Germinal theories in Łukasiewicz logic

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Germinal theories in Łukasiewicz logic Leonardo Manuel Cabrer a , Daniele Mundici b,∗ a

IRIF, University Paris–Diderot and CNRS, France Department of Mathematics and Computer Science “Ulisse Dini”, University of Florence, Viale Morgagni 67/A, I-50134 Florence, Italy b

a r t i c l e

i n f o

Article history: Received 25 September 2014 Received in revised form 11 October 2016 Accepted 9 November 2016 Available online xxxx MSC: primary 06D35 secondary 03B50, 22F05, 37A45 Keywords: MV-algebra Łukasiewicz logic Affine group over the integers Rational polyhedron Farey regular simplex Orbit invariant

a b s t r a c t Differently from boolean logic, in Łukasiewicz infinite-valued propositional logic Ł∞ the theory Θmax,v consisting of all formulas satisfied by a model v ∈ [0, 1]n is not the only one having v as its unique model: indeed, there is a smallest such theory Θmin,v , the germinal theory at v, which in general is strictly contained in Θmax,v . The Lindenbaum algebra of Θmax,v is promptly seen to coincide with the subalgebra of the standard MV-algebra [0, 1] generated by the coordinates of v. The description of the Lindenbaum algebras of germinal theories in two variables is our main aim in this paper. As a basic prerequisite of independent interest, we prove that for any models v and w the germinal theories Θmin,v and Θmin,w have isomorphic Lindenbaum algebras iff v and w have the same orbit under the action of the affine group over the integers. © 2016 Elsevier B.V. All rights reserved.

1. Introduction and statement of the main results We assume familiarity with Łukasiewicz infinite-valued propositional logic Ł∞ and MV-algebras. The monographs [6] and [14] provide all necessary background. 1.1. Germinal theories, differential semantics Following [14, Definition 2.16], a theory Θ(X1 , . . . , Xn ) in Łukasiewicz infinite-valued propositional logic Ł∞ is a set of formulas in the variables X1 , . . . , Xn containing all n-variable tautologies and having the * Corresponding author. E-mail addresses: [email protected] (L.M. Cabrer), [email protected]fi.it (D. Mundici). http://dx.doi.org/10.1016/j.apal.2016.11.009 0168-0072/© 2016 Elsevier B.V. All rights reserved.

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additional property that any formula which is a syntactic consequence of Θ is already a member of Θ. It is also assumed that Θ does not coincide with the totality of formulas in the variables X1 , . . . , Xn . Syntactic consequence in Ł∞ is defined in [6, 4.3.2] and characterized in [14, 1.9]. While the semantics of classical propositional logic is {0, 1}-valued, the semantics of Ł∞ is [0, 1]-valued. Upon writing {X1 ,...,Xn }

v ∈ [0, 1]n = [0, 1]

,

every model, also known as a “valuation”, v: {X1 , . . . , Xn } → [0, 1] is identified with a point in the unit n-cube, [14, 1.2]. Differently from classical propositional logic, where each point p ∈ {0, 1}n determines a unique theory having p as its only model, given v ∈ [0, 1]n the Ł∞ -theory Θmax,v consisting of all formulas Ł∞ satisfied by v is not the only one having v as its unique model: indeed, there is a smallest such theory Θmin,v , the germinal theory at v, which in general is strictly contained in Θmax,v . It is not hard to see that the Lindenbaum algebra of Θmax,v is uniquely isomorphic to the subalgebra of the standard MV-algebra [0, 1] generated by the coordinates of v (see Section 1.2 for details). The description of the Lindenbaum algebra of Θmin,v is our main concern in this paper. To this purpose we will make use of the differential semantics for Ł∞ developed in [15]. In that paper, the classical (Bolzano–Tarski, [19, footnote on page 417]) definition of “formula φ is a semantic consequence of theory Θ” is refined by taking into account the effect on φ of the stability of the truth-value of all θ ∈ Θ under small perturbations of the models satisfying Θ. These perturbations make no sense in boolean logic, where models are points lying in totally disconnected compact Hausdorff spaces, and formulas stand for boolean functions. But they do in Ł∞ , where models of n-variable formulas are points in [0, 1]n , and formulas stand for [0, 1]-valued continuous piecewise linear functions defined on [0, 1]n , as explained in the following lines. Generalizing the definition of the boolean function associated to a formula in classical propositional logic, ˆ [0, 1]n → [0, 1] is defined by induction on the number of connectives for any Ł∞ -formula φ the function φ: i is the ith coordinate function xi : [0, 1]n → [0, 1], and in φ: one first stipulates that, for all i = 1, . . . n, X  = 1 − ψ and ψ  ⊕ χ = ψ ⊕ χ . The set {ψˆ | ψ an Ł∞ -formula in n variables} equipped with then writes ¬ψ the pointwise MV-algebraic operations of the standard MV-algebra [0, 1] is the free n-generator MV-algebra F reen . (See [6, 3.1.4], where ψˆ is denoted ψ [0,1] .) An n-variable McNaughton function is a continuous piecewise linear function f : [0, 1]n → [0, 1], where each linear piece has integer coefficients, [6, 3.1.6]. (The number of linear pieces is always finite. The adjective “linear” is used in the affine sense.) McNaughton’s theorem [6, 9.1.5] then shows that F reen coincides with the MV-algebra M([0, 1]n ) of all n-variable McNaughton functions over [0, 1]n , equipped with the pointwise operations of the standard MV-algebra [0, 1]. The differential properties of McNaughton functions—notably the existence of their directional derivatives at any point of [0, 1]n —yield a novel semantic consequence |=∂ that turns out to coincide with syntactic consequence  . This is the content of the (strong) completeness theorem for |=∂ , [15, 3.9]. As a corollary, we have the following characterization: ˆ = 1 and A formula φ(X1 , . . . , Xn ) belongs to Θmin,v iff the function φˆ associated to φ satisfies φ(v) n ˆ ∂ φ(v)/∂u = 0 for all 0 = u ∈ R . In this paper we will focus on two-variable formulas and theories. Then models are identified with points v ∈ [0, 1]2 . Any theory Θmin,v (X1 , X2 ) will be said to be 2-germinal, because of its tight relationship with germinal ideals, to be introduced in the next few lines.

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1.2. Maximal and germinal ideals Following [14, §4], for every n ∈ {1, 2, . . .} and x ∈ [0, 1]n , the ideals hx and ox of the free MV-algebra M([0, 1]n ) are defined by hx = {f ∈ M([0, 1]n ) | f (x) = 0}, and ox = {f ∈ M([0, 1]n ) | f = 0 on some open set U ⊆ [0, 1]n containing x}. The sets hx and ox are respectively known as the maximal ideal and the germinal ideal of M([0, 1]n ) at x. Lindenbaum algebras of n-germinal theories Θmin,v (resp., Θmax,v ) precisely arise as quotients of M([0, 1]n ) by their germinal ideals ov (resp., their maximal ideals hv ). In particular, by [6, 4.2.7],  | ψ ∈ Θmin,v } = ov and conversely, {¬ψ | ψˆ ∈ ov } = Θmin,v . {¬ψ

(1)

For the sake of notational simplicity, throughout this paper we will work with quotient MV-algebras of germinal ideals rather than with their isomorphic copies given by Lindenbaum algebras of germinal theories. Among others, using [14, 4.9] one immediately sees that Θmin,v and Θmax,v are indeed the smallest and the largest theory whose only model is v. The quotient MV-algebra M([0, 1]n )/hx is easily described. By [6, 3.4.7], the map x → hx sends points of [0, 1]n one–one onto maximal ideals of M([0, 1]n ). It is well known, [6, 3.5], [14, 4.5], that M([0, 1]n )/hx is uniquely isomorphic to the subalgebra of the standard MV-algebra [0, 1] generated by the coordinates of x. In more detail, for each x = (x1 , . . . , xn ) the map f /hx → f (x) determines a unique isomorphism between M([0, 1]n )/hx and the subalgebra of [0, 1] whose universe is the set (Zx1 + · · · + Zxn + Z) ∩ [0, 1]. Let the totally ordered abelian group Gx ⊆ R be defined by Gx = Zx1 + · · · + Zxn + Z.

(2)

Stated otherwise, Gx is the subgroup of R generated by the coordinates of x together with 1. Gx inherits the natural order of R. By [6, 3.5.1], every n-generator simple MV-algebra is isomorphic to M([0, 1]n )/hx m for some x ∈ [0, 1]n . By [14, 4.5], for any x ∈ [0, 1]n and y ∈ [0, 1] , the MV-algebras M([0, 1]n )/hx and m M([0, 1] )/hy are isomorphic iff Gx = Gy . Letting now Γ be the categorical equivalence [11, 3.9] between lattice-ordered abelian groups with a distinguished (strong order) unit—unital -groups for short—and MV-algebras, for each x ∈ [0, 1]n we can write M([0, 1]n )/hx ∼ = Γ(Gx , 1).

(3)

1.3. Lindenbaum algebras of germinal theories Quotients of the free n-generator MV-algebra by its germinal ideals, (i.e., by (1), Lindenbaum algebras of germinal theories) have a more complicated description already for n = 1. See Example 5.5 in [14]. The aim of this paper is to give a complete description of these MV-algebras for n = 2. We will work concurrently with the following three kinds of algebras:

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(i) The free n-generator MV-algebra M([0, 1]n ). (ii) The free n-generator -group FREEn , [1,2,12]. For n ≥ 1, FREEn is the -group of continuous piecewise homogeneous linear functions f : Rn → R with integer coefficients and the pointwise operations of R, (see [1] for this representation). In particular, FREE1 is uniquely isomorphic to Z2 . Trivially, FREE0 = {0}. The (canonical) free generators of FREEn are the coordinate functions πi : Rn → R, (i = 1, . . . , n). (iii) The unital -group MR ([0, 1]n ) of continuous piecewise linear functions f : [0, 1]n → R with integer coefficients, and the pointwise operations of the -group R. The distinguished unit of MR ([0, 1]n ) is the constant map 1. See [1,2,11]. See [13, §3] for the freeness properties of MR ([0, 1]n ). Let Gn = GL(n, Z)  Zn denote the n-dimensional affine group over the integers. In Section 2 (Theorem 1), for all n and all x, y in the interior int[0, 1]n of [0, 1]n , we prove that the Lindenbaum algebras of Θmin,x and Θmin,y are isomorphic iff x and y have the same Gn -orbit. In Section 3, to any x ∈ R2 we assign an integer cx ≥ 1 in such a way that the pair (Gx , cx ) completely classifies the orbit of x. In other words, for all y ∈ R2 (Gy , cy ) = (Gx , cx ) iff y = γ(x) for some γ ∈ G2 . This is Theorem 8 in Section 4. In particular, when rank(Gx ) ∈ {1, 3}, the group Gx alone provides a complete invariant classifier of the orbit of x, because cx turns out to have value 1. We further let the integer dx be defined by dx = max{k ∈ Z | 1/k ∈ Gx }.

(4)

2

Let  denote lexicographic product, [2]. For all x ∈ int [0, 1] , the Lindenbaum algebras of 2-germinal theories Θmin,x are classified in Section 5 as follows: rank(Gx ) 3

M([0, 1]2 )/ox Γ(Gx  FREE0 , u)

value of the unit u of Gx  FREEi , (i = 0, 1, 2) u = (1, 0)

2 (dx > 1)

Γ( Gx  FREE1 , u)

u = (1, cx π1 ∨ (cx − dx )π1 )

2 (dx = 1)

Γ( Gx  FREE1 , u )

u = (1, π1 ∨ −π1 )

1

Γ( Gx  FREE2 , u)

u = (1, (dx − 1)π1 ∨ −π1 ∨ π2 ∨ −π2 )

2. Rational polyhedra, germinal quotients and GL(n, Zn )  Zn -orbits Fix n = 1, 2, . . . . Following [18], by a rational polyhedron in Rn we mean a finite union of simplexes in R with rational vertices. A simplicial complex Δ in Rn is a finite set of simplexes Ti in Rn , closed under taking faces, and having the further property that any two elements of Δ intersect in a common face, [18]. Δ is said to be rational if the vertices of all Ti ∈ Δ are rational. For every simplicial complex Δ, its support |Δ| ⊆ Rn is the pointset union of all simplexes of Δ. We say that Δ is a triangulation of |Δ|. Throughout we let Gn = GL(n, Z) Zn denote the group of affine transformations of the form x → U x +t for x ∈ Rn , where t ∈ Zn and U is an integer (n × n)-matrix with det(U ) = ±1. Gn is known as the (n-dimensional) affine group over the integers. By an “orbit” we will mean a Gn -orbit. We let orb(x) denote the orbit of x ∈ Rn . Thus n

orb(x) = {y ∈ Rn | y = γ(x) for some γ ∈ Gn }. The affine group Gn naturally arises in MV-algebra theory, because of the duality between finitely presented MV-algebras and rational polyhedra (this will be touched upon in Remark 17). In particular, as shown in this paper, the two-dimensional affine group G2 has a pervasive tool in the classification of 2-germinal theories.

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As a warm up let us prove: Theorem 1. For any n = 1, 2, . . . and points x, y lying in the interior of [0, 1]n the following conditions are equivalent: (i) M([0, 1]n )/ox ∼ = M([0, 1]n )/oy . (ii) y ∈ orb(x). Proof. (i)⇒(ii). By [14, 4.10] there are rational polyhedra P, Q ⊆ [0, 1]n such that x and y respectively lie in the interior of P and Q, and η(x) = y for some continuous invertible piecewise linear map η of P onto Q with integer coefficients, whose inverse has also integer coefficients. (η is known as a Z-homeomorphism of P onto Q, [14].) Let Δ be a triangulation of P such that the restriction ηS is linear for each simplex S in Δ. The existence of Δ is ensured by [14, 2.9]. Let S ∈ Δ be a maximal simplex of Δ such that x ∈ S. Since x lies in the interior of P , then S is an n-simplex. There is an integer (n × n)-matrix A together with b ∈ Zn satisfying η(z) = Az + b for each z ∈ S. Since η is a Z-homeomorphism, then A is invertible, the map z → Az + b belongs to Gn , and the point y = η(x) = Ax + b belongs to orb(x). (ii)⇒(i). The hypothesis yields γ ∈ Gn with y = γ(x). Let P ⊆ [0, 1]n be a rational polyhedron containing x in its interior. Then Q = γ(P ) is a rational polyhedron containing y in its interior. Since y ∈ int[0, 1]n , it is no loss of generality to assume P so small that Q is contained in [0, 1]n . Trivially, γP is a Z-homeomorphism of P onto Q. The desired isomorphism of M([0, 1]n )/ox onto M([0, 1]n )/oy is now provided by [14, 4.10]. 2 Regular simplexes and triangulations. For any point v = (v1 , . . . , vn ) ∈ Qn we let den(v) denote the smallest common denominator of the coordinates of v. The integer vector v = (den(v) · v1 , . . . , den(v) · vn , den(v)) = den(v)(v, 1) ∈ Zn+1

(5)

is called the homogeneous correspondent of v. Let k = 0, . . . , n. Then a k-simplex S ⊆ Rn is said to be regular if it is rational and the set of homogeneous correspondents of its k + 1 vertices can be completed to a basis of the free abelian group Zn+1 . A rational simplicial complex Δ is said to be regular (“unimodular” in [6]) if every simplex of Δ is regular. Regular complexes are the affine counterpart of the regular (also known as “nonsingular”, or “smooth”) fans of toric algebraic geometry, [7,16]. They have a pervasive role throughout this paper. Let conv(y1 , . . . , ym ) denote the convex hull of points y1 , . . . , ym ∈ Rn . The following result is a special case [14, 2.7]: Lemma 2. Let J = conv(v0 , v1 ) ⊆ Rn be a rational interval. Then the following conditions are equivalent: (i) J is regular. (ii) Any rational point t lying in the relative interior of J satisfies the inequality den(t) ≥ den(v0 ) +den(v1 ). Lemma 3. Suppose T = conv(v, a, b) ⊆ R2 is a regular 2-simplex, and a , b ∈ R2 are two points such that conv(a , b ) is a regular segment lying on the line through a and b. Then conv(v, a , b ) is a regular 2-simplex. Proof. Passing to homogeneous correspondents in the free abelian group Z3 ⊆ R3 , the result turns out to be a special case of the Steinitz exchange theorem [8, 8.3] for torsion free abelian groups. 2 The following is a routine exercise: Lemma 4. Let T = conv(v0 , v1 , v2 ) and T  = conv(v0 , v1 , v2 ) be regular 2-simplexes in R2 , with den(vi ) = den(vi ) for all i = 0, 1, 2. Then for a unique γ ∈ G2 we have γ(vi ) = vi for all i = 0, 1, 2.

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3. Orbit classification preliminaries: the invariant cx A rational hyperplane H ⊆ Rn is a set of the form H = {z ∈ Rn | h, z = r}, for some nonzero vector h ∈ Qn and r ∈ Q. Here · , · denotes scalar product. Following [5, §4], for every x ∈ Rn we define Fx =



{A ⊆ Rn | A is a rational hyperplane in Rn and x ∈ A}.

(6)

Clearly, if x ∈ Rn is rational (equivalently, if rank(Gx ) = 1) then Fx = {x}. Further, from Definition 3 and Lemma 14 in [5] we have min{den(v) | v ∈ Fx ∩ Qn } = max{k ∈ Z | 1/k ∈ Gx } = dx .

(7)

Following Definition 6 in [5], upon setting e = dim(Fx ) we define cx = the smallest possible denominator of the vertex of a regular n-simplex conv(v0 , . . . , vn ) ⊆ Rn with v0 , . . . , ve ∈ Fx . Note that if e = n then Fx = Rn and dx = cx = 1. The case e = 0 is dealt with by the following elementary lemma: Lemma 5. Suppose x ∈ R2 and rank(Gx ) = 1, that is, x ∈ Q2 . Then there exist points z1 , z2 ∈ Z2 such that the 2-simplex conv(x, z1 , z2 ) is regular. Thus in particular, cx = 1. ˜ and b = gcd(a1 , a2 ). Since gcd(b, a3 ) = 1, there exist m, l, m , l ∈ Z such that Proof. Let (a1 , a2 , a3 ) = x b = a1 m + a2 l

and

1 = bm + a3 l .

We have the following identities: ⎛

a1  ⎝ det a1 l /b − m l − l −l

⎞ ⎛ ⎞ a1 a2 a3 a2 a3 a2 l /b + m m + m 1 ⎠ = det ⎝ a1 l /b a2 l /b −m ⎠ m 1 −l m 1

= a1 mm + a2 m l + a3 l (

a2 a1 a1 m + a2 l m + l) = bm + a3 l = bm + a3 l = 1. b b b

Now set z1 = (a1 l /b − m l − l, a2 l /b + m m + m) and z2 = (−l, m).

2

The integer dx in the following lemma was defined in (4) and characterized in (7): Lemma 6. Suppose x ∈ R2 and rank(Gx ) = 2. Then there is a point r ∈ Q2 \ Fx together with rational points p, q ∈ Fx satisfying the following conditions: (i) (ii) (iii) (iv)

den(p) = den(q) = dx ; den(r) = cx ; conv(p, q, r) is a regular 2-simplex; x ∈ conv(p, q).

Any regular interval x ∈ I ⊆ Fx whose vertices have denominator equal to dx coincides with conv(p, q).

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Proof. Since rank(Gx ) = 2, the point x = (x1 , x2 ) is not rational and there exist integers a1 , a2 , a3 such that a1 x1 +a2 x2 +a3 = 0 and gcd(a1 , a2 , a3 ) = 1. The integer vector (a1 , a2 , a3 ) ∈ Z3 ⊆ R3 is uniquely determined up to scalar multiplication by ±1. It follows that dim(Fx ) = 1 and Fx = {(y1 , y2 ) ∈ R2 | a1 y1 +a2 y2 +a3 = 0}. Claim. dx = gcd(a1 , a2 ). Let us write d∗ for gcd(a1 , a2 ). Since gcd(d∗ , a3 ) = gcd(a1 , a2 , a3 ) = 1, then for suitable integers h and j we can write hd∗ − ja3 = 1. Thus, (ja1 /d∗ )x1 + (ja2 /d∗ )x2 + h = −a3 j/d∗ + h = 1/d∗ ∈ Gx , whence by Definition (4), d∗ ≤ dx . For the converse inequality, let b1 , b2 , b3 ∈ Z be such that b1 x1 + b2 x2 + b3 = 1/dx , as given by (4) and (2). Then gcd(b1 , b2 , b3 ) = 1, because dx is the largest element of the set {k ∈ Z | 1/k ∈ Gx }. Since gcd(a1 , a2 , a3 ) = 1 and dx b1 x1 + dx b2 x2 + dx b3 − 1 = 0, there exists an integer l such that l(a1 , a2 , a3 ) = (dx b1 , dx b2 , dx b3 − 1). In particular, la3 − dx b3 = −1, whence gcd(l, dx ) = 1. Since dx divides gcd(dx b1 , dx b2 ) = gcd(la1 , la2 ) = ld∗ , we finally obtain dx ≤ d∗ , which concludes the proof of our claim. Our claim yields integers l1 , l2 , m1 , m2 , m3 satisfying m1 a1 + m2 a2 + m3 a3 = 1 = gcd(a1 , a2 , a3 ) and l1 a1 + l2 a2 = dx = gcd(a1 , a2 ). Thus for each k ∈ Z the point

bk =

a3 l 1 a3 l 2 − ,− dx dx



+k

a2 a1 ,− 2 d2x dx



belongs to Fx . Since by our claim, dx is a divisor of both a1 and a2 , we easily see that den(bk ) = dx for all k. The identities ⎛

⎞ ⎛ bk −a3 l2 − k dax1 −a3 l1 + k dax2 ⎠ = det ⎝ −a3 l1 + (k + 1) a2 −a3 l2 − (k + 1) a1 det ⎝ b k+1 dx dx m1 m2 m3 m1 m2 ⎛ ⎞ −a3 l1 + k dax2 −a3 l2 − k dax1 dx a2 /dx −a1 /dx 0 ⎠ = det ⎝ m2 m3 [0.1cm]m1 ⎛ ⎞ −a3 l1 −a3 l2 dx = det ⎝ a2 /dx −a1 /dx 0 ⎠ m1

m2

⎞ dx dx ⎠ m3

m3

= m2 a2 + m1 a1 + m3 a3 (l1 a1 + l2 a2 )/dx = m1 a1 + m2 a2 + m3 a3 =1 show that for all k ∈ Z the 2-simplex conv(bk , bk+1 , (m1 /m3 , m2 /m3 )) is regular, whence so is the 1-simplex Ik = conv(bk , bk+1 ). Trivially, all intervals Ik have the same length. Further, any two distinct Ik , Ih with nonempty intersection intersect precisely in one (rational) vertex. The union of all Ik coincides with the line Fx . Since x ∈ Fx is not a rational point, for precisely one integer k0 we must have x ∈ Ik0 . Since for every k ∈ Z, Ik is regular and den(bk ) = dx , from Lemma 2 we obtain {y ∈ Fx ∩ Q2 | den(y) = dx } = {bk | k ∈ Z}. As a consequence, the two points p = bk0 and q = bk0 +1 are the only ones in Fx such that x ∈ conv(p, q) ⊆ Fx , den(p) = den(q) = dx and conv(p, q) is regular. 2 By definition of cx there are rational points v0 , v1 , v2 ∈ [0, 1] with v0 , v1 ∈ Fx , v2 ∈ / Fx , conv(v0 , v1 , v2 ) regular and den(v2 ) = cx . By Lemma 3, the 2-simplex conv(p, q, v2 ) is regular. Now set r = v2 to complete the proof. 2

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Remark 7. Items (i)–(iv) in Lemma 6 are also obtainable as the special cases for n = 2 of Lemmas 5 and 7 in [5]. However, the direct proof presented here is much shorter. 4. Orbit classification The following theorem will be the key tool for our classification of the Lindenbaum algebras of 2-germinal theories: Theorem 8. For any x ∈ R2 , orb(x) = {x ∈ R2 | (Gx , cx ) = (Gx , cx )}. Proof. If x ∈ orb(x) it is easy to see that (Gx , cx ) = (Gx , cx ). To prove the converse, let x = (x1 , x2 ) be a fixed but otherwise arbitrary point in R2 . Suppose the point x = (x1 , x2 ) ∈ R2 satisfies Gx = Gx and cx = cx . The proof that x belongs to orb(x) is divided into three cases, depending on the rank of Gx . rank(Gx ) = 1 In other words, x ∈ Q2 . Writing for short d = den(x), we immediately have Gx = Z d1 . It follows that x , too, is a rational point and den(x ) = d. Applying Lemma 5 to both x and x we obtain points z1 , z2 , z1 , z2 ∈ Z2 such that both 2-simplexes T = conv(x, z1 , z2 ) and T  = conv(x , z1 , z2 ) are regular. Since den(x) = den(x ) = d and den(z1 ) = den(z2 ) = den(z1 ) = den(z2 ) = 1, Lemma 4 yields a map γ ∈ G2 satisfying γ(x) = x , γ(z1 ) = z1 , γ(z2 ) = z2 , as desired to settle the present case. rank(Gx ) = 3 By assumption there exist integer (3 × 3)-matrices A and B such that A(x1 , x2 , 1) = (x1 , x2 , 1) and B(x1 , x2 , 1) = (x1 , x2 , 1). Then BA(x1 , x2 , 1) = (x1 , x2 , 1) and AB(x1 , x2 , 1) = (x1 , x2 , 1). Since rank(Gx ) = 3, B is the inverse of A and det(A) = ±1. The only integer solution of the equation ax1 + bx2 + c = 1 is a = 0, b = 0, and c = 1. Therefore, for suitable integers aij we can write ⎛

a11 A = ⎝ a21 0

a12 a22 0

⎞ a13 a23 ⎠ . 1

Letting A =



a11 a21

a12 a22



it follows that A (x1 , x2 ) + (a13 , a23 ) = (x1 , x2 ) and det(A ) = det(A) = ±1. We conclude that x ∈ orb(x). rank(Gx ) = 2 In this case, Fx ⊆ R2 coincides with the only rational line in R2 passing through x. By hypothesis there are integers a1 , a2 , a3 with gcd(a1 , a2 , a3 ) = 1, such that a1 x1 + a2 x2 + a3 = 0. Further, a1 and a2 are not both zero, and we can write Fx = {(y1 , y2 ) ∈ R2 | a1 y1 + a2 y2 + a3 = 0}. Recalling (4), let us write for short d = dx . Lemma 6 yields (uniquely determined) points p, q ∈ Fx with den(p) = den(q) = d, along with a point r ∈ / Fx with den(r) = c = cx , and a regular triangle T = conv(p, q, r) satisfying x ∈ conv(p, q). The

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denominators of p, q, r are equal to d, d, c respectively. Since T is regular, gcd(c, d) = 1, whence there exist integers l and m such that md − lc = 1.

(8)

Let the triangle T¯ = conv(¯ p, q¯, r¯) be defined by p¯ = (0, l/d), q¯ = (1/d, l/d), r¯ = (0, m/c). Then T¯ is regular and the denominators of its vertices are respectively equal to d, d, c. By Lemma 4 some α ∈ G2 maps p, q, r to p¯, q¯, r¯. We now turn to the point x and the group Gx = Gx . Again by Lemma 6, x lies in the convex hull of precisely two points p , q  ∈ Fx of denominator d such that there is a rational point r ∈ / Fx with den(r ) = c and a regular triangle conv(p , q  , r ). Since the denominators of the vertices p , q  , r are equal to d, d, c, by Lemma 4 some β ∈ G2 maps p , q  , r to p¯, q¯, r¯. To complete the proof that x belongs to orb(x) it is sufficient to exhibit a map γ ∈ G2 with γ(α(x)) = β(x ). For suitable irrational numbers ξ and ξ  we can write α(x) = (ξ, l/d) ∈ conv(¯ p, q¯) and β(x ) = (ξ  , l/d) ∈ conv(¯ p, q¯).

(9)

From rank(Gx ) = 2 and gcd(l, d) = 1 (which follows from (8)), we see that the abelian group Gx = Zξ + Z dl + Z is freely generated by ξ and 1/d, and is also freely generated by ξ  and 1/d, because Gx = Gx , Gx = Gα(x) , and Gx = Gβ(x ) . Consequently we can write

h 0

k 1



ξ 1/d



=

ξ 1/d

 for some h ∈ {−1, 1} and a unique k ∈ Z.

Let finally γ ∈ G2 be defined by

γ(x1 , x2 ) =

h 0

−ck 1



x1 x2



+

km 0

 .

By (8), hξ −ckl/d +km = hξ +k/d = ξ  , and hence the desired identity γ(α(x)) = β(x ) follows from (9). 2 Remark 9. The foregoing theorem is a special case of Theorem 15 in [5] stating that for every n = 1, 2, . . . and x ∈ Rn , orb(x) = {y ∈ Rn | (Gx , cx ) = (Gy , cy )}. The short proof given here for the case n = 2 avoids the machinery of [5]. 5. Lindenbaum algebras of 2-germinal theories 2

In this section we classify the Lindenbaum algebras of 2-germinal theories, Θmin,v , for all v ∈ int [0, 1] . By Theorems 1 and 8, the germinal quotient M([0, 1]2 )/ov only depends on the pair of invariants (Gv , cv ). Moreover, by Lemma 5 and the remark preceding it, M([0, 1]2 )/ov only depends on Gv in case rank(Gv ) ∈ {1, 3}. The three cases rank(Gv ) ∈ {1, 2, 3} will be handled separately. Case: rank(Gv ) = 3. In other words, v lies in no rational line. Thus every f ∈ M([0, 1]2 ) vanishing at v also vanishes on an open neighborhood of v. It follows that ov = hv , whence M([0, 1]2 )/ov = M([0, 1]2 )/hv . Thus by (3), the MV-algebra M([0, 1]2 )/ov = Γ(Gv , 1) = Γ(Gv  FREE0 , (1, 0)) is the subalgebra of the standard MV-algebra [0, 1] consisting of all possible values at v of McNaughton functions f ∈ M([0, 1]2 ). This is also the Lindenbaum algebra of Θmin,v = Θmax,v .

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Fig. 1. The regular simplicial complex Δv (for rank(Gv ) = 2 and dv > 1). Its maximal faces are the triangles conv(p, q, r) and conv(p, q, s) of Lemma 10. The points r, p, s are aligned.

Case: rank(Gv ) = 2. In other words, v lies on exactly one rational line L = Fv ⊆ R2 . By Lemma 6 we have points p, q, r ∈ Q2 with p, q ∈ L, r ∈ / L, v ∈ conv(p, q), den(p) = den(q) = dv , den(r) = cv , having the additional property that the 2-simplex conv(p, q, r) ⊆ R2 is regular. Lemma 10. There exists a point s ∈ R2 with the following properties: (a) (b) (c) (d)

conv(p, q, s) is a regular 2-simplex; conv(p, q, s) ∩ conv(p, q, r) = conv(p, q); if dv = 1, den(s) = 1; if dv > 1, den(s) = dv − cv .

Proof. Combining (7) and [5, Lemma 9], we obtain cv ≤ dv . Arguing by cases we first assume dv = 1. By Theorems 1 and 8, without loss of generality we may assume p = (0, 0), q = (1, 1) and r = (0, 1). Setting now s = (1, 0) it is easy to check that s satisfies (a)–(d). In case dv > 1, the inequality cv ≤ dv can be strengthened to cv < dv . Recalling the notation (5) for homogeneous correspondents in Z3 of rational points in Q2 , we let s be the only point in R2 satisfying s˜ = p˜ − r˜. Necessarily r, p, s are aligned. Then s and r are separated by the line L, and s satisfies (a)–(d). (See Fig. 1.) 2 The -group D v of Δv -dihedra, and its lattice order  . Let Δv be the regular simplicial complex in R2 whose maximal faces are conv(p, q, r) and conv(p, q, s). Then by a Δv -dihedron we mean a continuous function f : |Δv | → R which is linear over both triangles conv(p, q, r) and conv(p, q, s), and whose two pieces are given by linear polynomials with integer coefficients. Thus the value f (v) is an element of Gv , and f (p), f (q) ∈ Z d1 . We let Dv = the set of Δv -dihedra. For each h ∈ Gv we further set Dvh = {f ∈ Dv | f (v) = h}

and Dvh {r, s} = {f {r, s} | f ∈ Dvh }.

In Fig. 2 we depict a function f ∈ Dvh . The (pointwise) sum/subtraction of two Δv -dihedra is a Δv -dihedron. By contrast, the pointwise max/min of two Δv -dihedra need not be a Δv -dihedron. Thus let us introduce the following lexicographic order:

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Fig. 2. A Δv -dihedron f ∈ Dvh . The domain of f is the support |Δv | ⊆ R2 of the regular simplicial complex Δv of Fig. 1. The two linear pieces of f coincide over the segment conv(p, q). Each piece of f is represented by a linear polynomial with integer coefficients.

Given f, g ∈ Dv , we write f  g if either f (v) < g(v) or else, f (v) = g(v) and f (x) ≤ g(x) for each x ∈ |Δv |. 2

Since v ∈ int [0, 1] \Q2 , any two maps f, g ∈ Dv with f (v) = g(v) agree on the segment conv(p, q). Therefore, the lexicographic order  transforms the abelian group Dv into an -group. The constant function 1|Δv | is a (strong, order-) unit of Dv . Lemma 11. With the above notation we have: (i) The map λ: f ∈ D v → (f (v), f (r), f (s)) is an -isomorphism of Dv onto the -group Tv = Gv  1 1 × Z den(s) of triplets Z den(r)   1 1 , c ∈ Z den(s) (h, (b, c)) | h ∈ Gv , b ∈ Z den(r) , equipped with the lexicographic order: (h, (b, c)) ≤ (h , (b , c )) iff either h < h or else h = h , b ≤ b , and c ≤ c , addition/subtraction: (h, (b, c)) ± (h , (b , c )) = (h ± h , (b ± b , c ± c )), and the zero element (0, (0, 0)). (ii) The map λ is a unital -isomorphism of (Dv , 1|Δv |) onto (Tv , (1, (1, 1))). Proof. (i) For every h ∈ Gv there is (a linear) gh ∈ Dv with gh (v) = h. Thus Gv = {h ∈ R | f (v) = h for some f ∈ Dv }. 1 1 For any f ∈ Dv , the pair (f (r), f (s)) belongs to Z den(r) × Z den(s) because both linear pieces of f have integer coefficients. Thus λ maps Dv into Tv . 2 Since v ∈ [0, 1] \ Q2 , the restriction of any f ∈ Dv over the segment conv(p, q) is uniquely determined by the value f (v). Thus λ is a one–one map. Further, for all (m, n) ∈ Z2 there is f ∈ Dv0 satisfying f (r) = m/ den(r) and f (s) = n/ den(s). This follows from [14, 3.7(iii)], in view of the regularity of Δv . As a consequence, the set {λ(gh + f ) | f ∈ Dv0 } coincides

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1 1 1 1 with the set {h} × Z den(r) × Z den(s) . Having thus shown that Dvh {r, s} = Dv0 {r, s} = Z den(r) × Z den(s) for all h ∈ Gv , it follows that λ is onto Tv . It is now easy to see that λ an isomorphism of the underlying abelian groups of Dv and Tv . Perusal of the definition of  and of the lexicographic order of Tv completes the proof of (i). (ii) now easily follows by direct inspection. 2

Recall from Section 1.3(iii) the definition of the free unital -group MR ([0, 1]n ). Let oR,v denote the -ideal of MR ([0, 1]2 ) generated by the germinal ideal ov of M([0, 1]2 ). By [6, 7.2.2], ov = oR,v ∩ unit interval of MR ([0, 1]2 ) = oR,v ∩ M([0, 1]2 ).

(10)

Lemma 12. (i) For any e ∈ MR ([0, 1]2 ) there is precisely one |Δv |-dihedron e∨ such that e(v) = e∨ (v) and

∂e∨ (v) ∂e(v) = for all nonzero w ∈ R2 . ∂w ∂w

(11)

(ii) MR ([0, 1]2 )/oR,v and (Dv , 1|Δv |) are unitally -isomorphic. 2

Proof. (i) Since v ∈ int [0, 1] \ Q2 and e is continuous piecewise linear with integer coefficients, then for all small open segments J ⊆ conv(p, q) containing x, the value e(v) uniquely determines e over J. Further, for every small open disk D centered at v, the restriction eD will have two linear pieces, say, er : conv(p, q, r) ∩ D → R and es : conv(p, q, s)∩D → R. Let Hr ⊆ R2 (resp., Hs ) be the half-plane with boundary L containing r (resp., containing s). Over the semi-circular domains conv(p, q, r) ∩ D and conv(p, q, s) ∩ D, the maps er and es respectively agree with uniquely determined linear polynomials ar , as : R2 → R with integer coefficients. Let the map e∨ : |Δv | → R be given by e∨ = ar (Hr ∩ |Δv |) ∪ as (Hs ∩ |Δv |). Then e∨ is the only Δv -dihedron satisfying (11). By construction, e/oR,v = f /oR,v implies e∨ = f ∨ . (ii) We first prove that the map η: e/oR,v ∈ MR ([0, 1]2 )/oR,v → e∨ ∈ Dv constructed in (i) is one–one onto Dv . By (10) and [14, 4.3, 4.8], two functions f, g ∈ MR ([0, 1]2 ) satisfy f /oR,v = g/oR,v iff f and g agree on some open neighborhood of v iff f (v) = g(v) and the directional derivatives of f and g at v coincide along any direction w ∈ R2 iff f ∨ = g ∨ , from which the injectivity of η follows. To prove that η is onto Dv , given g ∈ Dv , the proof of [14, 3.2(i)⇒(iii)] yields is a regular triangulation Ω of the rational polyhedron 2 2 [0, 1] ∩ |Δv | such that g([0, 1] ∩ |Δv |) is linear over each simplex of Ω. Next, the proof of [14, 3.2(iii)⇒(ii)] 2 2 yields an extension f ∈ MR ([0, 1] ) of g([0, 1] ∩|Δv |). Evidently, η(f /oR,v ) = f ∨ = g, whence η is onto Dv . It is now easy to see that η is an isomorphism of the underlying groups of MR ([0, 1]2 )/oR,v and Dv , sending the germ 1/oR,v to the element 1|Δv | of Dv . There remains to be proved that η preserves the lattice structure. Observe that the inequality f /ov ≤ g/ov holds in the germinal quotient MR ([0, 1]2 )/oR,v iff either f (v) < g(v), or else f (v) = g(v) and f N ≤ gN pointwise on some open neighborhood N of v. This is equivalent to saying that either f ∨ (v) < g ∨ (v), or else f ∨ (v) = g ∨ (v) and f ∨ (x) ≤ g ∨ (x) for all x ∈ |Δv |. Having thus shown that the lattice order of the -group MR ([0, 1]2 )/oR,v corresponds to the lexicographic order  of Dv , we conclude that η is the desired unital -isomorphism of MR ([0, 1]2 )/oR,v onto (Dv , 1|Δv |). 2 Recalling (1), the following result is a classification of all Lindenbaum algebras of germinal theories Θmin,v when rank(Gv ) = 2:

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Fig. 3. The regular simplicial complex ∇v and the perimeter Λv of the quadrilateral |∇v | ⊆ [0, 1]2 . By definition, v = (1/d, 1/d) and p = (1/(d − 1), 1/(d − 1)).

Theorem 13. For all v ∈ int[0, 1]2 and rank(Gv ) = 2 we have: In case dv > 1, M([0, 1]2 )/ov ∼ = Γ(Gv  FREE1 , (1, cv π1 ∨ (cv − dv )π1 )) = Γ(Gv  Z2 , (1, (cv , dv − cv ))) ∼ and in case dv = 1, M([0, 1]2 )/ov ∼ = Γ(Gv  FREE1 , (1, π1 ∨ −π1 )). = Γ(Gv  Z2 , (1, (1, 1))) ∼ 2

Proof. Let the points p, q, r, s ∈ [0, 1] be as in Lemma 10. Then Lemmas 11 and 12 yield unital -isomorphisms MR ([0, 1]2 )/oR,v ∼ = (Dv , 1|Δv |) ∼ = (Tv , (1, (1, 1))) ∼ = (Gv  Z2 , (1, (den(r), den(s)))) ∼ = (Gv  FREE1 , (1, den(r)π1 ∨ − den(s)π1 )). From [6, 7.2.4] and (10) we have unital -isomorphisms M([0, 1]2 )/ov ∼ = Γ(Gv  FREE1 , (1, (den(r)π1 ∨ − den(s)π1 ))). = Γ(MR ([0, 1]2 )/oR,v ) ∼ If dv > 1, den(s) = dv − cv , whence M([0, 1]2 )/ov ∼ = Γ(Gv  FREE1 , (1, cv π1 ∨ (cv − dv )π1 )). If dv = 1, both cv and den(s) are equal to 1, whence M([0, 1]2 )/ov ∼ = Γ(Gv  FREE1 , (1, π1 ∨ −π1 )). The proof is complete. 2 In both cases the classification only depends on (Gv , cv ), in accordance with Theorems 1 and 8. From the minimality of Θmin,v we obtain: Corollary 14. Let Θ be a theory in two variables with only one model v ∈ int[0, 1]2 . Suppose rank(Gv ) = 2. Then the Lindenbaum algebra of Θ is isomorphic to a quotient of Γ(Gv  Z2 , (1, (cv , dv − cv ))) if dv > 1, and is isomorphic to a quotient of Γ(Gv  Z2 , (1, (1, 1))) if dv = 1. Case: rank(Gv ) = 1. In other words, v is a rational point, say of denominator d. Necessarily, d > 1, because v is assumed to lie in the interior of [0, 1]2 . Trivially, d = dv . By Theorem 8 it is no loss of generality to identify v with any arbitrary rational point of denominator d lying in the interior of [0, 1]2 . For convenience, 1 1 we let v = (1/d, 1/d). We also let p = ( d−1 , d−1 ), q = (0, 1), r = (0, 0), and s = (1, 0). Let ∇v be the regular simplicial complex whose maximal faces are the four triangles conv(v, p, q), conv(v, q, r), conv(v, r, s), conv(v, s, p). Further, let Λv be the perimeter of the quadrilateral |∇v |. (See Fig. 3.)

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Fig. 4. A radial ∇v -map f ∈ R0v . The domain of f is the rational polyhedron |∇v | ⊆ [0, 1]2 of Fig. 3. The open disk D is partitioned into l = 12 circular sectors centered at v. For each i = 1, . . . , l, gi is a linear function. The function f is the union of the linear functions g1 , . . . , gl , and is denoted g  in the proof of Lemma 16.

The -group Rv of radial ∇v -maps and its order  . By a radial ∇v -map we mean a continuous piecewise linear function f : |∇v | → R with integer coefficients, such that for each point x ∈ Λv , f is linear over the segment conv(v, x). (As always in this paper, the number of linear pieces of f is finite.) Let Rv = the set of radial ∇v -maps. For each h ∈ Gv we also write Rhv = {f ∈ Rv | f (v) = h},

Rhv Λv = {f Λv | f ∈ Rhv }.

(12)

Fig. 4 is a sketch of a map in R0v . The possible values at v of radial ∇v -maps are precisely the elements of Gv = Z d1 . The (pointwise) sum/subtraction of two radial ∇v -maps is a radial ∇v -map, but the pointwise max/min of two radial ∇v -maps need not be a radial ∇v -map. Thus we equip Rv with the lexicographic order  as follows: Given f, g ∈ Rv , we write f  g if either f (v) < g(v) or else, f (v) = g(v) and f (x) ≤ g(x) for each x ∈ |∇v |. Let (Rv , 1|∇v |) denote the unital -group of radial ∇v -maps with the distinguished unit given by the restriction to |∇v | of the constant function 1. Let further 1Λv denote the restriction to Λv of the constant function 1. Lemma 15. The map ω: f ∈ Rv → (f (v), f Λv ) ∈ Gv × (R0v Λv ) is a unital -isomorphism of (Rv , 1|∇v |) onto (Gv  (R0v Λv ), (1, 1Λv )). Proof. Since Gv = Z d1 , from [14, 3.7(iii)] it follows that Rhv Λv = R0v Λv for all h ∈ Gv ,

(13)

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and hence, ω maps Rv one–one onto Gv × (R0v Λv ). Direct inspection of the lexicographic orders of Rv and Gv  (R0v Λv ) shows that ω is an -isomorphism. The identity ω(1|∇v |) = (1, 1Λv ) is immediate. 2 Lemma 16. MR ([0, 1]2 )/oR,v ∼ = (Rv , 1|∇v |). Proof. Let us arbitrarily fix g ∈ MR ([0, 1]2 ), with the intent of describing its germ g/oR,v . For some closed disk D of a suitably small radius ρ > 0 centered at v, and some l ∈ N, D can be partitioned into circular sectors D1 , . . . , Dl of radius ρ, centered at v, such that for each i = 1, . . . , l, the restriction gDi coincides with the restriction to Di of a uniquely determined linear polynomial pi : R2 → R with integer coefficients. Each sector Di extends to a unique angle αi ⊆ R2 with vertex v. The intersection of two angles αi , αj , (i = j) is either {v} or a common side of αi , αj . For each i = 1, . . . , l, let the function gi : αi ∩ |∇v | → R be defined by gi = pi (αi ∩ |∇i |). Next let ∨ g : |∇v | → R be defined by g = g1 ∪ · · · ∪ gl . (See Fig. 4 for a sketch of this state of affairs.) Since g is a continuous piecewise linear function whose linear pieces have integer coefficients, then g is a member of Rv . By construction, g(v) = g (v) and

∂g (v) ∂g(v) = for all nonzero w ∈ R2 . ∂w ∂w

Any two f, g ∈ MR ([0, 1]2 ) with the same germ f /oR,v = g/oR,v have the same value at v and the same directional derivatives along all directions, whence f = g . It follows that the map μ: g/oR,v → g is a homomorphism of the underlying group of MR ([0, 1]2 )/oR,v into that of Rv , sending the germ 1/oR,v to the unit (1, 1∇v ) of Rv . A moment’s reflection shows that μ is one–one. Indeed, by (10) and [14, 4.3, 4.8], two functions f, g ∈ MR ([0, 1]2 ) have the same germ at v iff they agree on some open neighborhood of v iff f (v) = g(v) and their directional derivatives at v coincide along any direction w ∈ R2 iff f = g . To show that μ is onto Rv , let b ∈ Rv . The proof of [14, 3.2(i)⇒(iii)] yields is a regular triangulation Δ of the rational polyhedron |∇v | such that b is linear over each simplex of Δ. Next, the proof of [14, 3.2(iii)⇒(ii)] yields an extension ¯b ∈ MR ([0, 1]2 ) of b. Evidently, μ(¯b) = b, whence μ is onto Rv . To conclude the proof that μ is an -isomorphism, for all f, g ∈ MR ([0, 1]2 ) the inequality f /ov ≤ g/ov holds in the unital -group MR ([0, 1]2 )/oR,v iff either f (v) < g(v), or else f (v) = g(v) and f N ≤ gN over some open neighborhood N of v. This is equivalent to saying that either f (v) < g (v), or else f (v) = g (v) and f (x) ≤ g (x) for all x ∈ |∇v |. Thus the lattice order of MR ([0, 1]2 )/oR,v agrees with the lexicographic order of Rv . 2 The unital -group MR (P ) and its Γ-image M(P ). Generalizing the notation MR ([0, 1]n ) introduced in Section 1.3(iii) for the free unital -group, given a rational polyhedron P ⊆ [0, 1]n we let n

MR (P ) = {f P | f ∈ MR ([0, 1] )}. The distinguished unit of MR (P ) is the constant function 1 on P . The preservation properties of the Γ functor ensure that the MV-algebra Γ(MR (P )) coincides with the MV-algebra M(P ) = {f P | f ∈ M([0, 1]n )}. By [14, 3.2(i)⇔(ii)], M(P ) also coincides with the MV-algebra of all continuous piecewise

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linear [0, 1]-valued functions on P , where each piece has integer coefficients. Thus MR (P ) coincides with the unital -group of all continuous piecewise linear real-valued functions on P where each linear piece has integer coefficients. (The reader will remember that throughout this paper the adjective “linear” is understood in the affine sense.) Remark 17. The assignment P → M(P ) extends to a duality between rational polyhedra and continuous piecewise linear maps with integer coefficients (also known as Z-maps, see [14]) and finitely presented MV-algebras with homomorphisms. See [10, 4.12]. Related earlier notions and constructions can be found in [17, 35], [9, 5.1,5.2,6.4], and [4]. Lemma 18. With the notation of (12), recalling that d = den(v) = dv , we have: (i) (R0v Λv , 1Λv ) = MR (Λv ). (ii) The underlying -group of MR (Λv ) is -isomorphic to the free -group FREE2 via an -isomorphism sending the unit 1Λv of MR (Λv ) to the element (d − 1)π1 ∨ −π1 ∨ π2 ∨ −π2 of FREE2 . Proof. (i) For the inclusion (R0v Λv , 1Λv ) ⊆ MR (Λv ), let g ∈ (R0v Λv , 1Λv ). By the characterization of MR (P ) in the discussion preceding Remark 17, (or by the proof of [14, 3.2]), g can be extended to a function 2 g ∗ ∈ MR ([0, 1] ). The restriction g ∗ Λv is a member of MR (Λv ) and trivially coincides with g. For the converse inclusion, given f ∈ MR (Λv ), the proof of [14, 3.2, (i)⇒(iii)] yields a regular triangulation Δ of the rational polyhedron Λv such that f is linear over each segment of Δ. Focusing on any one of the four sides of the quadrilateral Λv , say, conv(p, q), let us pick a (necessarily regular) segment conv(y, z) in Δ, contained in conv(p, q). By Lemma 3, the regularity of conv(v, p, q) entails the regularity of conv(v, y, z). The totality of regular segments conv(yi , zi ) of Δ contained in Λv yields a regular triangulation of |∇v |, consisting of all triangles conv(v, yi , zi ) and their faces. There is precisely one (automatically continuous and piecewise linear) function f ∗ : |∇v | → R coinciding with f over Λv , having value 0 at v, having the additional property that for each w ∈ Λv , f ∗ is linear over the segment conv(v, w). To conclude the proof that f ∗ is a member of R0v we must only verify that each linear piece of f ∗ has integer coefficients. Again let us restrict attention to a regular triangle conv(v, yi , zi ), where as above, yi , zi ∈ Λv . The respective values of f at yi and zi are ki / den(yi ) and li / den(zi ) for some integers ki , li . These are also the values of f ∗ . Now the proof of [14, 3.7(iii)] shows that this, (as well as every) linear piece of f ∗ has integer coefficients. Thus f ∗ belongs to R0v . (ii) Let the function ξ1 : Λv → R be uniquely determined by the following stipulations: ξ1 is linear over each segment conv(p, q), conv(q, r), conv(r, s), conv(s, p); 1 1 ξ1 ( d−1 , d−1 )=

1 d−1 ;

ξ1 (0, 0) = −1; ξ1 (0, 1) = ξ1 (1, 0) = 0. Similarly let ξ2 : Λv → R be defined by: ξ2 is linear over each segment conv(p, q), conv(q, r), conv(r, s), conv(s, p); 1 1 ξ2 (0, 0) = 0 = ξ2 ( d−1 , d−1 );

ξ2 (0, 1) = 1; ξ2 (1, 0) = −1.

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In view of the regularity of each side of Λv , an application of [14, 3.7(iii)] ensures that every linear piece of both functions ξ1 and ξ2 has integer coefficients, whence ξ1 , ξ2 ∈ MR (Λv ). Let gen(ξ1 , ξ2 ) ⊆ MR (Λv ) denote the -group generated by ξ1 , ξ2 in MR (Λv ). Now observe that the -ideal generated by any linear combination (with integer coefficients) of ξ1, ξ2 in gen(ξ1 , ξ2 ) is strictly contained in some maximal -ideal of gen(ξ1 , ξ2 ). Therefore, from the main result of [12] it follows that the set {ξ1 , ξ2 } is free generating in gen(ξ1 , ξ2 ). We next show that gen(ξ1 , ξ2 ) = MR (Λv ). As a matter of fact, the four piecewise linear functions ξ2 ∨ 0, −ξ2 ∨ 0, ξ1 ∨ 0, −ξ1 ∨ 0 constitute a Schauder basis [14, 5.3] of the MV-algebra M(Λv ), with their respective multiplicities 1, 1, d −1, 1. By [14, 5.8(i)], these functions form a generating set of M(Λv ). The preservation properties of the Γ functor [11, §3], ensure that {ξ1 , ξ2 } is a generating set of the -group MR (Λv ). We conclude the proof by letting π1 , π2 be the canonical free generators of the free abelian -group FREE2 of piecewise homogeneous linear functions on R2 with integer coefficients introduced in Section 1.3(ii). Since, as we have proved, {ξ1 , ξ2 } is free generating in MR (Λv ), the map π1 → ξ1 , π2 → ξ2 extends to an -isomorphism η of FREE2 onto MR (Λv ). The inverse -isomorphism η −1 sends the unit 1Λv of MR (Λv ) to the element (d − 1)π1 ∨ −π1 ∨ π2 ∨ −π2 . 2 The following theorem classifies Lindenbaum algebras of germinal theories Θmin,v in case rank(Gv ) = 1: Theorem 19. Let the point v ∈ int[0, 1]2 be such that rank(Gv ) = 1. Then M([0, 1]2 )/ov ∼ = Γ(Gv  FREE2 , (1, (dv − 1)π1 ∨ −π1 ∨ π2 ∨ −π2 )). Proof. By hypothesis, v is a rational point, Gv = Z d1v and den(v) = dv > 1. We have unital -isomorphisms MR ([0, 1]2 )/oR,v ∼ = (Rv , (1, 1|∇v )), by Lemma 15 ∼ = (Gv  (R0v Λv ), (1, 1Λv )), by Lemma 16 ∼ = (Gv  (MR (Λv )), (1, 1Λv )), by Lemma 18(i) ∼ = (Gv  FREE2 , (1, (dv − 1)π1 ∨ −π1 ∨ π2 ∨ −π2 )), by 18(ii).

 ∼ In [6, 7.2.4] it is proved Γ (G,u) = j have unital -isomorphisms

Γ(G,u) j∩[0,u] ,

for any unital -group (G, u) and -ideal j of G. By (10), we

M([0, 1]2 ) ∼ Γ(MR ([0, 1]2 )) ∼ = =Γ ov oR,v ∩ M([0, 1]2 )



MR ([0, 1]2 ) oR,v

 .

This yields the desired isomorphism M([0, 1]2 )/ov ∼ = Γ(Gv  FREE2 , (1, (dv − 1)π1 ∨ −π1 ∨ π2 ∨ −π2 )). The proof is complete. 2 From the minimality of Θmin,v we obtain: Corollary 20. Let Θ be a theory in two variables with only one model v. Suppose v lies in the interior of [0, 1]2 and rank(Gv ) = 1. Then the Lindenbaum algebra of Θ is isomorphic to a quotient of the MV-algebra Γ(Gv  FREE2 , (1, (dv − 1)π1 ∨ −π1 ∨ π2 ∨ −π2 )).

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6. Concluding remarks, a conjecture and an open problem Several chapters of the monograph [14], and a wealth of recent MV-algebraic literature show that the theory of Łukasiewicz logic and its algebras goes far beyond non-classical logic and algebraic logic. Over the last few years algebraic topology, functional analysis, the regular fans of toric geometry and their associated regular complexes have provided key tools for the study of advanced topics in Łukasiewicz logic and MV-algebras. These topics range from the invariant measure [13] of maximal spectra of finitely presented MV-algebras (i.e., by duality, rational polyhedra), to the characterization of finitely generated projective MV-algebras and their dual Z-retracts, [4,21,22]. (Also see [14, §§5, 10, 14, 16, 17, 19]). Conversely, via the functors Γ and M and their adjoints, MV-algebra theory has found novel applications to diverse areas of mathematics. Here is a succinct selection of relevant papers appeared since the submission of the present paper: Categories, Morita equivalence, sheafs, dualities, [31,10,25,28] Differential geometry, [20,23] Discrete dynamical systems, [5,24] Functional analysis, AF algebras, Riesz spaces, [5,23,29,30,32] Game theory, [27] Inverse semigroups, [30] Lattice-ordered abelian groups, [3,9,21,22,25,28,29] Quantum structures, [26] Topology, [31]. In this paper the differential semantics of Łukasiewicz logic is combined with the Γ functor and the affine group over the integers to prove that the Lindenbaum algebra of the smallest Ł∞ -theory Θmin,x having just one model x (with x ∈ int[0, 1]2 ), is the Γ-image of the unital lexicographic product of the totally ordered abelian group Gx ⊆ R by a free -group over suitably many free generators. Germinal ideals yield an exemplary algebraic framework for the differential semantics of Łukasiewicz logic. They provide a geometrical representation of the Lindenbaum algebra of the germinal theory Θmin,x as the MV-algebra of germs at x of two-variable McNaughton functions. Letting x range over the totality of points in the interior of the unit square [0, 1]2 , all these MV-algebras are classified in terms of the G2 -orbit orb(x) of x. A single numerical invariant cx , together with the unital ordered abelian group Gx , turns out to completely classify orb(x). Owing to its preliminary character, this paper did not discuss the classification of Lindenbaum algebras of germinal theories in more than two variables. So we take leave of our readers with a conjecture and a problem. Conjecture. Fix n = 1, 2, . . . and a point v ∈ int[0, 1]n . Let the totally ordered abelian group Gv be as in (2). Then M([0, 1]n )/ov ∼ = Γ(Gv  FREEn+1−rank(Gv ) , (1, u)),

(14)

where (1, u) is a unit of the -group Gv  FREEn+1−rank(Gv ) only depending on the integers n, rank(Gv ), dv , cv . From Theorem 1 together with Theorem 15 in [5] (which generalizes Theorem 8 in the present paper), it follows that u in (14) only depends on n, Gv and cv . Our conjecture claims that u depends on Gv only via

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the two integers rank(Gv ) and dv . This is suggested by the case n = 2 (and n = 1), where the conjecture holds. By (3), the conjecture is also true for all n whenever rank(Gv ) = n + 1. Problem. Describe the unit (1, u) (assuming (14) to be true). Acknowledgements L.M. Cabrer’s research was partially supported by a Marie Curie Intra European Fellowship under the [European Community’s] Seventh Framework Programme (Grant Agreement No. 326202) and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 670624). We are grateful to the referee for her/his careful reading and valuable suggestions for improvement. References [1] W.M. Beynon, Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Canad. J. Math. 29 (1977) 243–254. [2] A. Bigard, K. Keimel, S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math., vol. 608, Springer, Berlin, 1977. [3] M. Busaniche, L.M. Cabrer, D. Mundici, Confluence and combinatorics in finitely generated unital lattice ordered groups, Forum Math. 24 (2) (2012) 253–271. [4] L. Cabrer, D. Mundici, Rational polyhedra and projective lattice-ordered abelian groups with order unit, Commun. Contemp. Math. 14 (3) (2012) 1250017, http://dx.doi.org/10.1142/S0219199712500174 (20 pages). [5] L.M. Cabrer, D. Mundici, Classifying orbits of the affine group over the integers, Ergodic Theory Dynam. Systems (2016), http://dx.doi.org/10.1017/etds.2015.45 (first published online 22 July 2015). [6] R.L.O. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Trends Log., vol. 7, Kluwer Academic Publishers, Dordrecht, 2000. [7] G. Ewald, Combinatorial Convexity and Algebraic Geometry, Grad. Texts in Math., vol. 168, Springer-Verlag, New York, 1996. [8] L. Fuchs, Abelian Groups, Pergamon Press, Oxford, 1967. [9] V. Marra, D. Mundici, The Lebesgue state of a unital abelian lattice-ordered group, J. Group Theory 10 (2007) 655–684. [10] V. Marra, L. Spada, The dual adjunction between MV-algebras and Tychonoff spaces, in: special Issue in Memoriam Leo Esakia, Studia Logica 100 (2012) 253–278. [11] D. Mundici, Interpretation of AF C∗ -algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986) 15–63. [12] D. Mundici, Free generating sets of lattice-ordered abelian groups, J. Pure Appl. Algebra 211 (2007) 400–403. [13] D. Mundici, The Haar theorem for lattice-ordered abelian groups with order-unit, Discrete Contin. Dyn. Syst. 21 (2008) 537–549. [14] D. Mundici, Advanced Łukasiewicz Calculus and MV-Algebras, Trends Log., vol. 35, Springer, Berlin, 2011. [15] D. Mundici, The differential semantics of Łukasiewicz syntactic consequence, Chapter 7, in: F. Montagna (Ed.), Petr Hájek on Mathematical Fuzzy Logic, in: Outst. Contrib. Log., vol. 6, Springer International Publishing, Switzerland, 2015, pp. 143–157. [16] T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Springer-Verlag, New York, 1988. [17] G. Panti, Multi-valued logics, in: P. Smets (Ed.), Quantified Representation of Uncertainty and Imprecision, Kluwer Academic Publishers, Dordrecht, 1998, pp. 25–74. [18] J.R. Stallings, Lectures on Polyhedral Topology, Lectures in Math., vol. 43, Tata Institute of Fundamental Research, Mumbay, 1968. [19] A. Tarski, On the concept of logical consequence, Chapter XVI, in: A. Tarski (Ed.), Logic, Semantics, Metamathematics, Clarendon Press, Oxford, 1956, Reprinted: Hackett, Indianapolis, 1983.

Added in proof: papers on MV-algebras outside algebraic logic (a short selection, 2014–2017) [20] L.M. Cabrer, Bouligand–Severi k-tangents and strongly semisimple MV-algebras, J. Algebra 404 (2014) 271–283. [21] L.M. Cabrer, Rational simplicial geometry and projective unital lattice-ordered abelian groups, arXiv:1405.7118, 28 May 2014. [22] L.M. Cabrer, Simplicial geometry of unital lattice-ordered abelian groups, Forum Math. 27 (2015) 1309–1344. [23] L.M. Cabrer, D. Mundici, Severi–Bouligand tangents, Frenet frames and Riesz spaces, Adv. in Appl. Math. 64 (2015) 1–20. [24] L.M. Cabrer, D. Mundici, Classifying GL(n, Z)-orbits of points and rational subspaces, Discrete Contin. Dyn. Syst. 36 (9) (2016) 4723–4738.

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[25] O. Caramello, A.C. Russo, The Morita-equivalence between MV-algebras and lattice-ordered abelian groups with strong unit, J. Algebra 422 (2015) 752–787. [26] A. Dvurečenskij, Quantum structures versus partially ordered groups, Internat. J. Theoret. Phys. 54 (12) (2015) 4260–4271 (first online 28 December 2014), http://dx.doi.org/10.1007/s10773-014-2479-9. [27] T. Kroupa, O. Majer, Optimal strategic reasoning with McNaughton functions, Internat. J. Approx. Reason. 55 (2014) 1458–1468. [28] M. Gehrke, S.J. van Gool, V. Marra, Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality, J. Algebra 417 (2014) 290–332. [29] S. Lapenta, I. Leustean, Scalar extensions for algebraic structures of Łukasiewicz logic, J. Pure Appl. Algebra 220 (4) (2016) 1538–1553. [30] M.V. Lawson, P. Scott, AF inverse monoids and the structure of countable MV-algebras, J. Pure Appl. Algebra 221 (2017) 45–74. [31] V. Marra, L. Reggio, Stone duality above dimension zero: axiomatising the algebraic theory of C(X), Adv. Math. 307 (2017) 253–287, arXiv:1508.07750, 1 October 2015. [32] D. Mundici, Hopfian -groups, MV-algebras and AF C∗ -algebras, Forum Math. 28 (6) (2016) 1111–1130 (published online on 1 May 2016), http://dx.doi.org/10.1515/forum-2015-0177.