spc( f )( p ))) = f ( a (spc( f )( p )) b (spc( f )( p )))= f ( a (spc( f )( p ))) f ( b (spc( f )( p ))) = f ( a )( p ) f ( b )( p ) . Thus f ( ab ) = f ( a ) f ( b ). Also if a ∈ b + c in V ( A ), given p ∈ spec( B ), since a (spc( f )( p )) ∈ b (spc( f )( p )) + c (spc( f )( p )) in the hyperfield K A (spc( f )( p )), we have f ( a )( p ) ∈ f ( b )( p ) + f ( c )( p ) in the hyperfield K B ( p ). Thus f ( a ) ∈ f ( b ) + f ( c ) in V ( B ). Therefore, f : V ( A ) → V ( B ) is a multiring morphism. Theuniqueness of f is stated and proved in the Lemma 4 . Claim. If A is a geometric von Neumann hyperring, then v A : A → V ( A ) is an isomorphism. Proof.
By Proposition 4 . v A : A → V ( A ) is a bijection. On the other hand, by Theorem 3 .
5, given a, b, c ∈ A , a ∈ b + c if, and only if, v A ( a ) ∈ v A ( b ) + v A ( c ). Thus v A is an isomorphism.Now if f : A → B is a morphism, B GvNH, then consider v ( f ) : V ( A ) → V ( B ). By the claim above, v B isisomorphism and we can consider f = v − B ◦ v ( f ). Note that f ◦ v A = v − B ◦ ( v ( f ) ◦ v A ) = v − B ◦ ( v B ◦ f ) = f .If f ′ : V ( A ) → B is another morphism with f ′ ◦ v A = f , then v B ◦ f ′ ◦ v A = v B ◦ f . Thus by Lemma4 . v B ◦ f ′ = v ( f ) and so f ′ = v − B ◦ v ( f ) = f . { algV(A) } Theorem 4.9.
Let A be a multiring and consider v : A → V ( A ) .i) The induced map spec ( V ( A )) → ( spec ( A )) const is homeomorphism.ii) The induced map sper ( V ( A )) ∼ = ( sper ( A )) const is homeomorphism.iii) Let p ∈ spec ( V ( A )) and q = v − ( p ) . The induced map K A ( q ) → K V ( A ) ( p ) is isomorphism.Proof. i) Using the characterization of prime ideals as morphism to Krasner hyperfield K = { , } , theuniversal property in Theorem 4 . V ( A )) → spec( A ) is a bijection.But by since V ( A ) is von Neumann hyperring, the topological space sper( V ( A )) is Boolean and thussper( V ( A )) → (sper( A )) const is homeomorphism.ii) Using the hyperfield 3 = {− , , } , the proof follows similarly to item i ). ii) Consider the inclusions maps i : A → K A ( q ) and j : V ( A ) → K V ( A ) ( p ). By propositions 1 . .
8, if f, g : K A ( q ) → K , K hyperfield, with f ◦ i = g ◦ i , then f = g . An analogous property holds for themap j : V ( A ) → K V ( A ) ( p ). Consider the following commutative diagram: A V ( A ) K A ( q ) K V ( A ) ( p ) vi jv q Furtheremore, by Theorem 4 .
8, exist morphism f : V ( A ) → K A ( q ) such that f ◦ v = i . Note that q = i − (0) = v − ( f − (0)) and thus f − (0) = p because v − : spec( V ( A )) → spec( A ) is injective. Thenby Proposition 1 . f p : K V ( A ) ( p ) → K A ( q ) such that A V ( A ) K V ( A ) ( p ) K A ( q ) v i f j f p is a commutative diagram.Then, combining this with the preceding diagram, we have f p ◦ v q ◦ i = f p ◦ j ◦ v = f ◦ v = i . Then f p ◦ v q = Id K A ( q ) ; on the other hand, we have v q ◦ f p ◦ j ◦ v = v q ◦ f ◦ v = v q ◦ i = j ◦ v . Thus, byTheorem 4 . v q ◦ f p ◦ j = j and then v q ◦ f p = Id K V ( A ) ( p ) . { RRMHFF } Lemma 4.10.
Let A be a real reduced multiring. Then for all p ∈ spec( A ), K A ( p ) is a real reducedhyperfield. Proof.
Since for all x ∈ A , x = x , the same is true for K A ( p ); but because it is a hyperfield, if a ∈ K A ( p )is non-zero, then a = 1. On the other hande, given x ∈ K A ( p ), by definition exist i / ∈ p and x ′ ∈ A such that x ′ − x ∩ p = ∅ and x ′ i ∈ i + i . Then x ′ i = i and in K A ( p ) we have x = x ′ = 1. Thus K A ( p ) is anreal reduced hyperfield. { GvnHRRM } Proposition 4.11.
Let A be a multiring.i) If A is a real reduced multiring, then V ( A ) also is.ii) If A is a von Neumann hyperring, then Q ( A ) also is. Proof. i) For every p ∈ spec( V ( A )), we have by Theorem 4 . K V ( A ) ( p ) ∼ = K A ( q ), where q = v − A ( p ).Thus by Lemma 4 . K V ( A ) ( p ) is real reduced hyperfield. But since 1 + a = { } in K A ( p ) for every p ∈ spec( V ( A )), the same is true for V ( A ) because it is geometric. The Proposition 3 .
11 implies that V ( A ) is real reduced hyperring.ii) The result is a direct consequence of Theorem 3 .
13 proof because if A is a vNH, then Q ( A ) ∼ = A/ m P A is a GvNH. { QVVQ } Theorem 4.12.
Let A be a semi-real multiring and consider the morphisms π A : A → Q ( A ) , v A : A → V ( A ) and v ( π A ) : V ( A ) → V ( Q ( A )) . Then V ( A ) is semi-real and the induced map Q ( V ( A )) → V ( Q ( A )) isisomorphism. roof. Since A is semi-real, by Proposition 1 .
15 and Theorem 4 . ∅ 6 = (sper( A )) const ∼ = sper( V ( A ))and thus V ( A ) semi-real. Now, to prove the induced map Q ( V ( A )) → V ( Q ( A )) is an isomorphism, weproceed with several applications of universal properties described in Theorem 4 . .
22. Firstnote that since V ( Q ( A )) is real reduced multiring by Proposition 4 .
11, the universal property of V ( A ) → Q ( V ( A )) (Theorem 1 .
22) guarantee the existence of f : Q ( V ( A )) → V ( Q ( A )) such that V ( A ) Q ( A ) Q ( V ( A )) V ( Q ( A )) V ( Q ( A )) Q ( V ( A )) v ( π A ) π V ( A ) Q ( v A ) v Q ( A ) f g is a commutative diagram. On the other hand, considering the morphism Q ( v A ) : Q ( A ) → Q ( V ( A )), byProposition 4 .
11 we have that Q ( V ( A )) is a GvNH and thus the universal property of Q ( A ) → V ( Q ( A ))(Theorem 4 .
8) ensures the existence of g : V ( Q ( A )) → Q ( V ( A )) such that g ◦ v Q ( A ) = Q ( v A ).Now we have to prove that g ◦ f = Id Q ( V ( A )) and f ◦ g = Id V ( Q ( A )) . Note that by Theorem 4 . , i ), wehave v ( π A ) ◦ v A = v Q ( A ) ◦ π A and by Theorem 1 . , i ) Q ( v A ) ◦ π A = π V ( A ) ◦ v A . A Q ( A ) A V ( A ) V ( A ) V ( Q ( A )) Q ( A ) Q ( V ( A )) π A v A v Q ( A ) v A π A π V ( A ) v ( π A ) Q ( v A ) Thus we have ( g ◦ f ) ◦ π V ( A ) ◦ v A = g ◦ v ( π A ) ◦ v A = g ◦ v Q ( A ) ◦ π A = Q ( v A ) ◦ π A = π V ( A ) ◦ v A . Then by the uniqueness of the universal property of v A and π V ( A ) we have g ◦ f = Id Q ( V ( A )) . Similarargument shows that f ◦ g = Id V ( Q ( A )) and so V ( Q ( A )) ∼ = Q ( V ( A )).Let A be a ring. Then V ( A ) is also a ring. In fact, given f, g ∈ V ( A ) and x, y ∈ f + g , we have forall p ∈ spec( A ) that x ( p ) , y ( p ) ∈ f ( p ) + g ( p ) in K A ( p ); but since A is a ring, K A ( p ) is also a ring and thus x ( p ) = y ( p ). Then x = y in V ( A ). An interesting corollary of the preceding theorem is that if a RRM isrepresentable by a ring using a Marshall quociente, then it is canonically representable by a von Neumannregular ring. { represent } Corollary 4.13.
Let A be a semi-real ring. Assume that exists a ring B and S ⊆ B multiplicative subsetsuch that Q ( A ) ∼ = B/ m S . Then Q ( A ) ∼ = V ( A ) / m (1 + P V ( A ) ) is represented by the von Neumann regularring V ( A ). Proof.
Since the Marshall quotient preserves hyperring (Proposition 1 . Q ( A ) ∼ = B/ m S is a real reducedhyperring and by Proposition 3 . Q ( A ) is a geometric von Neumann Hyperring. Thus by Theorem 4 . .
12 we have Q ( A ) ∼ = V ( Q ( A )) ∼ = Q ( V ( A )). Using the representation given in Theorem 3 .
13, theconclusion Q ( A ) ∼ = V ( A ) / m (1 + P V ( A ) ) follows. In Theorem 4 .
9, some algebraic properties of von Neumann hull were stated and proved. It is interestingto note that they are enough to characterize the hull. In other words, given a multiring A and a morphism f : A → V where V is a geometric von Neumann hyperring satisfying i ) the induced map spec( V ) → (spec( A )) const is homeomorphism and iii ) for each p ∈ spec( V ), the induced map K A ( q ) → K V ( p ), where = f − ( p ), is an isomorphism, then V ∼ = V ( A ). Furthermore, the functor V : Multi → GvNH preservesfinite product and since it is a left-adjoint it also preserves inverse limits.In the present work, we prove that von Neumann hull of a RRM is again a RRM (Proposition 4 .
11) (vonNeumann RRM were studied in [
Mir1 ] through the equivalent notion of von Neumann RS). This will beused to give applications for quadratic forms. More precisely, the concept of Witt ring will be generalizedfrom special groups to real semigroups in [
RM1 ] and we will show that V ( R ), the von Neumann regularhull of a real semigroup R , can also be constructed from W ( R ), the Witt ring of R . This will imply thatthere is canonical isomorphism between the Witt rings W ( R ) and W ( V ( R )).The above relations allows the analysis of the Witt ring with tools available for RS von Neumann asthe description of the isometry of forms through a pp-formula and the characterization of the transversalrepresentation in terms of isometry. This will be used to provide calculation of the graded Witt ring, anaxiomatization of the Witt rings in a convenient language L ω ,ω and classification of categorical quotientsof Witt rings. These results will be presented in [ RM1 ] and [
RM2 ]. References
AM.
P.Arndt, H.L. Mariano,
The von Neumann-regular Hull of (preordered) rings and quadratic forms ,South American Journal of Logic (2016), 201-244. CK.
C.C. Chang, H.J. Kiesler,
Model Theory , Studies in Logic and the Foundations of Mathematics,vol. 73, 1990.
DM1.
M. Dickmann, F. Miraglia,
Special Groups: Boolean-Theoretic Methods in the Theoryof Quadratic Forms , Memoirs of the AMS , American Mathematical Society, Providence, USA,2000.
DM2.
M. Dickmann, F. Miraglia,
On Quadratic Forms whose total signature is zero mod n . Solution toa problem of M. Marshall , Inventiones Mathematicae (1998), 243-278. DM3.
M. Dickmann, F. Miraglia,
Representation of reduced special groups in algebras of continuousfunctions , Quadratic forms - algebra, arithmetic and geometry , Contemp. Math., Amer. Math.Soc., Providence, RI, 2009.
DM4.
M. Dickmann, F. Miraglia,
Quadratic form theory over preordered von Neumann-regular rings , J.Algebra (2008), 1696-1732.
DP1.
M. Dickmann, A. Petrovich,
Real Semigroups and Abstract Real Spectra , Contemporary Mathemat-ics AMS (2004), 99-119.
DP2.
M. Dickmann, A.Petrovich,
Real Semigroups, Real Spectra and Quadratic Forms over Rings
Mar1.
M. Marshall,
Abstract Witt Rings , Queen’s Papers in Pure and Applied Mathematics ,Queen’s University, Canada, 1980. Jun.
J. Jun,
Algebraic Geometry over Hyperrings , arXiv:1512.04837v1, 2015, 37pp.
Mar2.
M. A. Marshall,
Spaces of Orderings and Abstract Real Spectra , Lecture Notes in Mathe-matics , Springer-Verlag, Berlin, Germany, 1996.
Mar3.
M. Marshall,
Real reduced multirings and multifields , Journal of Pure and Applied Algebra (2006), 452-468.
Mir1.
F. Miraglia,
An Introduction to Partially Ordered Structures and Sheaves , PolimetricaScientific Publishers, Contemporany Logic Series , Milan, 2007. Mir2.
F. Miraglia,
Boolean and von Neumann Regular Real Semigroups , preprint, 2019, 23pp.
RM1.
H.R.O. Ribeiro, H.L. Mariano,
Witt rings for Real Semigroups , in preparation.
RM2.
H.R.O. Ribeiro, H.L. Mariano,
Hulls for Real Semigroups and applications , in preparation.
RM.
H.R.O. Ribeiro, K.M.A Roberto, H.L. Mariano,
Functorial relationship between multiringsand the various abstract theories of quadratic forms , submitted, preliminary version available athttps://arxiv.org/pdf/1610.00816.pdf, 2020, 38 pp., submitted, preliminary version available athttps://arxiv.org/pdf/1610.00816.pdf, 2020, 38 pp.