Edit: Shortly after posting this, I found where the machinery I develop here was discussed in the literature. Real Algebraic Geometry by Bochnak, Coste, and Roy covers at least most of this material. I may eventually edit this to clean it up and adopt more standard notation, but don't hold your breath.
In algebraic geometry, an affine algebraic set is a subset of which is the set of solutions to some finite set of polynomials. Since all ideals of are finitely generated, this is equivalent to saying that an affine algebraic set is a subset of which is the set of solutions to some arbitrary set of polynomials.
In semialgebraic geometry, a closed semialgebraic set is a subset of of the form for some finite set of polynomials . Unlike in the case of affine algebraic sets, if is an arbitrary set of polynomials, is not necessarily a closed semialgebraic set. As a result of this, the collection of closed semialgebraic sets are not the closed sets of a topology on . In the topology on generated by closed semialgebraic sets being closed, the closed sets are the sets of the form for arbitrary . Semialgebraic geometry usually restricts itself to the study of semialgebraic sets, but here I wish to consider all the closed sets of this topology. Notice that closed semialgebraic sets are also closed in the standard topology, so the standard topology is a refinement of this one. Notice also that the open ball of radius centered at is the complement of the closed semialgebraic set , and these open balls are a basis for the standard topology, so this topology is a refinement of the standard one. Thus, the topology I have defined is exactly the standard topology on .
In algebra, instead of referring to a set of polynomials, it is often nicer to talk about the ideal generated by that set instead. What is the analog of an ideal in ordered algebra? It's this thing:
Definition: If is a partially ordered commutative ring, a cone in is a subsemiring of which contains all positive elements, and such that is an ideal of . By "subsemiring", I mean a subset that contains and , and is closed under addition and multiplication (but not necessarily negation). If , the cone generated by , denoted , is the smallest cone containing . Given a cone , the ideal will be called the interior ideal of , and denoted .
is partially ordered by . If is a set of polynomials and , then . Thus I can consider closed sets to be defined by cones. We now have a Galois connection between cones of and subsets of , given by, for a cone , its positive-set is (I'm calling it the "positive-set" even though it is where the polynomials are all non-negative, because "non-negative-set" is kind of a mouthful), and for , its cone is . is closure in the standard topology on (the analog in algebraic geometry is closure in the Zariski topology on ). A closed set is semialgebraic if and only if it is the positive-set of a finitely-generated cone.
Quotients by cones, and coordinate rings
An affine algebraic set is associated with its coordinate ring . We can do something analogous for closed subsets of .
Definition: If is a partially ordered commutative ring and is a cone, is the ring , equipped with the partial order given by if and only if , for .
Definition: If is closed, the coordinate ring of is . This is the ring of functions that are restrictions of polynomials, ordered by if and only if . For arbitrary , the ring of regular functions on , denoted , consists of functions on that are locally ratios of polynomials, again ordered by if and only if . Assigning its ring of regular functions to each open subset of endows with a sheaf of partially ordered commutative rings.
For closed , , and this inclusion is generally proper, both because it is possible to divide by polynomials that do not have roots in , and because may be disconnected, making it possible to have functions given by different polynomials on different connected components.
What is ? The Nullstellensatz says that its analog in algebraic geometry is the radical of an ideal. As such, we could say that the radical of a cone , denoted , is , and that a cone is radical if . In algebraic geometry, the Nullstellensatz shows that a notion of radical ideal defined without reference to algebraic sets in fact characterizes the ideals which are closed in the corresponding Galois connection. It would be nice to have a description of the radical of a cone that does not refer to the Galois connection. There is a semialgebraic analog of the Nullstellensatz, but it does not quite characterize radical cones.
Positivstellensatz 1: If is a finitely-generated cone and is a polynomial, then if and only if such that .
There are two ways in which this is unsatisfactory: first, it applies only to finitely-generated cones, and second, it tells us exactly which polynomials are strictly positive everywhere on a closed semialgebraic set, whereas we want to know which polynomials are non-negative everywhere on a set.
The second problem is easier to handle: a polynomial is non-negative everywhere on a set if and only if there is a decreasing sequence of polynomials converging to such that each is strictly positive everywhere on . Thus, to find , it is enough to first find all the polynomials that are strictly positive everywhere on , and then take the closure under lower limits. Thus we have a characterization of radicals of finitely-generated cones.
Positivstellensatz 2: If is a finitely-generated cone, is the closure of , where the closure of a subset is defined to be the set of all polynomials in which are infima of chains contained in .
This still doesn't even tell us what's going on for cones which are not finitely-generated. However, we can generalize the Positivstellensatz to some other cones.
Positivstellensatz 3: Let be a cone containing a finitely-generated subcone such that is compact. If is a polynomial, then if and only if such that . As before, it follows that is the closure of .
proof: For a given , , an intersection of closed sets contained in the compact set , which is thus empty if and only if some finite subcollection of them has empty intersection within . Thus if is strictly positive everywhere on , then there is some finitely generated subcone such that is strictly positive everywhere on , and is finitely-generated, so by Positivstellensatz 1, there is such that .
For cones that are not finitely-generated and do not contain any finitely-generated subcones with compact positive-sets, the Positivstellensatz will usually fail. Thus, it seems likely that if there is a satisfactory general definition of radical for cones in arbitrary partially ordered commutative rings that agrees with this one in , then there is also an abstract notion of "having a compact positive-set" for such cones, even though they don't even have positive-sets associated with them.
An example of cone for which the Positivstellensatz fails is , the cone of polynomials that are non-negative on sufficiently large inputs (equivalently, the cone of polynomials that are either or have positive leading coefficient). , and is strictly positive on , but for , .
However, it doesn't really look is trying to point to the empty set; instead, is trying to describe the set of all infinitely large reals, which only looks like the empty set because there are no infinitely large reals. Similar phenomena can occur even for cones that do contain finitely-generated subcones with compact positive-sets. For example, let . , but is trying to point out the set containing and all positive infinitesimals. Since has no infinitesimals, this looks like .
To formalize this intuition, we can change the Galois connection. We could say that for a cone , , where is the field of hyperreals. All you really need to know about is that it is a big ordered field extension of . is the set of hyperreals that are bigger than any real number, and is the set of hyperreals that are non-negative and smaller than any positive real. The cone of a subset , denoted will be defined as before, still consisting only of polynomials with real coefficients. This defines a topology on by saying that the closed sets are the fixed points of . This topology is not because, for example, there are many hyperreals that are larger than all reals, and they cannot be distinguished by polynomials with real coefficients. There is no use keeping track of the difference between points that are in the same closed sets. If you have a topology that is not , you can make it by identifying any pair of points that have the same closure. If we do this to , we get what I'm calling ordered affine -space over .
Definition: An -type over is a set of inequalities, consisting of, for each polynomial , one of the inequalities or , such that there is some totally ordered field extension and such that all inequalities in are true about . is called the type of . Ordered affine -space over , denoted is the set of -types over .
Compactness Theorem: Let be a set of inequalities consisting of, for each polynomial , one of the inequalities or . Then is an -type if and only if for any finite subset , there is such that all inequalities in are true about .
proof: Follows from the compactness theorem of first-order logic and the fact that ordered field extensions of embed into elementary extensions of . The theorem is not obvious if you do not know what those mean.
An -type represents an -tuple of elements of an ordered field extension of , up to the equivalence relation that identifies two such tuples that relate to by polynomials in the same way. One way that a tuple of elements of an extension of can relate to elements of is to equal a tuple of elements of , so there is a natural inclusion that associates an -tuple of reals with the set of polynomial inequalities that are true at that -tuple.
A tuple of polynomials describes a function , which extends naturally to a function by is the type of , where is an -tuple of elements of type in an extension of . In particular, a polynomial extends to a function , and is totally ordered by if and only if , where and are elements of type and , respectively, in an extension of . if and only if , so we can talk about inequalities satisfied by types in place of talking about inequalities contained in types.
I will now change the Galois connection that we are talking about yet again (last time, I promise). It will now be a Galois connection between the set of cones in and the set of subsets of . For a cone , . For a set , . Again, this defines a topology on by saying that fixed points of are closed. is ; in fact, it is the topological space obtained from by identifying points with the same closure as mentioned earlier. is also compact, as can be seen from the compactness theorem. is not (unless ). Note that model theorists have their own topology on , which is distinct from the one I use here, and is a refinement of it.
The new Galois connection is compatible with the old one via the inclusion , in the sense that if , then (where we identify with its image in ), and for a cone , .
Like our intermediate Galois connection , our final Galois connection succeeds in distinguishing and from and , respectively, in the desirable manner. consists of the type of numbers larger than any real, and consists of the types of and of positive numbers smaller than any positive real.
Just like for subsets of , a closed subset has a coordinate ring , and an arbitrary has a ring of regular functions consisting of functions on that are locally ratios of polynomials, ordered by if and only if , where is a representation of as a ratio of polynomials in a neighborhood of , either and , or and , and if and only if . As before, for closed .
is analogous to from algebraic geometry because if, in the above definitions, you replace "" and "" with "" and "", replace totally ordered field extensions with field extensions, and replace cones with ideals, then you recover a description of , in the sense of .
What about an analog of projective space? Since we're paying attention to order, we should look at spheres, not real projective space. The -sphere over , denoted , can be described as the locus of in .
For any totally ordered field , we can define similarly to , as the space of -types over , defined as above, replacing with (although a model theorist would no longer call it the space of -types over ). The compactness theorem is not true for arbitrary , but its corollary that is compact still is true.
should be thought of as the -sphere with infinitesimals in all directions around each point. Specifically, is just , a pair of points. The closed points of are the points of , and for each closed point , there is an -sphere of infinitesimals around , meaning a copy of , each point of which has in its closure.
should be thought of as -space with infinitesimals in all directions around each point, and infinities in all directions. Specifically, contains , and for each point , there is an -sphere of infinitesimals around , and there is also a copy of around the whole thing, the closed points of which are limits of rays in .
and relate to each other the same way that and do. If you remove a closed point from , you get , where the sphere of infinitesimals around the removed closed point becomes the sphere of infinities of .
More generally, if is a totally ordered field, let be its real closure. consists of the Cauchy completion of (as a metric space with distances valued in ), and for each point (though not for points that are limits of Cauchy sequences that do not converge in ), an -sphere of infinitesimals around , and an -sphere around the whole thing, where is the locus of in . does not distinguish between fields with the same real closure.
This Galois connection gives us a new notion of what it means for a cone to be radical, which is distinct from the old one and is better, so I will define to be . A cone will be called radical if . Again, it would be nice to be able to characterize radical cones without referring to the Galois connection. And this time, I can do it. Note that since is compact, the proof of Positivstellensatz 3 shows that in our new context, the Positivstellensatz holds for all cones, since even the subcone generated by has a compact positive-set.
Positivstellensatz 4: If is a cone and is a polynomial, then if and only if such that .
However, we can no longer add in lower limits of sequences of polynomials. For example, for all real , but , even though is radical. This happens because, where is the type of positive infinitesimals, for real , but . However, we can add in lower limits of sequences contained in finitely-generated subcones, and this is all we need to add, so this characterizes radical cones.
Positivstellensatz 5: If is a cone, is the union over all finitely-generated subcones of the closure of (again the closure of a subset is defined to be the set of all polynomials in which are infima of chains contained in ).
Proof: Suppose is a subcone generated by a finite set , and is the infimum of a chain . For any , if for each , then for each , and hence . That is, the finite set of inequalities does not hold anywhere in . By the compactness theorem, there are no -types satisfying all those inequalities. Given , , so ; that is, .
Conversely, suppose . Then by the compactness theorem, there are some such that . Then , is strictly positive on , and hence by Positivstellensatz 4, such that . That is, is a chain contained in , a finitely-generated subcone of , whose infimum is .
Ordered commutative algebra
Even though they are technically not isomorphic, and are closely related, and can often be used interchangeably. Of the two, is of a form that can be more easily generalized to more abstruse situations in algebraic geometry, which may indicate that it is the better thing to talk about, whereas is merely the simpler thing that is easier to think about and just as good in practice in many contexts. In contrast, and are different in important ways. The situation in algebraic geometry provides further reason to pay more attention to than to .
The next thing to look for would be an analog of the spectrum of a ring for a partially ordered commutative ring (I will henceforth abbreviate "partially ordered commutative ring" as "ordered ring" in order to cut down on the profusion of adjectives) in a way that makes use of the order, and gives us when applied to . I will call it the order spectrum of an ordered ring , denoted . Then of course can be defined as . should be, of course, the set of prime cones. But what even is a prime cone?
Definition: A cone is prime if is a totally ordered integral domain.
Definition: is the set of prime cones in , equipped with the topology whose closed sets are the sets of prime cones containing a given cone.
An -type can be seen as a cone, by identifying it with , aka . Under this identification, , as desired. The prime cones in are also the radical cones such that is irreducible. Notice that irreducible subsets of are much smaller than irreducible subsets of ; in particular, none of them contain more than one element of .
There is also a natural notion of maximal cone.
Definition: A cone is maximal if and there are no strictly intermediate cones between and . Equivalently, if is prime and closed in .
Maximal ideals of correspond to elements of . And the cones of elements of are maximal cones in , but unlike in the complex case, these are not all the maximal cones, since there are closed points in outside of . For example, is a maximal cone, and the type of numbers greater than all reals is closed. To characterize the cones of elements of , we need something slightly different.
Definition: A cone is ideally maximal if is a totally ordered field. Equivalently, if is maximal and is a maximal ideal.
Elements of correspond to ideally maximal cones of .
also allows us to define the radical of a cone in an arbitrary partially ordered commutative ring.
Definition: For a cone , is the intersection of all prime cones containing . is radical if .
Conjecture: is the union over all finitely-generated subcones of the closure of (as before, the closure of a subset is defined to be the set of all elements of which are infima of chains contained in ).
Definition: An ordered ringed space is a topological space equipped with a sheaf of ordered rings. An ordered ring is local if it has a unique ideally maximal cone, and a locally ordered ringed space is an ordered ringed space whose stalks are local.
can be equipped with a sheaf of ordered rings , making it a locally ordered ringed space.
Definition: For a prime cone , the localization of at , denoted , is the ring equipped with an ordering that makes it a local ordered ring. This will be the stalk at of . A fraction () is also an element of for any prime cone whose interior ideal does not contain . This is an open neighborhood of (its complement is the set of prime cones containing ). There is a natural map given by , and the total order on extends uniquely to a total order on the fraction field, so for , we can say that at if this is true of their images in . We can then say that near if at every point in some neighborhood of , which defines the ordering on .
Definition: For open , consists of elements of that are locally ratios of elements of . is ordered by if and only if near (equivalently, if at ).
, and this inclusion can be proper. Conjecture: as locally ordered ringed spaces for open . This conjecture says that it makes sense to talk about whether or not a locally ordered ringed space looks locally like an order spectrum near a given point. Thus, if this conjecture is false, it would make the following definition look highly suspect.
Definition: An order scheme is a topological space equipped with a sheaf of ordered commutative rings such that for some open cover of , the restrictions of to the open sets in the cover are all isomorphic to order spectra of ordered commutative rings.
I don't have any uses in mind for order schemes, but then again, I don't know what ordinary schemes are for either and they are apparently useful, and order schemes seem like a natural analog of them.