The spectrum problem for Abelian $\ell$-groups and MV-algebras

Abstract

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian $\ell$-groups. As a first main result, we show that a topological space X is the prime spectrum of an MV-algebra if and only if X is spectral, and the lattice K(X) of compact open subsets of X is a closed epimorphic image of the lattice of “cylinder rational polyhedra” (a natural generalization of rational polyhedra) of $[0,1]^Y$ for some set Y. As a second main result we extend our results to Abelian $\ell$-groups. That is, a topological space X is the prime spectrum of an Abelian $\ell$-group if and only if X is generalized spectral, and the lattice K(X) is a closed epimorphic image of the lattice of “cylinder rational cones” (a generalization of rational cones) in ${{\mathbb {R}}}^Y$ for some set Y. Finally, we axiomatize, in monadic second order logic, the Belluce lattices of free MV-algebras (equivalently, the lattice of cylinder rational polyhedra) of dimension 1, 2 and infinite, and we study the problem of describing Belluce lattices in certain fragments of second order logic.

Appendices

Appendices

Appendix A: Axioms for $K_2$

We propose the following list of axioms.

We did not check whether the list is redundant.

We postulate the existence of a set $\Pi \subseteq K_2$ and an atom O (the origin) with the following properties.

Intuitively, $\Pi$ is a set of parallelograms and is used to define the relations of congruence and parallelism between segments.

1.1 Basics

Axiom A1

$K_2$ is a bounded lattice and contains the minimum and the maximum of $L_2$. The binary infimum of $K_2$ coincides with the binary infimum of $L_2$.

Note that the binary suprema in $K_2$ and $L_2$ differ in general. Intuitively the infimum in $K_2$ is intersection, the supremum is the convex hull, and the atoms are the rational points of the unit square.

In this appendix the capital letters $A,B,C,\dots$ denote atoms unless otherwise specified.

Definition

A segment is the supremum of two atoms. A triangle is a supremum of three atoms. A line is a maximal segment.

For convenience we denote the segment (i.e. supremum) of A and B by AB rather than $A\vee B$. Similarly for triangles ABC, etc.

Our segments are not oriented, so we identify AB with BA.

Definition

A zero segment is simply an atom.

Axiom A2

For every segment there is an atom disjoint with it. Every nonzero segment is contained in a unique line. If $AB=CD$ then $\{A,B\}=\{C,D\}$.

1.2 Betweenness

Definition

(betweenness). We let $A<B<C$ if $B\le AC$ and $B\not = A,C$.

Axiom A3

For every two atoms A, B, The binary relation $A< X< Y < B$ is a strict total order between the atoms of AB different from A and B.

1.3 Congruence, parallelism and comparison of segments

Because we are in two dimensions, we can use parallelograms; then two segments are congruent when they are two opposite sides of a parallelogram, unless the two segments are contained in a same line (in which case we need two parallelograms to establish that they are congruent). So we give the following definitions:

Definition

AB is strictly congruent to CD if the supremum of A, B, C, D lies in $\Pi$ and $AB\wedge CD=0$.

Definition

The segment AB is congruent to the segment CD (written $AB=cCD$) if and only if there are E, F such that AB, EF and CD, EF are strictly congruent. (this is a kind of congruence relation inspired by the axiomatization of real vector spaces in [36]).

Definition

AB is parallel to CD if two nonzero subsegments of them are congruent or either AB or CD is zero.

Axiom A4

Congruence and parallelism between nonzero segments are equivalence relations. Any two segments contained in the same segment are parallel. All zero segments are congruent to each other.

Definition

(segment comparison). We let $AB>CD$ (as segments) if and only if AB and CD are parallel and there is E such that $A<E<B$ and $AE=cCD$. This notation should not be confused with the order of the lattice $K_2$.

Axiom A5

Any two parallel segments are either comparable or congruent. Zero segments are smaller than nonzero segments.

Axiom A6

parallel segment comparison is irreflexive and transitive.

1.4 Parallel sum of segments

Definition

(parallel sum). We write $AB+pCD=A'C'$ if AB and CD are parallel, and there is an atom $B'$ such that $A'B'=cAB$, $B'C'=cCD$ and $A'<B'<C'$.

Axiom A7

The parallel sum, when it exists, is unique up to congruence. The parallel segment sum, as a partial operation on parallel segments, is commutative and associative, and preserves the relations of comparison and congruence. Zero segments are neutral for parallel sum.

Partiality of parallel sum holds due to truncation problems (for instance we cannot make the parallel sum of two lines because this sum should be longer than both, whereas lines have maximal length). However, note that if $A<C<B$ then $AB=AC+pCB$. More generally, by induction on n, we can prove that if $A<C_1<\dots<C_n<B$, then $AB=AC_1+pC_1C_2+p\dots +pC_nB$.

1.5 Division of segments

In this subsection we want to put sufficiently strong axioms so to have the divisibility of a segment into any finite number of congruent parts. Again we use the fact that we are in two dimensions, and we can state an axiom analogous to Thales Theorem of plane euclidean geometry.

Axiom A8

Let AB, CD be two parallel segments. Then $AB\wedge CD$ is zero, or an atom, or a segment.

Axiom A9

Let AB, CD be non parallel segments. Then $AB\wedge CD$ is zero or an atom.

Axiom A10

(Playfair axiom, inspired by the parallel axiom of elementary geometry in the form of Playfair). Let TUV be a triangle and let $A\in VU$. There is a segment $AA'$ parallel to TV and such that $A'\in TU$.

Axiom A11

(Thales axiom, inspired by the Thales Theorem of elementary geometry). Let TUV be a triangle and $V\le A\le B\le C\le D\le U$. Consider the segments $AA'$, $BB'$, $CC'$, $DD'$ parallel to TV and ending in TU. Then $T\le A'\le B'\le C'\le D'\le U$. If $AB=cCD$ then $A'B'=cC'D'$. If $AB\wedge CD=0$ then $A'B'\wedge C'D'=0$. If $AB\wedge CD$ is an atom then $A'B'\wedge C'D'$ is an atom.

Axiom A12

Every segment AB can be divided in two congruent parts, that is, there is $C\le AB$ such that $AC=cCB$.

The axiom implies divisibility in $2^n$ congruent parts for every integer $n\ge 2$, by induction.

More generally, we can prove:

Lemma 12.1

We can divide any (nonzero) segment AB in n congruent parts for every integer $n\ge 3$.

Proof

Consider a triangle ABC, and subdivide BC in $2^n$ congruent segments; let $B=P_1<P_2\dots <P_{2^n+1}=C$ the extremes of these segments. Then one can draw n lines parallel to $P_nA$ and passing through $P_1,\dots ,P_n$ respectively; the intersection of these n lines with AB give n congruent segments by Thales axiom. $\square$

1.6 The rationality axiom

In this subsection we want to establish that the ratio between two parallel segments (where the second is nonzero) is always a rational number.

Definition

AB is a submultiple of CD if there is a finite set F of points of CD such that every segment with extremes in two consecutive points of F is congruent to AB.

Note that the definition is given in monadic second order logic, because the set F is finite if and only if every nonempty subset of F has a minimum and a maximum in the betweenness order of CD.

Axiom A13

(rationality axiom). Any two parallel segments have a common submultiple.

By the previous axioms, every segment is a rational multiple of any other segment parallel to it, and the ratio of two parallel segments (where the second is nonzero) is always well defined. We denote the ratio between AB and CD by AB : CD.

1.7 Assigning coordinates to atoms and coordinate lines

Intuitively, the elements of $K_2$ live in the unit square of the (rational) Cartesian plane xy. In particular, atoms should have two rational coordinates between 0 and 1, and lines parallel to an axis x or y should have a rational coordinate.

Definition

An atom is extreme if it is not interior to any segment.

Axiom A14

There are exactly four extreme atoms in $K_2$. One of them is O.

Definition

Two extreme atoms A, B are adjacent if no atom of the segment AB is internal to a triangle.

Axiom A15

There are two extremes $O_x,O_y$ adjacent to O; we call $OO_x$ the x axis and $OO_y$ the y axis. The axes are lines.

Intuitively, the previous two lines are the axes x and y of the Cartesian plane.

Axiom A16

Two lines, each of them parallel to some different coordinate line, meet in a single atom.

Axiom A17

For every coordinate line l and every atom A outside l there is a unique line $l'$ parallel to l and passing through A.

Definition

The x-th projection of an atom A, written $A_x$, is the intersection of the x axis with the line parallel to the y axis and passing through A, which is an atom and is unique by the previous axioms. Likewise we define the y-th projection.

Definition

The x-coordinate of an atom A is the ratio between the segment $OA_x$ and the x axis. Likewise we define the y coordinate.

1.8 Atoms and coordinates

Lemma 12.2

The coordinate assignment gives a bijection between atoms and pairs of rational numbers between 0 and 1.

Proof

Any atom has a pair of coordinates associated to its projections on the axes x and y. Conversely, given two rational numbers $p,q\in [0,1]$, a point with coordinates (p, q) is obtained (as usual) by intersection of two lines parallel to the x and y axis, the first line passing through the point of the x axis with coordinate p, the second line through the point of the y axis with coordinate q.

$\square$

Lemma 12.3

• Finite sets of atoms contained in any segment AB are definable in monadic second order logic in $K_2$.

• Finite sets of atoms are definable in monadic second order logic in $K_2$.

Proof

The first point holds because atoms in a segment AB are totally ordered by the relation $A<x<y<B$.

The second point holds because a set F of atoms is finite if and only if the projections of the elements of F range over a finite set. $\square$

1.9 Segments and coordinates

Definition

Two segments AB, CD are equioriented if $O<A_x<B_x$ if and only if $O<C_x<D_x$, and the same holds for y.

Axiom A18

Let AB, CD be parallel, equioriented segments. Then AB is a multiple of CD if and only if and there is a positive integer n such that $A_xB_x=nC_xD_x$ and $A_yB_y=nC_yD_y$.

Note that the previous axiom is monadically expressible. In fact, in order to express the existence of n above in monadic second order logic, we can say that there are finite sets $F_1,F_2$ of atoms in $A_xB_x$ and $A_yB_y$ and a bijection $\gamma$ between $F_1$ and $F_2$, such that the subsegments between consecutive points are congruent to $C_xD_x$ and $C_yD_y$ respectively. The finiteness of $F_1,F_2$ can be imposed by saying that every nonempty subset of them has a maximum and a minimum in the betweenness order, and the bijection $\gamma$ can be realized with a set of segments with one extreme in $F_1$ and the other in $F_2$.

Axiom A19

$B\le AC$ (in the order of $K_2$) if and only if $A_xC_x\ge A_xB_x$, $A_yC_y\ge A_yB_y$, and there is D such that AB, AD, AC are equioriented, and AB and AC are multiples of AD.

By the previous two axioms, the relation $B\le AC$ depends only on the coordinates of A, B, C.

1.10 Reduction to atoms and segments

Axiom A20

Every element of $K_2$ is the supremum of a finite set of atoms.

Axiom A21

(from convex polygons to segments). Let $F,F'$ be finite sets of atoms. Then $sup\ F\le sup\ F'$ if and only if every set $T\subseteq K_2$ containing $F'$ and closed under segment (that is if $A,B\in T$, then every atom in AB is in T) contains F.

The previous axiom in a sense reduces the calculation of the convex hull of n points to an iterated calculation of the segment between two points. Note that the axiom is expressible in monadic second order logic, and indeed, it seems crucial for the description of $K_2$ in monadic second order logic.

1.11 The final theorem for $K_2$

Theorem 12.4

Let $K'$ be a lattice satisfying the axioms of this appendix. By Lemma 12.2 there is a function $\beta _2$ which maps each atom of $K'$ to the unique atom of $K_2$ with the same pair of coordinates. Let us extend $\beta _2$ to $K'$ by letting $\beta _2(sup\ F)=sup\ \beta _2(F)$, where F is any finite set of atoms. Then $\beta _2$ is a well defined lattice isomorphism from $K'$ to $K_2$.

Proof

Recall that the relation $B\le AC$ between three atoms depends only on the coordinates of A, B, C. Hence, by induction on n, also the fact that an atom A is below the supremum of n atoms $B_1,\dots ,B_n$ depends only on the coordinates of $A,B_1,\dots ,B_n$. And by a further induction on m, the fact that the supremum of $A_1,\dots ,A_m$ is below the supremum of $B_1,\dots ,B_n$ depends only on the coordinates of $A_1,\dots , A_m,B_1,\dots ,B_n$. Since $\beta _2$ respects the coordinates of the atoms, we have $sup\ F\le sup\ G$ if and only if $sup\ \beta _2(F)\le sup\ \beta _2(G)$, and $sup\ F\le sup\ G$ if and only if $\beta _2(sup\ F)\le \beta _2(sup\ G)$. So $\beta _2$ is monotonic and injective. Moreover $\beta _2$ is surjective on atoms, and since every element of $K_2$ is a finite supremum of atoms, $\beta _2$ is surjective on $K_2$. So $\beta _2$ is bijective and its inverse is monotonic, so $\beta _2$ is an isomorphism. $\square$

Appendix B: Axioms for $K_X$

We note that the axioms for $K_2$ can be modified so to axiomatize every single $K_n$ for every integer $n>2$. We avoid details for simplicity. So we pass directly to the infinite dimensional case, and we axiomatize the lattice $K_X$ when X is infinite.

We propose the following list of axioms.

We did not check whether the list is redundant.

The main difference with $K_2$ is that the lattice $K_2$ is atomic, whereas $K_X$ is atomless. However, in $K_X$ we have a kind of substitute for atoms, which we call pseudoatoms, which can be appropriately axiomatized.

1.1 Basics

We postulate the existence of sets $X,Par,\Pi \subseteq K_X\subseteq L_X$ satisfying the following axioms. Intuitively:

• X is the set of hyperplanes of the form $\{f\in [0,1]^X\mid f(i)=0\}$ for some $i\in X$;

• Par is the set of hyperplanes (parallel to elements of X) of the form $\{f\in [0,1]^X\mid f(i)=a\}$ for some $i\in X$ and for some rational a between 0 and 1;

• a pseudoatom is a finite nontrivial intersection of elements of Par, so it has the form

  $$\begin{aligned} A=\{f\in [0,1]^X\mid f(i_1)=a_1,\dots ,f(i_n)=a_n\} \end{aligned}$$

  where $\{i_1,\dots ,i_n\}\subseteq X$ has size $n\ge 2$;

• $\Pi$ is a set of parallelograms whose vertices are pseudoatoms, and these parallelograms are used, as in “Appendix A”, to define congruence and parallelism of segments.

First of all we postulate:

Axiom B1

$K_X$ with the order induced by $L_X$ is a bounded lattice and contains the minimum and the maximum of $L_X$. The infimum of $K_X$ coincides with the infimum of $L_X$.

Note that the suprema in $K_X$ and $L_X$ differ in general.

1.2 On Par and pseudoatoms

Axiom B2

$X\subseteq Par$. Zero (the minimum of $K_X$) does not belong to Par and is not a pseudoatom. the infimum of two elements of Par is zero or a pseudoatom. the infimum of two pseudoatoms is zero or a pseudoatom.

Axiom B3

Every set containing the binary infima of elements of Par and closed under binary infimum contains all pseudoatoms.

The previous two axioms imply that pseudoatoms coincide with finite infima of at least two elements of Par. Note that the second axiom is expressed in monadic second order logic.

Axiom B4

For every pseudoatom A there is a unique subset T(A) of Par such that A is the infimum of T(A).

Note that the set T(A) is finite and has size at least 2.

Definition

We say that $H,H'\in Par$ are parallel (as hyperplanes) if they are equal or disjoint.

Axiom B5

Parallelism in Par is an equivalence relation. Every element of Par is parallel to a unique element of X.

Axiom B6

Suppose that A, B are pseudoatoms and no element of T(A) is disjoint from any element of T(B). Then $A\wedge B$ is a pseudoatom.

The axiom implies that every finite subset of Par containing no pair of disjoint elements has an infimum which is a pseudoatom.

Definition

$T_X(A)$ is the set of elements of X parallel to some element of T(A).

Definition

Two pseudoatoms A, B are called compatible if $T_X(A)=T_X(B)$.

Now we can replace with pseudoatoms (taken in a fixed compatibility class) the atoms occurring in finite dimensional lattices ${{\,\mathrm{Poly}\,}}([0,1]^n)$, and replace the axioms in dimension 2 of the previous appendix with axioms in codimension n where, intuitively, the “atoms in codimension n” are the pseudoatoms A such that T(A) has n elements.

The segment of two compatible pseudoatoms A, B is denoted by AB.

Axiom B7

Let A, B, C, D be compatible pseudoatoms. If $AB=CD$ then $\{A,B\}=\{C,D\}$.

1.3 Betweenness

Same as the homonymous subsection of “Appendix A”, up to replacing atoms with compatible pseudoatoms.

1.4 Congruence, parallelism and comparison of segments

Same as the homonymous subsection of “Appendix A”, up to replacing atoms with compatible pseudoatoms.

1.5 Parallel sum of segments

Same as the homonymous subsection of “Appendix A”, up to replacing atoms with compatible pseudoatoms.

1.6 Division of segments

Same as the homonymous subsection of “Appendix A”, up to replacing atoms with compatible pseudoatoms.

1.7 The rationality axiom

Same as the homonymous subsection of “Appendix A”, up to replacing atoms with compatible pseudoatoms.

In particular, we can define the ratio of two parallel segments AB, CD, where A, B, C, D are compatible pseudoatoms, assuming $C\not =D$.

1.8 Assigning coordinates to elements of Par and pseudoatoms

The assignment of coordinates changes and is more complicate with respect to “Appendix A”. Parallel sum, congruence, comparison and parallelism are defined for segments with extremes pseudoatoms, but not for “segments with extremes in Par”. The problem is that we do not have “big enough” parallelograms to define these notions. So we choose an indirect way of defining coordinates of elements of Par.

Definition

We define an element $H\in Par$extremal if it does not disconnect the space, that is, for every $U,V\in K_X$ disjoint from H, UV is also disjoint from H.

Axiom B8

Every element of X is extremal.

Axiom B9

For every $H\in X$ there is only another extremal $H'\in Par$ parallel to H, which will be called an anti-coordinate hyperplane.

Intuitively, $H=\{f\in [0,1]^X\mid f(i)=0\}$ for some $i\in X$, and $H'=\{f\in [0,1]^X\mid f(i)=1\}.$

Axiom B10

For every $H\in Par$ and A pseudoatom disjoint from H there is a unique $H'\in Par$ parallel to H and containing A.

Definition

For every pseudoatom A, let $O_A$ (the origin relative to A) be the intersection of $T_X(A)$. Note that $O_A$ is a pseudoatom compatible with A.

Axiom B11

Let A be a pseudoatom and $H\in T_X(A)$. The AH-axis is a segment $O_AB$, where B is the infimum of $T'(A)$, and $T'(A)$ is T(A) where the element parallel to H is replaced by the unique extremal hyperplane disjoint from H. Note that B is a pseudoatom compatible with A.

Definition

If $H\in T_X(A)$, $A\in H'$ and $H'$ is parallel to H, we define $A_H$ the intersection of $H'$ with the AH-axis. $A_H$ is also called the H-th projection of A.

We would like to define parallel sum, congruence, comparison and parallelism between “segments with extremes in the elements of Par parallel to the same $H\in X$”, so to assign “coordinates” to elements of Par. The problem is that segments with hyperplane extremes are “too big”. So we follow an indirect way, by considering the intersection of the elements of Par with coordinate axes relative to any pseudoatom A. To do it, since we have many possibilities for A, we add the following independence axiom:

Axiom B12

(independence of parallel sum of hyperplane segments). Let $H_i\in Par,1\le i\le 6$ parallel to the same $H\in X$. Let A be a pseudoatom such that $H\in T_X(A)$. Let $A_i=H_i\wedge O_AB$. If $A_1A_2=A_3A_4+pA_5A_6$, then the same holds for any other $A'$ such that $H\in T_X(A')$.

The same independence from A holds for congruence of coordinate segments associated to elements of Par:

Axiom B13

(independence of comparison of hyperplane segments). Let $H_i\in Par,1\le i\le 4$ parallel to the same $H\in X$. Let A such that $H\in T_X(A)$. Let $A_i=H_i\wedge O_AB$. If $A_1A_2>A_3A_4$, then the same holds for any other $A'$ such that $H\in T_X(A')$.

We add also an axiom on independence of betweenness:

Axiom B14

Let $H_i\in Par,1\le i\le 3$ parallel to the same $H\in X$. Let A such that $H\in T_X(A)$. Let $A_i=H_i\wedge O_AB$. If $A_1>A_2>A_3$, then the same holds for any other $A'$ such that $H\in T_X(A')$.

Definition

If A is any pseudoatom and $H\in T_X(A)$, the H-coordinate of A is the ratio between the segments $O_AA_H$ and $O_AB$, and the coordinate of $H'\in Par$ parallel to H is the H-th coordinate of the point $H'\wedge O_AB$.

By the previous axioms, the coordinate of any element $H'\in Par$ is a unique, well defined rational number between 0 and 1. For every $H\in X$, every rational number between 0 and 1 is the coordinate of some element $H'\in Par$ parallel to H.

1.9 Pseudoatoms and coordinates

Lemma 12.5

The coordinate assignment gives a bijective function between pseudoatoms A and functions from finite subsets of X to the set of the rational numbers between 0 and 1.

Proof

Any pseudoatom A has a finite set of coordinates, one for each $H\in T_X(A)$. Conversely, let f be a function from a finite subset G of X to the rational numbers between 0 and 1. For every $g\in G$ there is a hyperplane $H_g$ parallel to X with coordinate f(g). Then the intersection of all $H_g$ is a pseudoatom whose g-th coordinate is f(g). $\square$

1.10 Definability of finiteness

Lemma 12.6

• Finite sets of compatible pseudoatoms contained in a segment AB are definable in monadic second order logic in $K_X$.

• Finite sets of compatible pseudoatoms are definable in monadic second order logic in $K_X$.

Proof

The first point holds because compatible pseudoatoms in a segment AB are totally ordered by the relation $A<x<y<B$.

The second point holds because a set of compatible pseudoatoms is finite if and only if the projections of its elements range over a finite set.

$\square$

1.11 Segments and coordinates

This subsection is analogous to the homonymous subsection of “Appendix A”. The difference is that a pseudoatom can have any finite number of coordinates rather than two.

As usual, $A,B,C,D,\dots$ are assumed to be compatible pseudoatoms.

Definition

Two segments AB, CD are equioriented if $O<A_H<B_H$ is equivalent to $O<C_H<D_H$ for every $H\in T_X(A)$.

Axiom B15

Let AB, CD be equioriented segments. Then AB is a multiple of CD if and only if and there is a positive integer n such that $A_HB_H=nC_HD_H$ for every $H\in T_X(A)$.

In order to express the existence of n as above in monadic second order logic, we can say that there is a set F (necessarily finite) which intersects every segment $A_HB_H$ (for $H\in T_X(A)$) in finitely many points, the number of these points is independent from H (via bijections given by suitable sets of segments with extremes in F), and every two consecutive members of F on each axis H span a segment congruent to $C_HD_H$.

Axiom B16

$B\le AC$ if and only if $A_HC_H\ge A_HB_H$ for every $H\in T_X(A)$ and there is D such that AB, AD, AC are equioriented and AC, AB are multiples of AD.

By the previous two axioms, the relation $B\le AC$ among compatible pseudoatoms depends only on the coordinates of A, B, C.

1.12 The dimension reduction axiom

The following axiom in a sense reduces the dimension (and increases the codimension) of pseudoatoms:

Axiom B17

Let A be a pseudoatom. Then $T_X(A)$ is a proper subset of X. Let $H\in X{\setminus } T_X(A)$ and let $H'$ be the extremal element of Par parallel to H and different from H. Then A is the supremum of $A\wedge H$ and $A\wedge H'$.

1.13 From convex bodies to segments

Axiom B18

For every nonzero element e of $K_X$, there is a pseudoatom A such that e is the supremum of a finite set of pseudoatoms compatible with A.

By the previous axioms we have:

Lemma 12.7

Finite subsets of $K_X$ are definable in monadic second order logic.

Proof

By the previous axiom and the dimension reduction axiom, a subset F of $K_X$ is finite if and only if there is a finite set G of compatible pseudoatoms such that every element of F is supremum of a subset of G.

$\square$

Axiom B19

(from convex bodies to segments). Let $F,F'$ be finite sets of compatible pseudoatoms. Then $\sup \ F\le sup\ F'$ if and only if every set T of pseudoatoms containing $F'$ and closed under compatible segment (that is if $A,B\in T$ are compatible with the elements of $F'$, then every element below AB compatible with A is in T) contains F.

The previous axiom reduces the calculation of the convex hull of a compatible finite set of pseudoatoms to a finite iteration of the betweenness relation $A\le BC$. Note that the axiom is expressible in monadic second order logic.

1.14 The final theorem for $K_X$

Theorem 12.8

Let $K'$ be a lattice satisfying the axioms of this appendix, where X is a set of given infinite cardinality $\lambda$. By Lemma 12.5 there is a function $\beta _\lambda$ which maps each pseudoatom in $K'$ to the unique pseudoatom of $K_X$ with the same coordinates. Let us extend $\beta _\lambda$ to $K'$ by letting $\beta _\lambda (sup\ F)=sup\ \beta _\lambda (F)$, where F is any finite set of compatible pseudoatoms. Then $\beta _X$ is a well defined isomorphism from $K'$ to $K_X$.

Proof

Recall that the relation $B\le AC$ between three compatible pseudoatoms depends only on the coordinates of A, B, C. Hence, by induction on n, also the fact that a pseudoatom A is below the supremum of n pseudoatoms $B_1,\dots ,B_n$ depends only on the coordinates of $A,B_1,\dots ,B_n$. And by a further induction on m, the fact that the supremum of $A_1,\dots ,A_m$ is below the supremum of $B_1,\dots ,B_n$ depends only on the coordinates of $A_1,\dots , A_m,B_1,\dots ,B_n$, assuming all of them are compatible. If $A_i$ and $B_j$ are not compatible, we can use the dimension reduction axiom and we can express each of them as a supremum of a single finite subset E of compatible pseudoatoms, and the coordinates of the elements of E depend only on the coordinates of the $A_i$ and $B_j$, so we can assume $A_i$ and $B_j$ compatible.

Since $\beta _\lambda$ respects the coordinates of the pseudoatoms, for every two finite sets F, G of compatible pseudoatoms we have $sup\ F\le sup\ G$ if and only if $\beta _\lambda (sup\ F)\le \beta _\lambda (sup\ G)$. So $\beta _\lambda$ is monotonic and injective. Moreover $\beta _\lambda$ is surjective on pseudoatoms, and since every element of $K_\lambda$ is a finite supremum of pseudoatoms, $\beta _\lambda$ is surjective on $K_\lambda$. So $\beta _\lambda$ is bijective and its inverse is monotonic; summing up, $\beta _\lambda$ is an isomorphism. $\square$