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The logic of causally closed spacetime subsets

The causal structure of space-time offers a natural notion of an opposite or orthogonal in the logical sense, where the opposite of a set is formed by all points non time-like related with it. We show that for a general space-time the algebra of subsets that arises from this negation operation is a complete orthomodular lattice, and thus has several of the properties characterizing the algebra physical propositions in quantum mechanics. We think this fact could be used to investigate causal structure in an algebraic context. As a first step in this direction we show that the causal lattice is in addition atomic, find its atoms, and give necesary and sufficient conditions for ireducibility.


I. INTRODUCTION
The logical propositions that represent physical questions in quantum mechanics are of the form 'the state of the system is an eigenvector of the operator O with eigenvalue λ', for an observable O. The non commutativity of the observable operators gives place to a non Boolean logic for the propositions, in contrast to the case of the propositions about the phase space of a system in classical physics. This is mathematically described by the orthomodular lattice structure of projectors in Hilbert space, which is often called quantum logic [1,2].
The algebra of propositions about the space in classical physics is Boolean. Consider for example a particle, and the set of propositions of the form 'the particle is in the set S'. The opposite in this logic to the proposition given by the subset S of the space is given by −S, the set of all points that do not belong to S. A positive answer to S implies a negative one to −S. The intersection and union of sets corresponds to the conjunction and disjunction of logical propositions about the physical system. The algebra of propositions becomes the same as the Boolean algebra of subsets of the space in set theory. This structure is inherent to the space, because it will be the same for any system. However, as we will see in the following, the situation when including time is different. In the relativistic context the kinematics imposed by the causal structure will induce a non Boolean character for the algebra of propositions given by certain space-time subsets (see [3] for the case of Galilean physics). A particle can not be concentrated at a given point p in space-time because it existed in the past and will exist in the future of p, but we can think that the answer to the proposition given by a point is true if the particle passes through p and false if not. The causal structure gives a definite prescription of what the opposite means in this case, because if the particle passes through p it can not pass through any point spatially separated from p. All the points non related by a time-like curve to a given set S will form the causal and logical complement of S.
A surprising fact about the algebra of propositions that arises along this line of thought from the causal structure of space-time alone, independently of the propagating physical system, is that it is also an orthomodular lattice. Though, we think, not widely known, this very interesting point was shown for Minkowski space some time ago in ref. [4].
In this work we will show how to define the lattice of causally closed sets L(M) for a general space-time M, even non causally well behaved, and prove that it is a complete orthomodular lattice. In addition, we will show that the lattice is atomic, and find its atoms. They are the points for causally well behaved space-times, but in general they can be of three different kinds. We will also find the necessary and sufficient conditions for this lattice to be irreducible.
The lattice L(M) can have interest in two senses. We think that this lattice structure can be more fundamental than the causal structure of the space-time, much in the spirit of the studies of the postulates of quantum mechanics from the lattice perspective [2]. In this sense this paper can be consider as a step in the study of the causal structure of space-times from an algebraic point of view. In fact, we find that certain undesirable aspects of the causal structure of general space-times as closed time-like curves can be eliminated or localized in the structure of L(M), this lattice being largely independent of the details of the metric in the bad sectors of space-time. Two space-times related by a conformal transformation will have the same causal structure and lattice L(M), however the converse can be false.
The other aspect of interest is the possible relation of L(M) with quantum mechanics and quantum field theory, motivated by the intriguing fact that causal structure and Hilbert space share an important piece of its mathematical structure. We will not develop in this sense here. We note, however, that the causally closed sets in L(M) appear naturally in the algebraic approach to quantum field theory. In that context, we have a set of C * -algebras associated to space-time sets. To a set in M and its causal complement correspond two C * -algebras that commute. However, the whole relation between the lattice structure and the operations among local algebras seems not to be completely clear at the moment [5]. We will say a bit more in Section III.
To make this article as much self contained as possible, in the following Section we include a brief introduction to orthomodular lattices and some relevant mathematical preliminaries. Further details can be found in refs. [2,6,7]. In Section III we define the orthocomplemented complete lattice L(M) and discuss several related possibilities for constructing causal set lattices. In Section IV we show that L(M) is orthomodular. In Section V we show that it is atomic and find the atoms, which are an important piece in the understanding of the general structure of the lattice that will allows us to characterize irreducibility. In the last Section we make some comments and discuss directions open to future work.

II. ORTHOMODULAR LATTICES
The family of subsets B(U) of any set U forms a Boolean algebra when considered with the relation of inclusion, and the operations of intersection, union, and complement (denoted ⊆, ∩, ∪ and − respectively). This is at the base of the relation between set theory and classical logic, where the propositions can be restated in terms of the subsets of a given set U in set theory. To a given proposition P we assign a subset S with the interpretation that an element belongs to S if and only if it satisfies the proposition P. Thus, the operations of implication, conjunction, disjunction, and negation between propositions are interpreted as the relation of inclusion, and the operations of intersection, union, and complement between subsets respectively.
In quantum mechanics the space of individuals is given by a Hilbert space H. The corresponding propositions or questions that are answered with a yes or a no, are given by projection operators P , P 2 = P . These operators must be observables, that is, autoadjoint operators, P = P † , such that their domain is all H, and in consequence they are also bounded. The projection operators are in one to one correspondence with the closed subspaces V P of H, such that P ( V P ) = V , and P (V ⊥ P ) = 0. Here we use the notation W ⊥ for the subspace orthogonal to W , which is always closed.
The interpretation is that the proposition P is true for the elements of V P and false for the elements of V ⊥ P . Here is possible to talk in terms of propositions (projection operators) or in terms of closed linear subspaces indistinctly, as it happens in the case of classical logic, where it is possible to talk in terms of propositions or in terms of sets.
The set C(H) of closed subspaces of a Hilbert space can be furnished with a structure that resembles the one in B(U), where the relation of inclusion and the operation of intersection between linear subspaces are the set inclusion and intersection, but where the set complement is replaced by the operation of taking the orthogonal subspace, and the union of two sets by the sum of subspaces. The resulting structure is called an orthomodular lattice. In what follows we will describe this structure in detail.
The inclusion between subsets is an order relation, that makes C(H) (and B(U)) a partially order set (poset), In C(H) or B(U) there always exist elements that are the greatest lower bound (g.l.b.) and the lowest upper bound (l.u.b) of two given elements with respect to the order relation given by ⊆. In B(U) they are given by the set intersection and union while in C(H) by the set intersection and sum of linear subspaces. We will denote generically the g.l.b. and l.u.b. of two elements A and B as A ∧ B, and A ∨ B respectively, and call them the meet and the join of A and B.
The meet and the join are associative and symmetric. A poset where the join and the meet of any two elements always exist is called a lattice. A lattice is in addition complete if the meet and the join exist for arbitrary families of elements.
A lattice L have a unit I and a zero element O if A ⊆ I and O ⊆ A for every element A ∈ L.
An element A in a lattice has a complement B if it is A ∧ B = O and A ∨ B = I. The lattice is called orthocomplemented (or ortholattice) if there exist a unary operation ⊥: L → L that assigns to every element A a complement A ⊥ , and in addition In an orthocomplemented lattice it can be deduced that what shows that the negation is a dual map with respect to the operations ∧ and ∨.
Both, B(U) and C(H) are complete and orthocomplemented. However, for the lattice B(U) holds the distributivity law while it is easy to see that it does not hold for C(H). An orthocomplemented distributive lattice is called a Boolean algebra. However, in the lattice C(H) holds a relaxation of the distributivity, called orthomodularity Orthomodularity is equivalent to the simpler relation For finite dimensional Hilbert spaces a stronger form holds, the modular law, whose definition, as the one of distributivity, makes no use of the orthocomplementation operation. Distributivity implies modularity and, for an orthocomplemented lattice, modularity implies orthomodularity. We see that the orthomodular law (7) means that distributivity holds under special circumstances for the elements involved. In the lattice of closed linear subspaces of the Hilbert space, C(H), when using the representation of projection operators, these conditions can be rewritten as that the projector P A commutes with P B and P C . In a general orthocomplemented lattice two elements A and B are said to commute if The relation of commutativity is symmetric in orthomodular lattices, and the Boolean subalgebras are formed by mutually commuting elements. We will need a few additional definitions. With two lattices L 1 and L 2 we can construct the direct product of lattices L 1 × L 2 defined in the Cartesian product of sets, and where the operations are done component by component. The product of orthomodular lattices is orthomodular. A lattice is irreducible if it can not be written as a direct product of lattices. An orthocomplemented lattice is irreducible if and only if its centre, that is, the set of elements that commute with all other elements, is equal to {I, O}.
An atom of a lattice is a non zero element A such that A and O are the only elements included in A. A lattice is atomic if every non zero element contains an atom. An orthomodular atomic lattice is also atomistic, that is, every non zero element is the join of the atoms that contains. Both, B(U) and C(H) are atomic, its atoms being the points in U and the unidimensional vector spaces in H.

Closure and Galois connection
We will now describe a fundamental method for constructing complete orthocomplemented lattices that will be useful later [6]. The orthocomplemented lattice will be constructed with a subset (not a sublattice) of a given complete lattice L (not necessarily orthocomplemented). The idea is to start with a would be complement operation ⊥ and then to require the elements in the new lattice to be the A ∈ L that satisfy ⊥⊥ A = A. This is imposed by eq.(4). This is similar to the case of a Hilbert space, where, if we start with B(H) and propose the orthogonal to be the orthocomplement operation, the resulting lattice is C(H), because the orthogonal of every set of vectors is a closed linear subspace.
Given a complete lattice L a Galois connection in L is a unary operation g in L such that The operation c = g • g is called a closure operation, and the elements A of L that satisfy A = A c are called closed with respect to the closure operation c. The closure c has the following properties If we want the Galois connection g to be an orthocomplementation we must pick up the elements of the lattice such that the closure operation is the identity operator in them, as is enforced by eq.(4). Indeed, if we further have the set of closed elements L c of L with respect to c, forms a complete orthocomplemented lattice where the order and the meet are the order and the meet in L, the complement is given by g and the join is the join in L followed by the operation of closure c. Finally, we add that there is a very natural way to construct a Galois connection in the lattice B(U). Let R be a symmetric reflexive relation between points in the set U, with no further requirements. Thus pRq ⇔ qRp and pRp. The operation ⊥: X → X ⊥ given by that is, the set of all points not related by R with no point of X, is a Galois connection that satisfies eqs. (11,12) and (16, 17).

III. CAUSAL SET LATTICES
The space-time we will consider is a (Hausdorff) manifold M of dimension d ≥ 2 with a pseudometric tensor g µν with signature (+, −, .., −), smooth and non singular everywhere. We do not ask the space-time M to be time orientable nor to satisfy any causality condition (for example to be free from closed time-like curves). For a review of causal structure see refs. [8,9,10]. Given a definite naming of the local light cones of a point p as future and past light cones, its causal future and past, I + (p) and I − (p), are defined as the set of points that can be reached by time-like curves starting at p with tangent vector in the future and past light cone of p respectively. When referring to time-like curves we always mean differentiable curves with continuous tangent vector except a finite number of points where the tangent can have a discontinuity, but keeping always in the same local light cone. These discontinuities can always be removed by modifications in arbitrarily small neighborhoods. The past and future of p are open sets. We will use the symbol I + (S) ∪ I − (S) for the open set q∈S (I + (q) ∪ I − (q)) that always makes sense, even in non time-orientable space-times.
The natural orhtocomplement operation for sets in M based on the causal structure identifies all that is not time-like related to a given set with its complement. To be explicit, we can define in M a symmetric reflexive relation that relies only on the causal structure. Given two points p and q of M we write p ∼ q, and say that p and q are time-connected or just connected, if there is time-like curve that passes through p and q. Thus, p ∼ q is equivalent to p ∈ I + (q) ∪ I − (q) ∪ {q}. All points that are not time-connected to p must be in the complement of p. Starting from this complement operation, as it was described in the previous Section, we can construct a complete orthocomplemented lattice coming from the Galois connection in B(M) given by eq.(18). We call it the lattice of the causally closed sets L(M). The orthocomplement is given by  The sets generated by subsets in achronal surfaces form Boolean subalgebras.
The meet in L(M) is just the set intersection. The join of a family of elements is given by the set union followed by the causal closure (the double orthogonal) (20) The join of two sets is exemplified in fig. 2. For two spatially separated sets A and B the set A ∨ B will be just the set union A ∪ B, while otherwise A ∨ B will contain also at least all the time-like curves that connect points from A to B. In general, for a set A in M, the causal closure A ⊥⊥ is bigger than the domain of dependence of A, taking this latter as the set of points such that all past (future) inextendible time-like curves from the point intersects the given set [9] (see fig. 2). The lattice L(M) is not distributive nor modular in the general case, as it is shown for Minkowski space in fig. 3(a) and fig.3(b), while it is orthomodular as will be shown in the next Section and is illustrated in fig. 3(c).
As mentioned in the Introduction, the lattice L(M) has a logical interpretation in terms of propositions for classical particles. The proposition corresponding to a space-time subset S is given by "the particle passes through S". However, for an arbitrarily chosen set S the logical opposite of that proposition is "the particle does not passes through S". In general this is not of the form "the particle passes through T " for some space-time subset T . The prescription of taking causally closed sets is just what is needed to have a closed algebra of propositions coming from space-time subsets in the above sense. In contrast to the case of the Boolean logic of space subsets, here only some subsets of space-time have an opposite in terms of space-time subsets, and the resulting algebra is non Boolean. The logical interpretation of the operation ∧ between two propositions corresponding to the sets A and B is given by the maximal space-time proposition that implies "the particle passes through A and B", while the proposition corresponding to A ∨ B is given by the minimal space-time proposition implied by " the particle passes through A or B".
It is immediate that when the manifold M is non connected the lattice L(M) is the direct product of the lattices L(M i ) of each connected component M i of M. In this case the lattice is reducible. We will see a sufficient condition for irreducibility in Section V.
One can try to construct other lattices based on the causal structure of M. For example, the basic symmetric and reflexive relation between points used to define the orthocomplement can be taken as the relation pSq, that holds if there is a space-like curve passing where here A means the usual topological closure of A. This satisfies eqs. (11,12) and (16, 17), and lead to a complete orthocomplemented lattice L O (M) of open sets. However, again due to details in the borders of the sets, the lattice is not orthomodular, as is shown in fig. 4. It can be shown, at least for globally hyperbolic space-times, that distributivity in L O (M) holds under a little more stringent conditions than the required by orthomodularity. In fact, instead of eq.(8) we have It seems essential for orthomodularity that the lattice should contain along with the diamond shaped sets at least the null surfaces. Once we have null surfaces in the lattice its intersection will generate lower dimensional sets as lines and points. This makes difficult a correspondence between the elements in an orthomodular lattice of causal sets with C * -algebras in the algebraic approach to quantum field theory, because the points should be assigned non trivial algebras (two points can generate a set that includes an open set) (see the discussion in [5]). The lattice L O (M) would be a better candidate, but it is only approximately orthomodular in the sense of eq.(22) (weaker conditions for distributivity can also be found).
Form now on we will only refer to L(M). From its definition we see that it does not change with conformal transformations, and thus it is a property of the conformal structure.

IV. ORTHOMODULARITY OF L(M)
We will now prove that L(M) is orthomodular in any space-time. First, we will prove the following previous result. Suppose Therefore, a set S in L(M) contains all time-like segments between points in S and also the time-like border of S, that is, the border of S accessible from S by time-like curves. The points in the space-like border of S may or may not belong to S. Using Lemma 1 we prove To start we assume that B ⊆ A. Let p ∈ A ∧ (B ∨ A ⊥ ), so p ∈ A and p ∈ B ∨ A ⊥ . Therefore no time-like curve connects p with A ⊥ nor with B ⊥ ∧ A. We have to prove that p ∈ B. Then, let us suppose that p / ∈ B and show that it leads to a contradiction. This assumption implies that there is a time-like curve γ that connects p with a point q ∈ B ⊥ . As p is not connected with A ⊥ nor with B ⊥ ∧ A it is q / ∈ A and q / ∈ A ⊥ . A segment of γ with end points p and q intersects B ⊥ and A in non empty sets, which according to Lemma 1 are closed segments. These are disjoint, because otherwise p could be connected with a point in B ⊥ ∧ A. Thus, there is a point r in the segment of γ between p and q that is not in B ⊥ nor in A. As r / ∈ B ⊥ , r can be connected with a point s ∈ B. Thus, as r is in the segment of γ between p and q it is either q connected with s or p connected with s. However, the point q ∈ B ⊥ can not be connected with s ∈ B. Then p ∈ A is connected with s ∈ B ⊆ A by a curve passing through r. Then, by Lemma 1, it is r ∈ A, what shows a contradiction. Therefore p ∈ B and orthomodularity holds.
Of course, there are orthomodular lattices that are not of the form L(M) for a spacetime M. In the following Section we go a step further in the characterization of causal structures from the lattice theoretical point of view by showing that L(M) is atomic, and finding its atoms. In the remaining part of this Section we will extract what are the essential properties of the causal structure used in the proof of orthomodularity (see ref. [11] for a related discussion).
For constructing an orthocomplemented lattice we used a symmetric reflexive relation between points in the space-time with no further properties. Now we can read from the previous theorem what additional structure of the causal relation ∼ that we used in the definition of L(M) is central in the proof of orthomodularity. Basically, what we need is a set of curves in a set M, locally one to one functions of (possible infinite) open intervals of real numbers to M, that we can call time-like curves. Given a point p and any curve γ with γ(x) = p, the points γ(y) with y = x belong to two sets, non necessarily disjoint, that we can call the past and future sets of p. We have either all γ(y) for y > x belong to the future of p and all γ(y) for y < x belong to the past of p or vice versa. If a point q is in the future set of p, then either all the future set of q or the past set of q are included in the future set of p, and, in addition, there is a neighborhood of q (in the sense induced by the real numbers) in all time-like curves passing through q that belongs to the future set of p. The same can be said regarding the past set of p.
These geometrical properties, that are immediate for the time-like curves in a general space-time M, are sufficient in order to have an orthomodular lattice generated by the relation ∼. We see that certain sort of transitivity condition is essential for the causal relation but, for non orientable space-times no global transitive relation can be defined. There is also a condition of continuity. None of these is respected by the relation N defined in the preceding Section, while the relation C does not respect the continuity condition.  L(M), p ⊥⊥ = p, then it is an atom. This is the case for all points in Minkowski space. We will now analyze which of the points of M are atoms in the general case.
As we noted before, if two points p and q belong to S in L(M), then all points in timelike curves connecting p and q are also in S. If there is a time-like segment γ connecting a point p with itself, then p can not be an atom because γ ⊆ p ⊥⊥ . There are two types of such curves, as shown in fig. 5. One, the closed time-like curves, that are totally included in the intersection I + (p) ∩ I − (p), where a segment of the curve between p and p start and end with its tangent vector in the same light cone of p. The other type, that we will call vertex curves, where a segment between p and p start and end with tangent vectors at p in different light cones of p. In this later case we will call p a vertex. For the existence of these types of curves the manifold must not be simply connected [10]. In the vertex curve case we see that the space-time is not time orientable. However, there are time non orientable space-times without vertex curves. The vertex curves mark space-times where the notion of future and past have no sense for a single observer. The points p that belong to closed time-like curves or are vertex points can be characterized as all points that belong to I + (q) ∩ I − (q) for some q of M (any q of γ in the open segment between p and p for example). We will call it the set  an atom. We will show that if p is not an atom then p ∈ B. Let us suppose p ⊥⊥ = p. Thus, there is a point q ∈ p ⊥⊥ with q = p. The point q must be time connected with p otherwise it would be q ∈ p ⊥ . Therefore p ∈ I + (q) ∪ I − (q) and a connected, time orientable, open neighborhood U of p is included in I + (q) ∪ I − (q). Thus, as every point in U is connected with q ∈ p ⊥⊥ , it must be that U ∩ p ⊥ is empty. In other words p ⊥ is separated from p. We have shown that in the closed set M − B all points are atoms. Next we will show that a different kind of atoms can exist in space-times that have both types of defects, closed time-like and vertex curves. Consider the time-like curve β of fig. 6. We will call such a curve a bridge. It is a closed time-like curve but also a past and future vertex curve for all points in the curve. The essential property of a bridge is that for any point p in it one can construct a time-like curve that passes through p with tangent in any light cone and passes through any point q in the bridge with tangent in the any light cone of q. Thus any two points time connected with two points in the bridge β are also connected to each other. Then we have the following Lemma 3: the set β ⊥⊥ generated by any bridge β is an atom Let p ∈ β (see fig. 6). If q ∈ p ⊥⊥ then q is connected with p. The set q ⊥⊥ is included in p ⊥⊥ and if they are different it must be p / ∈ q ⊥⊥ . In that case p must be connected with q ⊥ . But that is not possible because as p belongs to a bridge it would lead to q connected with q ⊥ . Thus q ⊥⊥ = p ⊥⊥ for every q ∈ p ⊥⊥ and p ⊥⊥ is an atom that includes β.
We will call the set β ⊥⊥ generated by a bridge β a bridge atom. Any bridge atom contains an open set. We also have that two atoms cannot intersect. Therefore in a paracompact manifold there can be at most a numerable amount of bridge atoms. The bridge atoms must be totally included in the set of bad points B.
We have shown what happens to the sets p ⊥⊥ for p ∈ M − B or p in β ⊥⊥ where β is a bridge. In both cases p ⊥⊥ is an atom. We will see that this is not always the case. First consider the case where p is a vertex. We have Lemma 4: The set p ⊥⊥ where p is a vertex contains a point in M − B or a bridge atom Let γ(z), z ∈ [0, 1], and γ(0) = γ(1) = p be a vertex curve of the vertex p. Let us suppose that every point in γ is in B. For every value of the parameter z ∈ [0, 1] we define the future light cone of γ(z) at z as the light cone of γ(z) given by the tangent vector of γ at z. Thus, the direction of the tangent vector at the point z = 1 will mark the future light cone of p at z = 1, but it will be a different light cone of the point p than the future light cone at z = 0. For any z such that the point q = γ(z) is a vertex, we can define it as a future or a past vertex according the vertex curve based at q starts with tangent in the future or the past light cone of q at z. Let us call W + ⊆ [0, 1] and W − ⊆ [0, 1] the set of values of the parameter that have future and past vertices respectively. These sets are non empty as 0 has a future vertex and 1 has a past vertex. If w 1 has a past vertex and w 2 has a future vertex and w 1 ≤ w 2 , then both vertex curves together with γ([w 1 , w 2 ]) will form a bridge. Let the infimum of W − be w − and the supremum of W + be w + , and assume w + ≤ w − . Then let us consider now the case of points in closed time-like curves. Let p be a point in C and C p be the connected component of C that includes p. It is C p ⊆ p ⊥⊥ . If C p is a connected component of M then C p will be an atom. If this is not the case, let q be a point in the border of C p . The point q can only be a vertex or a point in M − B. Therefore if p ⊥⊥ contains a point in the border of C p it will contain an atom. But any point in the border of C p that can be reached by a time-like curve passing through a point in p ⊥⊥ must belong to p ⊥⊥ because of Lemma 1. Thus, if there are no point or bridge atoms in p ⊥⊥ we have C p = I + (C p )∪I − (C p ). A point in the set −C p can not be joined by a time-like curve to C p , and, as every point is in the future or the past of a different point, it is −C p = I + (−C p ) ∪I − (−C p ). Therefore C p and −C p are open disjoint sets that cover M, and C p is a connected component of M.
Resuming, we have shown That is, every element is the join of the atoms that includes. Thus, from the algebraic point of view all the bad points of the space-time disappear from the algebra, except the discrete bridge atoms and connected components included in C (see fig. 7). These later obviously commute with all L(M) and then belong to the centre. We can see that the same is true for the bridge atoms as follows. Lemma 5: the bridge atoms belong to the centre of L(M) If an atom does not belong to β ⊥ for a bridge β it must be connected with β. If the atom is a point p then as p is connected with β it implies that p is a vertex, what is not consistent with p being an atom. Then let the bridge χ be connected with β, it is easy to see that both will generate the same set, β ⊥⊥ = χ ⊥⊥ (they form a unique bridge curve). Thus an atom is either equal to β ⊥⊥ or is included in β ⊥ , and therefore commutes with β ⊥⊥ . As L(M) is atomistic, and every element is the join of its atoms, β ⊥⊥ will commute with all the elements of the lattice [7].
We have shown that if the manifold is not connected or if it contains at least two atoms and a bridge then L(M) is reducible. We will complete the following If there is a point p inM −Ŝ −Ŝ ⊥ it will be an atom that does not commute with S (see eq.(10)). Thus, if S is in the centre of L(M), it must beM =Ŝ ∪Ŝ ⊥ , andŜ ∩Ŝ ⊥ = ∅. Let q be a point in M, then, as the atoms that generate q ⊥⊥ are in S or in S ⊥ , we have q ⊥⊥ = (q ⊥⊥ ∩ S) ∨ (q ⊥⊥ ∩ S ⊥ ) = (q ⊥⊥ ∩ S) ∪ (q ⊥⊥ ∩ S ⊥ ), the last equality coming from the form of the join for orthogonal sets. Thus, q belongs to S ∪ S ⊥ , S ∪ S ⊥ = M, S ∩ S ⊥ = O. Then, all future and past of all points of S must be in S, otherwise S would be connected with S ⊥ , and, as for a point r ∈ S there is always a different point s in a neighborhood that is in I + (r) ∪ I − (r), and so r ∈ I + (s) ∪ I − (s) with s ∈ S. Therefore S = I + (S) ∪ I − (S) is an open set. The same can be said for S ⊥ , S ⊥ = I + (S ⊥ ) ∪ I − (S ⊥ ). Therefore, M is non connected. Identify   FIG. 8. The picture on the left shows a space-time constructed over a cylinder with no closed time-like curves but that is not strongly causal. This means that there is a point (all points in B) such that each of its neighborhoods is intersected by a time-like curve more than once. All points are atoms. The structure of the lattice L(M) is shown on the right. It is the horizontal sum of the lattices generated by the atoms in the sets A, B and C. The lattice generated by B is Boolean. The join of an atom in A with an atom in C (or in B) is the whole space-time, since all points are either in the future of a point in C or in the past of a point in A.
The case where L(M) is nontrivial and irreducible is the most interesting for physical space-times. In that case there is a non countably number of atoms formed by the points in M − B, that can be consider the relevant space-time from the algebraic point of view.

VI. DISCUSSION
We have already mentioned that the lattice C(H) of closed linear subspaces of the Hilbert space is complete, atomic, irreducible and orthomodular. It has a further property called the covering law. An element A in a lattice is said to cover B if B ⊆ A and if for every element C such that B ⊆ C ⊆ A it is C = A or C = B. The covering law means that given an atom X and an element A with A ∧ X = O, then X ∨ A covers A. There exist reconstruction theorems stating that a complete, atomic, irreducible, orthomodular lattice, satisfying the covering law, can be represented as a lattice of closed linear subspaces of a vector space with a Hermitian form [2]. Under minor assumptions regarding the field of scalars for the vector space it is the lattice of a Hilbert space. Thus, a lattice with this set of properties characterize almost uniquely Hilbert spaces.
The lattice L(M) is also complete, atomic and orthomodular, and, in most of the physically interesting cases, irreducible. However, the covering law does not apply. For example, the join of two points connected by a time-like curve in Minkowski space does not cover any of them. This is the essential difference with C(H), that obstructs the immersion of the lattice in a linear structure [2,7].
It would be very interesting to explore what additional algebraic properties does the lattice L(M) have, and, if possible, to find a set of properties that characterize the lattices of causally closed sets in the sense of a reconstruction theorem as the already mentioned. Such a possibility is suggested by the fact that the causal structure for stably causal spacetimes determines not only its conformal structure, but also the topological and differential structure of the manifold [9,12]. The next step in the construction of a dictionary between the causal structure and the algebraic setting is to translate into the lattice theoretical language the different causality conditions such as strong causality and stable causality. Taub-NUT like space-times or the space-time of fig. 8 are not strongly causal and the structure of the resulting lattices is that of a horizontal sum of several lattices. Given two lattices L 1 and L 2 , their horizontal sum is the lattice formed by the set union of L 1 and L 2 , where the unit and zero elements are identified (see fig. 8), so that the meet and join of an element of L 1 with an element of L 2 are O and I respectively. As a different kind of example we note that the Galilean space-time gives place to a causal lattice formed by the horizontal sum of the Boolean algebras corresponding to the space at different times. All this suggest that the appearance of horizontal sums is characteristic of non causally well behaved space-times. We postpone the general analysis for a future paper [13].
Another application of the lattice framework could be found in the study of the asymptotic infinity. The spatial corner have a very simple expression in the algebraic context, being simply the image of L(M) under an endomorphism of orthomodular lattices [13].
The causal structure is more fundamental in a logical sense than other aspects of spacetime such as the metric. As we have already recalled, it also determines the topological and differential structure in well behaved space-times. In part motivated by this fact, there are in the literature approaches to quantum gravity that use a poset or lattice structure representing a causal order in space-time as a fundamental object. One is based on an axiomatic generalization of the history approach to quantum mechanics to a situation where the time evolution is less rigid than in ordinary quantum theory [14]. Another one, parts from discrete posets that would represent the causal structure at the Planck scale, and looks at the possible dynamics and the large scale structure that emerges [15]. It would be very interesting to see if the discussion in this paper, also focused in a lattice structure coming from the causal order, could add to these various invertigations.

VII. ACKNOWLEDGMENTS
I would like to thank C. Rovelli for useful discussions. This work was supported by CONICET, Argentina.