Resolutions of Convex Geometries

Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions -- compounds of hypergraphs, as in Chein, Habib and Maurer (1981), and compositions of set systems, as in Mohring and Radermacher (1984) -- , resolutions of convex geometries always yield a convex geometry. We investigate resolutions of special convex geometries: ordinal and affine. A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine. A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones. We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements. Several open problems are listed.


Introduction
Convex geometries, named after Edelman and Jamison (1985), are finite mathematical structures that capture combinatorial features of convexity from various settings. A 'resolution' of convex geometries, as we define it here, is a procedure that builds a new convex geometry from given ones. Intuitively, this procedure replaces each element of a given 'base' convex geometry by another convex geometry, called a 'fiber', and consistently defines a new family of convex sets on the union of the fibers.
Resolutions of convex geometries are related to a similar construction in the field of choice theory, namely 'resolutions of choice spaces', 1 recently introduced by Cantone, Giarlotta and Watson (2020). This is hardly surprising in view of an enlightening result of Koshevoy (1999), who shows that there is a one-to-one correspondence between convex geometries and special choice spaces, called 'path independent'. Our construction is also reminiscent of the 'compounds of hypergraphs' (Chein et al., 1981) or 'compositions of set systems' (Möhring and Radermacher, 1984). There is, however, a salient difference between the two constructions: a composition of convex geometries may fail to be a convex geometry, whereas a resolution of convex geometries is always a convex geometry.
In this paper, we examine resolutions of two special types of convex geometries: ordinal and affine. We also investigate 'primitive' convex geometries, that is, convex geometries that cannot be obtained as resolutions of smaller convex geometries. In particular, we show that the notion of a primitive convex geometry generalizes the classical notion of a primitive poset. We also perform some computations related to 'small' convex geometries, showing that among the 6 convex geometries on three elements only 1 is primitive, and exactly 12 of the 34 geometries on four elements are primitive.
The paper is organized as follows. Section 2 collects preliminary facts on convex geometries. Section 3 introduces resolutions of convex geometries, and compares them to compositions. Section 4 deals with resolutions of affine and ordinal convex geometries. Section 5 provides a taxonomy of all primitive convex geometries having at most 4 elements. Section 6 collects several questions and open problems.

Background on Convex Geometries
In this section we provide the basic notions and tools for our analysis.

Definitions and Examples
We start with one of the many equivalent definitions of a convex geometry, a notion that is originally due to Edelman and Jamison (1985). Unless otherwise specified, X is a finite nonempty set, which consists of points or elements, depending on the context. As usual, 2 X denotes the family of all subsets of X.
Definition 2.1. A convex geometry on a finite nonempty set X is a collection G of subsets of X satisfying the following three axioms: (G1) ∅ P G; (G2) G is closed under intersection: if F and G are in G, then F X G is in G; (G3) G is upgradable: for any G in GztXu, there exists x in XzG such that G Y txu P G.
Here X is the ground set of G, and the sets in G are called convex. We slightly abuse terminology, and also call the pair pX, Gq a convex geometry. A convex geometry pX, Gq is nontrivial if |X| ě 2, and trivial otherwise.
Since X is finite, Axioms (G1) and (G3) of a convex geometry pX, Gq imply that X " Ť G P G. Thus, for any A P 2 X , the family of convex sets G such that A Ď G always contains X. This fact, along with Axiom (G2), ensures the soundness of the following notion: Definition 2.2. Let pX, Gq be a convex geometry. For any A P 2 X , the convex hull of A in X is the smallest convex superset of A, that is, Whenever the family G is clear from context, we shall often simplify notation, and write convpAq in place of conv G pAq.
Example 2.4. (Ordinal convex geometries) Let pX, ďq be a nonempty finite poset. Further, let G be the collection of all ideals of pX, ďq, that is, G :" G P 2 X p@x, y P Xq px ď y^y P Gq ùñ x P G ( . Then pX, Gq is a convex geometry, which is the ordinal convex geometry derived from the partial order ď (or, equivalently, from the poset pX, ďq).
Ordinal convex geometries have a very simple characterization: Theorem 2.5 (Edelman and Jamison, 1985). A convex geometry pX, Gq is ordinal if and only if G is closed under union. Further, if pX, Gq is a convex geometry with G closed under union, then there is exactly one partial order ď on X such that G consists of the ideals of ď: in fact, x ď y if and only if x P convptyuq, for all x, y P X.
The partial order ď in Theorem 2.5 is said to be associated to the ordinal convex geometry G. Notice that, for all z, t P X, we have Example 2.6. (Affine convex geometries) Let X be a nonempty finite set of points in some real affine space R d . Moreover, let G be the collection of all sets obtained as intersections of X with convex subsets of R d , that is, Then pX, Gq is a convex geometry, which we call affinely embedded. The family G is the geometry induced on the subset X of R d . For any subset A of X, we have conv G pAq " X X conv R pAq, where conv R denotes the convex hull in R d . A convex geometry is affine if it is isomorphic to some affinely embedded convex geometry.
A special feature of affine convex geometries is that they are atomistic, which means that all their one-element sets are convex (Edelman and Jamison, 1985). The problem of algorithmically characterizing affine convex geometries is nontrivial: on this topic, see Hoffmann and Merckx (2018).
Substructures of arbitrary convex geometries are defined with a procedure similar to the construction of affine convex geometries in Example 2.6: Definition 2.7. Let pZ, Gq be a convex geometry, and ∅ ‰ X Ď Z. The convex geometry H induced on X by G consists of all intersections of X with elements of G. 3 In this case, we also say that pX, Hq is a subgeometry of pZ, Gq.

Extreme Elements
The notion of an 'extreme element' of a set is central in the theory of convex geometries (see, for instance, Edelman and Jamison, 1985).
Definition 2.8. Let pX, Gq be a convex geometry, and A P 2 X . An element a P A is an extreme element of A if a R convpAztauq. We write ex G pAq for the set of extreme elements of A, or simply expAq when there is no risk of confusion. The function ex G : 2 X Ñ 2 X is the extreme operator on pX, Gq.
Given a convex geometry pX, Gq, any nonempty set A Ď X always has at least one extreme element; in particular, ex G ptxuq " txu for any x P X. Observe also that the extreme operator ex G : 2 X Ñ 2 X is such that ex G p∅q " ∅ and ∅ ‰ ex G pAq Ď A for all A P 2 X zt∅u; that is, according to Footnote 1 or to Definition 2.15 below, the pair pX, ex G q is a choice space.
Example 2.9. We determine the extreme elements of some sets with respect to the convex geometries of Example 2.3. For A " tx, zu and B " ty, zu, we have: The next four lemmas collect several properties of the extreme operator, which will be used in later sections. Since the first three lemmas are well-known, we shall only prove the fourth.
Lemma 2.10. Let pX, Gq be a convex geometry. The following properties are equivalent for any A P 2 X and a P A: (1) a P expAq; (2) convpAztauq Ĺ convpAq; (3) pDG P Gq`Aztau Ď G^a R G˘.
Theorem 2 in Monjardet and Raderanirina (2001) yields Lemma 2.11. Let pX, Gq be a convex geometry. For any A P 2 X , we have: The following properties are given in Edelman and Jamison (1985): Lemma 2.12. Let pX, Gq be a convex geometry. For any G P G and E P 2 X , we have: Finally, we prove Lemma 2.13. Let pX, Gq be a convex geometry. For any A, B P 2 X , we have: Proof. To prove (i), suppose A Ď B, and let a P A X expBq. By Lemma 2.10, there exists G in G such that Bztau Ď G and a R G. Then Aztau Ď G and a R G, hence a P expAq again by Lemma 2.10.
To prove (ii), let p P expAqXB. Toward a contradiction, suppose expAYBqXB is empty. Then, expA Y Bq Ď A Y B yields expA Y Bq Ď A, which in turn implies convpA Y Bq Ď convpAq by Lemma 2.11(ii). It follows that B Ď convpAq. On the other hand, Lemma 2.11(i) gives p P expconvpAqq. By Lemma 2.12(i), it follows that convpAqztpu is a convex set. Since the latter set includes pAY Bqztpu but does not contain p, Lemma 2.10 entails p P expA Y Bq. We conclude that p P expA Y Bq X B " ∅, a contradiction. Lemma 2.13(i) has a simple rephrasing that we will often use: If an element of a set A is not extreme in A, then it cannot be extreme in any superset of A.
Remark 2.14. Lemma 2.13(ii) cannot be strengthened (as its proof might suggest) by requiring that the inclusion expAq X B Ď expA Y Bq X B holds. Consider, for instance, the convex geometry G 5 on X " tx, y, zu defined in Example 2.3. Then, for A :" tx, zu and B :" X, we have The next definition, due to Plott (1973), introduces 'path independent choice spaces', which are important structures in mathematical economics.
Definition 2.15. A function c : 2 X Ñ 2 X such that, for any A P 2 X , is called a choice correspondence; in this case, the pair pX, cq is a choice space.
If, in addition, c satisfies the property of path independence, i.e., for any A, B P 2 X , then pX, cq is a path independent choice space.
Path independent choice spaces come up in the analysis of convex geometries because of the following striking result: Theorem 2.16 (Koshevoy, 1999). If pX, Gq is a convex geometry, then pX, exq is a path independent choice space; in particular, for all A, B P 2 X , we have Conversely, if pX, cq is a path independent choice space, then there is a unique convex geometry G c on X whose extreme operator coincides with c, namely G c " G P 2 X`@ A P 2 X˘`c pAq " cpGq ùñ A Ď G˘( . Theorem 2.16 allows one to establish properties of convex geometries by translating properties of path independent choice spaces. For instance, the two implications in Lemma 2.13 can also be derived from path independence: see, e.g., Moulin (1985) for (i), and Cantone et al. (2020) for some rephrasing of (ii). Several additional properties of the extreme operator of a convex geometry pX, Gq can also be derived as analogous features of choice correspondences. For instance, the following property (Aizerman Property) holds for any A, B P 2 X : In fact, Aizerman and Malishevski (1981) show that the join of the property in Lemma 2.13(i) 4 and Aizerman Property is equivalent to path independence.
We close this section with two remarks about the main examples of convex geometries that will be examined in this paper.
Remark 2.17. Let pX, Gq be an ordinal convex geometry having ď as associated partial order. It is immediate to check that the extreme operator coincides with the operator of maximization w.r.t. ď, that is, for all nonempty A Ď X, expAq " maxpA, ďq :" ta P A p@a 1 P Aq a ď a 1 ùñ a " a 1 u .
In other words, the extreme elements of a set in an ordinal convex geometry are exactly the non-dominated ones. The choice space pX, exq that arises in view of Theorem 2.16 is said to be 'rationalizable' by the partial order ď. 5 Remark 2.18. For a finite subset X of a real affine space R d as in Example 2.6, let G be the convex geometry induced on X. Then the extreme elements of a subset A of X are the vertices of the polytope conv R pAq. In other words, expAq is composed of all the points a of A with the property that some linear functional on R d is maximized on A at a but at no other point of A.

Resolutions of Convex Geometries
Here we introduce the main notion of this paper, namely resolutions of convex geometries. This is an application of a general notion of resolution, which captures the concept of expanding a mathematical structure by substituting each of its elements by structures of the same type.
Historically, resolutions were originally defined in a topological setting by Fedorcuk (1968), and then extensively studied by Watson (1992). Resolutions have proven to be extremely useful in set-theoretic topology, providing a unified point of view for many seemingly different topological spaces.
Very recently, the notion of resolution has been adapted to the field of choice theory by Cantone et al. (2020). Upon restricting this notion to path independent choice spaces, and in view of the structural bijection provided by Theorem 2.16, resolutions of convex geometries naturally arise.

Definition and Examples
Definition 3.1. Let pX, G X q be a convex geometry, the base geometry, and tpY x , G x q x P Xu a family of convex geometries, the fiber geometries, where the sets Y x 's are pairwise disjoint and also disjoint from X. Let and define the projection by π : Z Ñ X , z Þ Ñ x for all x P X and z P Y x . 5 The rationalizability of a choice space is a well-known notion within the classical theory of revealed preferences, pioneered by the economist Samuelson (1938). Formally, a choice space pX, cq is rationalizable if there is an acyclic binary relation À (not necessarily a partial order) on X such that, for any A P 2 X , the equality cpAq " maxpA, Àq " ta P A pEa 1 P Aq a ă a 1 u holds. We refer the reader to the collection of papers by Suzumura (2016) for a vast account of the topic of rationalizable choices. For some related notions of bounded rationality, which use binary relations or similar tools to explain choice behavior in different ways, see the recent survey by Giarlotta (2019) and references therein.
The resolution of pX, G X q into tpY x , G x q x P Xu is the pair pZ, G Z q, where G Z consists of all subsets A of Z satisfying the following three requirements: We shall use the suggestive notation pZ, G Z q " pX, G X q h pY x , G x q xPX , and generically call pZ, G Z q a resolution of convex geometries. A resolution of convex geometries is nontrivial if both the base and at least one fiber have more than one element. (See Figure 1, where each element x of the base X " t1, 2, 3, 4, 5u is resolved into a fiber Y x , to obtain a nontrivial resolution.) Similarly, we define the composition pZ, C Z q " pX, G X q a pY x , G x q xPX of the base pX, G X q into the fibers pY x , G x q, x P X, by letting C Z be the family of all subsets of Z that satisfy (R1) and (R2) (but not necessarily (R3)). The nontriviality of a composition of convex geometries is defined similarly to resolutions. Figure 1: A nontrivial resolution pZ, G Z q " pX, G X q h pY x , G x q xPX , having X " t1, . . . , 5u as base set; each fiber Y x is above the corresponding element x P X.
Compositions of convex geometries are special cases of 'compound of hypergraphs' (Chein et al., 1981), and of (the equivalent notion of) 'compositions of set systems' (Möhring and Radermacher, 1984), hence also of the more abstract 'set operads' (Méndez, 2015). Contrary to what happens for resolutions, compositions do not differentiate fibers according to the status of their index (whereas resolutions do because of Requirement (R3)). This is the reason why a composition of convex geometries may fail to be a convex geometry, as the next example shows.
Example 3.2. Let X " t1, 2u, G X " t∅, t1u, Xu, Clearly, pX, G X q and pY i , G i q, for i " 1, 2, are all convex geometries; in fact, they are even ordinal. Their (nontrivial) composition C Z , defined on Z " ta 1 , b 1 , a 2 u, contains both ta 1 , a 2 u and tb 1 , a 2 u but not their intersection ta 2 u, and so it fails to be a convex geometry. Instead, the resolution pZ, G Z q " pX, G X q h pY i , G i q 2 i"1 is a convex geometry (in fact, an ordinal one), because G Z " t∅, ta 1 u, tb 1 u, ta 1 , b 1 u, Zu.
As announced in the Introduction, we have: Theorem 3.3. A resolution of convex geometries is a convex geometry.
Proof. Let pX, G X q be a convex geometry, and tpY x , G x q x P Xu a family of convex geometries such that the sets Y x 's are pairwise disjoint and disjoint from X. To prove that also pZ, G Z q " pX, G X q h pY x , G x q xPX is a convex geometry, we show that G Z satisfies Axioms (G1)-(G3) in Definition 2.1 by using Requirements (R1)-(R3) in Definition 3.1. To simplify notation, here we abbreviate ex G X into ex.
(G2) To prove that G Z is closed under intersection, suppose F and G are arbitrary elements of G Z , and so they satisfy (R1)-(R3). We show that F X G satisfies (R1)-(R3) as well.
For (R1), we prove πpF X Gq P G X . Set E :" pπpF q X πpGqqzπpF X Gq.
Elementary computations yield πpF X Gq " pπpF qzEq X pπpGqzEq. Since G X is closed under intersection, it suffices to show that both πpF qzE and πpGqzE are in G X .
The proof that the inclusion E Ď expπpGqq holds is similar, and so the Claim is established.
Since πpF q, πpGq P G X by (R1), now the Claim and Lemma 2.12(iii) allow us to conclude what we were after, namely πpF qzE and πpGqzE are in G X .
Finally, to establish (R3), let x P πpF X Gqz expπpF X Gqq. Then x P πpF q, and moreover x R expπpF qq by Lemma 2.13(i) (for A " πpF X Gq and On the other hand, if F is a union of fibers, then, since F ‰ Z and πpF q P G X , there is x in XzπpF q such that πpF q Y txu P G X . There exists y in Y x such that tyu P G x . Then F Y tyu P G Z (because x P expπpF Y tyuq).
This completes the proof.
Remark 3.4. Let pZ, G Z q " pX, G X q h pY x , G x q xPX be a resolution of convex geometries. For any A P 2 X , we have hence, in particular, for any x P X, is vacuously satisfied, because the unique element of txu is extreme.) Also, the resolution is atomistic if and only if the base and all fibers are atomistic. Moreover, for any x P X, the equivalence , S intersects each fiber in exactly one element), then the convex geometry induced by pZ, G Z q on S is isomorphic to pX, G X q.
We shall show in Section 4 that the two main classes of convex geometries examined in this paper, namely ordinal (Example 2.4) and affine (Example 2.6), behave in a radically different way with respect to resolutions: in fact, ordinality is preserved whereas affineness is not. Specifically, the resolution of ordinal convex geometries is again ordinal, and this procedure encapsulates a well-known construction on posets (Theorem 4.14). On the other hand, we shall provide an elementary example of a resolution of affine convex geometries that fails to be affine (Example 4.2).
We conclude these preliminaries with a technical result, which describes the extreme operator and the convex hull operator in a resolution. and Proof. Let A P 2 Z . We prove (5). To start, we show that the right-hand side of (5) belongs to G Z . Indeed, it satisfies (R1), because its projection is equal to conv G X pπpAqq, and so it is in G X . Moreover, it satisfies (R2) and (R3), too: use the special case ex G X pconv G X pπpAqqq " ex G X pπpAqq of Lemma 2.11(i). This proves that the right-hand side of (5) is convex. Therefore, to complete the proof of (5), it suffices to show that any convex set G in G Z that includes A also includes the right-hand side.
To that end, let G P G Z be such that A Ď G. Take any . This proves that (5) holds.
Next, we prove (6). For the forward inclusion, let z P ex G Z pAq. Set x :" πpzq. To prove the claim, we show that Then the set πpGq is convex in G X and includes πpAqztxu but not x, which again implies This completes the proof of the forward inclusion in (6).
For the reverse inclusion, let x P ex G X pπpAqq and z P ex Gx pAXY x q. It suffices to show z R conv G Z pAztzuq, which we do by applying (5) to Aztzu in two exhaustive

Primitivity vs Resolvability. Shrinkable Sets
Using resolutions, convex geometries can be partitioned in two classes: Definition 3.6. A convex geometry is primitive (or irresolvable) if it cannot be obtained as a nontrivial resolution; otherwise, it is resolvable.
How can we directly recognize whether a given convex geometry is primitive or resolvable? Definition 3.1 implies that the convex geometry resulting from a resolution is partitioned into fibers. Then a naive algorithm to test primitiveness should check one by one all nontrivial partitions of the convex geometry, derive the corresponding base and fibers, and stop when their resolution is the initial convex geometry. Obviously, this algorithm is highly ineffective. A better algorithm can be derived from Theorem 3.8 below. To state it, we need the following notion of 'shrinkability': Definition 3.7. Let pZ, Gq be a convex geometry. A subset S of Z is shrinkable (in Z) if it is a nontrivial fiber in some nontrivial resolution producing pZ, Gq.
Observe that if S is shrinkable in X, then 1 ă |S| ă |X|. Clearly, a convex geometry is primitive if and only if it has no shrinkable set. The following characterization of shrinkability yields a more effective test for the primitiveness of a convex geometry: Theorem 3.8. Let pZ, G Z q be a convex geometry. The following statements are equivalent for a subset S of Z such that 1 ă |S| ă |Z|: (i) S is shrinkable; (ii) S satisfies the following two properties: then the result holds trivially). Then s P ex G X pπpGqq by (R3), hence Lemma 2.12(i) yields πpGzY s q " πpGqztsu P G X , which proves that (R1) holds for GzY s . Furthermore, GzY s satisfies Requirement (R2), because so does G. Thus, to prove that GzY s P G Z , it remains to show that also Requirement (R3) holds for GzY s . To that end, observe that Lemma 2.13(i) yields πpGzY s q X ex G X pπpGqq Ď ex G X pπpGzY s qq. It follows that any non-extreme element of πpGzY s q is also non-extreme in πpGq. Hence GzY satisfies (R3) because so does G. For (ii) ùñ (i). Suppose S is a subset of the convex geometry pZ, G Z q such that 1 ă |S| ă |Z|, and Properties (S1) and (S2) hold for all G, H P G Z . To show that S is shrinkable, we shall express pZ, G Z q as a nontrivial resolution having S as its unique nontrivial fiber.
Select s R Z. Let σ : ZzS Ñ W be a bijection from ZzS onto a set W that is disjoint from Z Y tsu, and denote x z :" σpzq for all z P ZzS. As base set, take X :" W Y tsu. Notice that X X Z " ∅ by construction, and |X| ě 2 because S ‰ Z by hypothesis. As fiber sets, take Y s :" S, and Y xz :" tzu for all where |X| ě 2 and |Y s | ě 2. As usual, let π : Z Ñ X be the (projection) map defined by πpzq :" x z if z P ZzS, and πpzq :" s if z P S. Next, we endow both the base set X and the fiber sets Y x 's with convex geometries induced by G Z .
Concerning the base, let G X be the family of subsets of X defined by We now check that pX, G X q is a convex geometry. To begin with, observe that πp∅q " ∅ P G X , that is, Axiom (G1) holds.
To prove Axiom (G2) (closure under intersection), we let πpGq, πpHq P G X , where G, H P G Z , and show that πpGq X πpHq P G X . We know that πpG X Hq Ď πpGq X πpHq. If equality holds, then we are immediately done, since G X H P G Z .
Otherwise, we must have ∅ ‰ G X S ‰ S, ∅ ‰ H X S ‰ S, pG X Hq X S " H, and πpGq X πpHq " πpG X Hq Y tsu. Now apply Property (S1) to G and next Property (S2) to G and H, to derive that the set K :" pGzSq Y pH X Sq is an element of G Z . Since K XH P G Z , and πpGqXπpHq " πpGXHqYtsu " πpK XHq, we conclude that πpGq X πpHq is an element of G X also in this case, as claimed. This shows that G X satisfies Axiom (G2).
To prove Axiom (G3), let πpGq ‰ X, with G P G Z . By Axiom (G3) for pZ, G Z q, there is z 1 P ZzG such that G 1 :" G Y tz 1 u P G Z . If G X S is either empty or equal to S, there holds πpz 1 q R πpGq, and the set πpGq Y tπpz 1 qu " πpG 1 q P G X is what we were looking for. On the other hand, suppose ∅ ‰ G X S ‰ S. If z 1 R S, then πpG 1 q " πpGq Y tx z 1 u, and we are done. Otherwise, there is z 2 P ZzG 1 such that G 2 :" G 1 Y tz 2 u P G Z . Again, if z 2 R S, the claim holds. Otherwise, we iterate. Since πpGq ‰ X and X is finite, we shall eventually get G n :" G n´1 Y tz n u P G Z such that πpG i q " πpGq for all i " 1, . . . , n´1, and z n R S. Then πpG n q " πpGq Y tx zn u proves that G X satisfies (G3).
To complete the definition of the resolution, we provide each fiber Y x with the convex geometry G x induced by G Z , that is, Below we show that pZ, G Z q " pX, G X q h pY x , G x q xPX , thus completing the proof.
To start, we prove that any convex set G from G Z is also convex in the resolution pX, G X q h pY x , G x q xPX , that is, G satisfies Requirements (R1)-(R3). Clearly πpGq P G X by the very definition of G X , thus (R1) holds for G. Furthermore, the intersection of G with an arbitrary fiber Y x is convex in Y x by definition of G x , so G also satisfies (R2). Finally, if G X S " ∅ or G X S " S, then (R3) holds trivially for G. On the other hand, if ∅ ‰ G X S ‰ S, then we can apply Property (S1) to S " Y s to get GzS P G Z . Since tsu " πpGqzπpGzSq, we obtain that s is an extreme element in the convex set πpGq P G X . We conclude that G satisfies (R3) also in this case, and so G is convex in the resolution pX, G X q h pY x , G x q xPX .
For the reverse inclusion, we show that if K is a nonempty convex set in pX, G X q h pY x , G x q xPX , then K P G Z . Since πpKq P G X by Requirement (R1) of a resolution, there is G P G Z such that πpKq " πpGq. Clearly, K X Y x " G X Y x for all x P W . If also K X S " G X S holds, then we have K " G, and we are immediately done. Since the latter fact trivially holds if K and S are disjoint, we may assume that K XS is different from both ∅ and GXS (so GXS is nonempty, too). Next we appeal to Requirement (R2), and get K X S P G s , hence, by the very definition of G s , there is H P G Z such that K X S " H X S ‰ ∅. It follows that K " pGzSq Y pH X Sq. To conclude our proof, we deal separately with the following two exhaustive cases.
(1) In case S Ę G, we have G X S ‰ S, hence Property (S1) yields GzS P G Z . Since G X S ‰ ∅, Property (S2) entails K " pGzSq Y pH X Sq P G Z , as required.
(2) In case S Ď G, we have S Ę K, because K X S differs from G X S. Then, by Requirement (R3), s is extreme in πpKq. The latter set is convex in X, and moreover it is equal to πpGq. We derive that πpGqztsu is also convex in G X , and next that GzS belongs to G Z (because πpGqztsu must be the image by π of a convex set from G Z , which must be GzS). By (S2) we conclude K " pGzSq Y pH X Sq P G Z , as required.
Remark 3.9. If in (S2) we replace G X S ‰ ∅ with |G X S| " 1, then we get an equivalent condition. Indeed, assume the modified (S2) is true, and let us prove (S2) in case |G X S| ą 1. Since both GzS and G are convex by assumption, there exists some s P G X S such that G 1 :" pGzSq Y tsu P G Z (this follows from the axioms of a convex geometry). By assumption the modified (S2) holds for G 1 . As we have pGzSq Y pH X Sq " pG 1 zSq Y pH X Sq, it follows that (S2) holds for G. This completes the proof of the equivalence of the two conditions. Finally, observe that in the modified (S2), the element s forming G X S is an extreme element of G, because G and Gztsu are convex.
Properties (S1) and (S2) are logically independent, even on some (small) affine convex geometry pZ, Gq. The next example shows such independence, at the same time exhibiting a resolvable, affine convex geometry.
Example 3.10. Let Z consist of four points a, b, c, d in the real affine plane, where c belongs to the segment having b and d as extreme points, and a is outside the line passing through the other three points (see Figure 2). Consider the affine convex geometry G Z induced on Z.
To show that (S1) and (S2) are mutually independent properties, let S 1 " ta, bu and S 2 " ta, cu. Then, S 1 satisfies (S1) (because all subsets of ZzS 1 are convex) but not (S2) (take G " ta, du and H " tbu). On the other hand, S 2 does not satisfy (S1) (take G " tb, c, du) but satisfies (S2) (because there are only two sets in Z that are not convex, namely tb, du and ta, b, du).
Observe also that S " tb, c, du is a shrinkable set in pZ, G Z q, which is therefore a resolvable convex geometry. In fact, it is easy to check that pZ, G Z q is the resolution with base pt1, 2u, 2 t1,2u q and fibers pY 1 , G 1 q and pY 2 , G 2 q, where Here, the base, the fibers, and the resolution are affine convex geometries. For compositions (or compounds) of hypergraphs, 'committees' play a role akin to our shrinkable sets. According to Chein et al. (1981), a committee in a hypergraph pX, Gq (where by definition G Ď 2 X ) is any subset S of X satisfying a property similar to (S2) in Theorem 3.8, where the antecedent is G X S ‰ ∅ and H X S ‰ ∅. Cantone et al. (2020) characterize shrinkable sets in the more general setting of choice spaces. In view of Koshevoy's result (Theorem 2.16) and a simplification due to the path independence of the associated choice space, this directly yields an alternative test for the primitiveness of a convex geometry which employs the extreme operator.
Theorem 3.11. Let pZ, Gq be a convex geometry. The following statements are equivalent for a subset S of Z such that 1 ă |S| ă |Z|: (i) S is shrinkable; (ii) S satisfies the following properties for any A P 2 Z : Moreover, when pZ, Gq is atomistic, (T1) implies (T3), and so Properties (T1) and (T2) characterize the shrinkability of S.
We now establish the independence among the three Properties (T1)-(T3).
Example 3.12. As in Example 3.10, consider the affine convex geometry induced on four points a, b, c and d in the real affine plane, with c between b and d and moreover a outside the line through b, c and d. If we let S " ta, b, cu, then S does not satisfy (T1) (for A " ta, b, cu) but it satisfies (T2) and (T3). If we now let S " ta, du, then S satisfies (T1) and thus also (T3), but not (T2) (for A " ta, b, cu). Thus even in affine convex geometries, (T1) and (T2) are each one independent of the other two properties.
To show that (T3) is independent of (T1) and (T2), consider the ordinal convex geometry derived from the partial order ď on Z " ta, b, cu with a ă b, a ă c and no other strict comparison. Then the subset S " ta, bu satisfies (T1) and (T2), but not (T3) (take A " ta, cu).

Extreme Resolutions and Extremely Shrinkable Sets
Here we study resolutions whose nontrivial fibers are all indexed by extreme elements of the base.
Definition 3.13. A resolution pZ, G Z q " pX, G X q h pY x , G x q xPX is extreme when for each x P X, if x R ex G X pXq then |Y x | " 1.
As we shall see in Theorem 4.3, any resolution of convex geometries that happens to be affine is also extreme. The next result uses extremeness to further clarify the link between compositions and resolutions.
Theorem 3.14. Let pZ, G Z q " pX, G X q h pY x , G x q xPX be a resolution of convex geometries, and pZ, C Z q " pX, G X q a pY x , G x q xPX the composition with the same base and fibers. Then G Z Ď C Z , and G Z " C Z holds if and only if the resolution is extreme.
Proof. Definition 3.1 readily yields G Z Ď C Z . Next we show that if the resolution is extreme, then C Z Ď G Z . Indeed, in an extreme resolution, any subset A of Z satisfies Requirement (R3), because an element x that is non-extreme in πpAq is also non-extreme in X, hence Y x has only one element and is contained in A.
Conversely, we show that if the resolution is not extreme, then C Z Ď G Z does not hold. By assumption, there exists some x in Xz ex Gx pXq such that the fiber Y x contains at least two elements. By the definition of a convex geometry, G Yx contains a nonempty convex set G distinct from Y x . Now it is easy to check that pZzY x q Y G belongs to C Z but not to G Z .
Remark 3.15. Theorem 3.14 has a direct extension to a large family of set systems, as we now explain. A set system is a pair pX, F q, where X is a nonempty set, and F is a nonempty collection of subsets of X. A set system pX, F q is simple if Ť F " X; in particular, pX, F q is plain if it is simple and moreover F ‰ t∅, Xu whenever |X| ą 1.
Compositions of set systems are defined exactly as compositions of convex geometries (Definition 3.1). To define resolutions of set systems, we only need a notion of extreme element in set systems, and then again copy from Definition 3.1. Lemma 2.10 suggests the following definition: given a set system pX, F q, an element x in a subset A of X is extreme in A when there exists some F P F such that Aztxu Ď F and x R F (compare with, for instance, Ando, 2006). Now consider a composition pZ, F Z q " pX, F X q a pY x , F x q xPX and a resolution pZ, C Z q " pX, F X q h pY x , F x q xPX of set systems (with the same base and fibers). As for convex geometries (Theorem 3.14), the resolution F Z is a subcollection of the composition C Z . For plain set systems, the arguments in the proof of Theorem 3.14 show that the equality F Z " C Z occurs exactly when the following property is satisfied: for any nontrivial fiber Y x and convex set F in F X containing x, there holds x P ex F X pF q. (Observe that if X P F X , it suffices to require this property to hold for F " X, because an element x that is extreme in X is also extreme in any subset of X containing x.) To recognize which convex geometries can be written as a nontrivial extreme resolution, we introduce and characterize a variant of shrinkability.
Definition 3.16. Let pZ, G Z q be a convex geometry. A subset S of Z with at least two elements is extremely shrinkable (in Z) if it is a fiber in some nontrivial extreme resolution producing pZ, G Z q. Whenever Z contains such a set, we say that pZ, G Z q is extremely resolvable.
Theorem 3.17. Let pZ, G Z q be a convex geometry. The following statements are equivalent for a shrinkable set S Ď Z: (i) S is extremely shrinkable; (ii) for any resolution pX, G X q h pY x , G x q xPX equal to pZ, G Z q, if S coincides with a fiber Y x , then x P ex G X pXq; (iii) for at least one resolution pX, G X q h pY x , G x q xPX equal to pZ, G Z q, if S coincides with a fiber Y x , then x P ex G X pXq; (iv) ZzS P G Z .
(i) ùñ (iv). If S is extremely shrinkable in pZ, G Z q, then by definition there is an extreme resolution pX, G X q h pY x , G x q xPX equal to pZ, G Z q such that S " Y x for some x P X. Since |S| ě 2, the hypothesis yields x P ex Gx pXq. We claim that T P G Z . Requirement (R1) in Definition 3.1 holds for T by the equivalence (4) in Remark 3.4. Next, as T XY x " ∅ and T XY x 1 " Y x 1 for x 1 P Xztxu, we derive that (R2) holds for T as well. Since T also satisfies (R3), the implication (i) ùñ (iv) is fully proved.
(iv) ùñ (ii). Suppose T P G Z . In the resolution considered in (ii), the assumption implies πpT q " Xztxu P G X . It follows that x P ex Gx pXq.
(ii) ùñ (i). Suppose (ii) holds. As S is assumed to be shrinkable, there exists (as in the proof of Theorem 3.8) a resolution in which one fiber equals S and all the other fibers have size 1. By (ii), the projection of S is an extreme element of the base. As a consequence, the resolution is extreme.
A characterization of shrinkable subsets in terms of convex sets appeared in Theorems 3.8. We derive a simpler characterization for extremely shrinkable sets.
Theorem 3.18. Let pZ, G Z q be a convex geometry. The following statements are equivalent for a subset S of Z such that 1 ă |S| ă |Z|: (i) S is extremely shrinkable; (ii) S satisfies the following properties for any G, H P G Z : Proof. (i) ùñ (ii). Assume S is extremely shrinkable. To prove (V1), suppose G P G Z and |G X S| " 1. By Theorem 3.17, the hypothesis entails ZzS P G Z , whence GzS " G X pZzSq P G Z . Since S is shrinkable, Property (S2) from Theorem 3.8 yields (taking H :" Z) G Y S P G Z . To prove (V2), let H P G Z . Then, for G :" Z in (S2), we get pZzSq Y pH X Sq P G Z , as desired.
(ii) ùñ (i). Assume (V1) and (V2) hold. We first derive the shrinkability of S by establishing Properties (S1) and (S2) in Theorem 3.8. Property (S1) holds because GzS " G X pZzSq P G Z follows from (V2) with H " ∅. To prove (S2), let G, H P G Z be such that G X S ‰ ∅ and GzS P G Z . In view of Remark 3.9, we can assume |G X S| " 1. Using we derive from both (V1) and (V2) that the latter set lies in G Z . This proves that S is shrinkable. Finally, observe that S is also extremely shrinkable, because ZzS P G Z follows from (V2) with H :" ∅, hence we can make use of Theorem 3.17.
Remark 3.19. In case an extremely shrinkable set S is convex in G Z , Property (V2) becomes equivalent to the following one: The reason is that tH X S H P G Z u " tH 1 P G Z H 1 Ď Su whenever S P G Z .

Resolutions of Special Convex Geometries
Here we examine resolutions of the two classes of convex geometries introduced in Section 2.1: ordinal and affine. We start with the affine case.

Affine Convex Geometries
Recall from Example 2.6 that a convex geometry is affine if and only if it is isomorphic to a convex geometry induced on a finite subset of a real affine space.
Lemma 4.1. If a resolution is an affine convex geometry, then the base and the fibers of the resolution are also affine convex geometries.
Proof. The result follows at once from Remark 3.4: the fibers are subgeometries of the resolution, and the base is isomorphic to a subgeometry of the resolution.
The converse of Lemma 4.1 does not hold: Examples 4.2 and 4.10 below show that resolutions of affine convex geometries need not be affine.
All pairs pX, G X q and pY i , G i q, for i " 1, 2, 3, are affine convex geometries. A simple computation shows that the resolution of pX, G X q into tpY i , G i q i P Xu is the convex geometry pZ, G Z q " pX, G X q h pY i , G i q iPX , where Z " ta, b, c, du, and G Z " 2 Z z ta, du, ta, b, du, ta, c, du ( . Figure 3, which describes the lattice pG Z , Ďq of convex sets.) However, the convex geometry pZ, G Z q is not affine. This readily follows from the fact that b, c P conv G Z pta, duq, but b R conv G Z pta, cuq and c R conv G Z pta, bu: indeed, if pZ, G Z q were affinely embedded in some real affine space, we would have b and c on the segment ra, ds, and so either b P ra, cs or c P ra, bs.

(See
Observe that the non-affine resolution pZ, G Z q of Example 4.2 is also nonextreme, since 2 R ex G X pXq and yet |Y 2 | ą 1. In fact, to conclude that pZ, G Z q is non-affine, it suffices to check that it is non-extreme, as the next result guarantees.   (Theorem 4.4), which gives a sufficient condition for a resolution to be extreme. This condition is the well-known exchange property, due to Levi (1951) and Sierksma (1984), applied to a convex geometry pZ, G Z q: for any A P 2 Z and p P Z,  Proof. Let pZ, G Z q " pX, G X q h pY x , G x q xPX be a resolution of convex geometries that satisfies the exchange property. Denote by E the set of all extreme points of the base set X. Toward a contradiction, assume there is w P XzE such that |Y w | ě 2, and let a, b be two distinct elements of Y w . Since w P X " conv G X pEq, there is a minimal subset V of E such that w P conv G X pV q. For each v P V , denote by V vÐw the set pV ztvuq Y twu. By the minimality of V , we have w P ex G X pV vÐw q for all v P V .
Now let T Ď Z be a transversal for the family tY v v P V u, that is, |T X Y v | " 1 for each v in V . Observe that ex G X pπpT qq " ex G X pV q " V by Lemma 2.12(i). Therefore, Lemma 3.5 yields and so, since w P conv G X pV qzV and a, b P Y w , we deduce Since a P conv G Z pT q, the exchange property entails where T tÐa stands for pT zttuq Y tau. Since b P conv G Z pT q, it follows b P conv G Z pT uÐa q for some u P T .
Let r " πpuq. Another application of Lemma 3.5 yields whence, by (7) and (9), we deduce b P conv Gw pT uÐa X Y w q " conv Gw ptauq , which in turn implies b R ex Gw pta, buq. The roles of a and b being exchangeable, we also have a R ex Gw pta, buq. It follows that ex Gw pta, buq " ∅, a contradiction.
Corollary 4.5. Let pZ, G Z q be an affine convex geometry. For any shrinkable set S Ď Z, we have: (i) By Lemma 4.1, the base pX, G X q is an affine convex geometry, which implies that any one-element set txu in 2 X belongs to G X . By (3) in Remark 3.4, we conclude S " Y x P G Z .
(ii) To start, we prove ZzS P G Z . By Theorem 4.3, the resolution pZ, G Z q " pX, G X q h pY x , G x q xPX is extreme, and so S " Y x is extremely shrinkable. An application of Theorem 3.17 readily yields ZzS P G Z . Then, for arbitrary G P G Z , we get GzS P G Z , because GzS " G X pZzSq.
Finally, we obtain a characterization of all affine convex geometries that are resolvable (hence of primitive ones): Corollary 4.6. The following statements are equivalent for an affine convex geometry pZ, G Z q: (1) pZ, G Z q is resolvable; (2) there is a shrinkable subset S of Z; (3) there is an extremely shrinkable subset S of Z; (4) there is a subset S of Z, with 1 ă |S| ă |X|, satisfying the following properties for any G, H P G Z : Proof. Simply observe that as a consequence of Theorem 4.3, affine resolutions fall under the application of Theorem 3.18.
The primitivity of a convex geometry is characterized by the non-existence of a shrinkable set. However, even for affine convex geometries, this characterization is not computationally effective, because Properties (V1) and (V2) are to be checked for all convex sets of the given geometry. We wonder whether there are more instructive answers to the next two, related problems.
Problem 4.7. Given an affine convex geometry pZ, Gq, characterize when a subset of Z is shrinkable.
Problem 4.8. Geometrically characterize when an affine convex geometry is primitive.
By Theorems 4.3 and 3.14, Problems 4.7 and 4.8 are also problems about compositions of affine convex geometries. Although they appear to be central problems, we were unable to find any mention of them in the literature.
The next result states an equivalent (geometric) formulation of Property (T1) in Theorem 3.11. This reformulation is in the spirit of the answers we would like to obtain for Problems 4. 7 and 4.8. In what follows, 'conv G Z ' denotes the convex hull operator in a convex geometry G Z , whereas 'conv R ' is used for the standard convex hull in the affine space R d .
Proposition 4.9. Let Z be a finite subset of R d , and pZ, Gq the convex geometry induced on Z. The following statements are equivalent for any set S Ď Z such that 1 ă |S| ă |Z| : (i) S satisfies Property (T1) in Theorem 3.11, namely for all A P 2 Z , A simple consequence of Condition (ii) is that S lies in the relative boundary 6 of conv R pZq. However, Condition (ii) asserts more than that.
(ii) ùñ (i): Suppose there exist proper faces F 1 , F 2 , . . . , F k of the convex polytope conv R pZq such that S " Z X pF 1 Y F 2 Y¨¨¨Y F k q. If any face F i equals conv R pZq, then S " Z, and so S satisfies (T1). Thus we may assume that all F i 's are proper faces of conv R pZq. Let A P 2 Z ; we shall show that expA X Sq Ď expAq. Given w in expAXSq, we know by (ii) that w belongs to some face F i of conv R pZq, with moreover Z X F i Ď S. If w P expAq does not hold, then there exists a subset B of Aztwu such that w P conv R pBq. Such a minimal subset B of Aztwu is formed by the vertices b 1 , b 2 , . . . , b ℓ of a simplex containing w in its relative interior. Then all b j 's belong to F i , because the proper face F i of conv R equals the intersection of conv R pZq with some hyperplane supporting conv R pZq. It follows that, for all j's, we have b j P A X F i Ď A X Z X F i Ď A X S, contradicting the initial assumption w P expA X Sq. Claim: If some point w of S is in the relative interior of any face F (proper or not) of conv R pZq, then Z X F Ď S and the face F is proper.
Proof of Claim. Toward a contradiction, assume there is f 0 P pZ X F qzS. The line passing through f 0 and w must meet the relative boundary of F on the side of w opposite to f 0 . Thus there exist vertices f 1 , f 2 , . . . , f k of the face F such that w belongs to the relative interior of the simplex with vertices f 0 , f 1 , . . . , f k . Notice tf 0 , f 1 , f 2 , . . . , f k u Ď Z (because all vertices of conv R pZq must be in Z). We split the analysis in the only two possible cases.
Case 1: f i is in S, for some i P t1, 2, . . . , ku. Set A " tw, f 0 , f 1 , . . . , f k u, and notice f i P expAq X S together with w P expA X Sqz expAq. This contradicts the assumption that S satisfies (T1).
Case 2: tf 0 , f 1 , . . . , f k u Ď ZzS. By our assumption |S| ě 2, there is v P Sztwu. Consider two subcases for the possible position of the point v. First, if v R conv R ptf 0 , f 1 , f 2 , . . . , f k uq, then we set A " tv, w, f 0 , f 1 , f 2 , . . . , f k u. Notice v P expAq X S and w P expA X Sqz expAq, again a contradiction with S satisfying (T1). Second, if v P conv R ptf 0 , f 1 , f 2 , . . . , f k uq, there is a point x on the relative boundary of the simplex T " conv R ptf 0 , f 1 , . . . , f k uq such that w P sv, xr. Let now A be formed by the points v, w and the vertices of the minimal face of the simplex T which contains x. Again we get a contradiction because v P expAq X S and w P expA X Sqz expAq.
To complete the proof of the Claim, simply observe that F must be proper, because otherwise we would have Z " S.
From the Claim, we derive that S contains the intersection of Z with any face of conv R pZq containing in its relative interior at least one point of S. Thus S includes the intersection of Z with the union of all such faces. The reverse inclusion also holds, because any point w of S belongs to both Z and the relative interior of the smallest face of conv R pZq containing w.
We are still missing a translation of Property (T2) in Theorem 3.11. This translation appears to be of some interest in view of the fact that, along with the translation of (T1) obtained in Proposition 4.9, it would deliver a solution to Problem 4.7.
The next example illustrates another type of obstruction to the affineness of a resolution.
Example 4.10. Let X " t1, 2, 3u , G X " 2 X ztt1, 3uu , Y 1 " tau , The resolution of pX, G X q into tpY i , G i q i P Xu is the convex geometry pZ, G Z q " pX, G X q h pY i , G i q iPX , where Z " ta, b, c, du, and Although all convex geometries pX, G X q and pY i , G i q, i " 1, 2, 3, are affine, their resolution pZ, G Z q is not. Indeed, we have b P conv G Z pta, cuq X conv G Z pta, duq, along with c R conv G Z ptb, du and d R conv G Z ptb, cu, which is impossible in any affine geometry. (If pZ, G Z q were affinely embedded, we would have in some real affine space c and d on the line through a and b, on the side of b opposite to a. However, this implies c P rb, ds or d P rb, cs.) The crucial assumptions in the last example are that 2 lies between 1 and 3, and that the fiber Y 3 contains more than one element. We generalize them in the next proposition (where p plays the role of 2). Proposition 4.11. Suppose a resolution pZ, G Z q " pX, G X q h pY x , G x q xPX of convex geometries is affine, with G Z the geometry induced on the subset Z of some real affine space R d . Assume that the base contains elements p, p 1 , . . . , p n`1 such that p P conv G X ptp 1 , p 2 , . . . , p n`1 uq and p R conv G X pT q for any proper subset T of tp 1 , p 2 , . . . , p n`1 u. For i " 1, 2, . . . , n`1, let q i be any point in the fiber Y p i . Then all fibers Y p i lie in the affine subspace of dimension n generated by the points q 1 , q 2 , . . . , q n`1 , and so all fibers pY p i , G p i q are isomorphic to convex geometries affinely embedded in a real affine space of dimension n.
Proof. As mentioned in Lemma 4.1, the base pX, G X q and all fibers pY x , G x q are also affine geometries. By Theorem 4.3, the fiber Y p contains just one point, say q. Note that q P conv R ptq 1 , q 2 , . . . , q n`1 uq in R d (the reason is that the projection on the base of the convex hull conv G Z tq 1 , q 2 , . . . , q n`1 u in Z must be convex in the base X and at the same time contain p 1 , p 2 , . . . , p n`1 ; thus by our assumptions the projection contains also p). Moreover, q cannot be in the convex hull of less than n`1 of the points q 1 , q 2 , . . . , q n`1 (because the projection p of q does not lie in the convex hull in X of less than n`1 of the p i 's). Thus in R d , the point q is in the relative interior of the simplex with vertices q 1 , q 2 , . . . , q n`1 . To derive the thesis for i " 1 (the arguments are similar for the other values of i), note that q 1 lies in the affine hull of the points q, q 2 , q 3 , . . . , q n . As this result holds for any point in the fiber Y p 1 in place of q 1 , we deduce that the fiber Y p 1 is included in the affine hull of q, q 2 , q 3 , . . . , q n`1 , which is the same as the affine hull of q 1 , q 2 , q 3 , . . . , q n`1 .
We know of several other necessary conditions for an affine convex geometry to be primitive, but none of them is both necessary and sufficient. We leave Problems 4.7 and 4.8 unsolved.

Ordinal Convex Geometries
Recall from Example 2.4 and Theorem 2.5 that a convex geometry pZ, G Z q is ordinal if and only if G Z is closed under union, or, equivalently, G Z consists of all ideals of some unique partial order ď on Z (the partial order associated to G Z ). Remark 2.17 readily yields that the equivalence holds for all z, z 1 P Z. Here we show that (1) a resolution of ordinal convex geometries is always ordinal, and (2) its associated partial order is the 'resolution' (as in the next definition) of the partial orders associated to the base and the fibers.
Definition 4.12. Let X be a finite base set, and tY x x P Xu a family of finite, pairwise disjoint fiber sets disjoint from the base set. Furthermore, let R X be a binary relation on X, and R x a binary relation on Y x for each x P X. Set Z :" Ť xPX Y x , and call projection the mapping π : Z Ñ X, with πpzq " x when z P Y x . The resolution of pX, R X q into tpY x , R x q x P Xu is the pair pZ, R Z q, where R Z is the binary relation on Z defined by for all z, z 1 P Z. With a slight abuse of terminology, we shall also say that R Z is the resolution of R X into the family tR x x P Xu. We use a notation similar to the one employed for convex geometries, namely pZ, R Z q " pX, R X q m pY x , R x q xPX .
A binary relation on a finite set is primitive when it cannot be obtained as a nontrivial resolution of relations, and is resolvable otherwise. 7 Remark 4.13. Resolutions of binary relations are well-known, often under a different name: see, for instance, Dörfler (1971). For the special case of partial orders, they are called sums by Hiraguchi (1951), lexicographic sums by Trotter (1992, page 24), and ordered sums by Harzheim (2005, page 85). Observe that Bang-Jensen and Gutin (2001, page 8) use the term 'composition' in place of 'resolution'. In this paper, we employ the term 'resolution' for binary relations not only to avoid confusion, but also in an attempt to use a common name for the codification of the same concept in different mathematical settings.
It is well-known that the resolution of binary relations is a partial order exactly when the base relations and the fiber relations are all partial orders: see Hiraguchi (1951), Trotter (1992), or Harzheim (2005). The main result of this section (Theorem 4.14) proves two things: (1) a resolution of convex geometries is ordinal if and only if so are its base and its fibers; (2) there is a tight connection between resolutions of ordinal convex geometries and resolutions of partial orders.
Theorem 4.14. A resolution of convex geometries is an ordinal convex geometry if and only if its base and all its fibers are ordinal convex geometries. Furthermore, the partial order associated to the resolved convex geometry is equal to the resolution of the partial order associated to the base into the family of partial orders associated to the fibers.
Proof. Let pZ, G Z q " pX, G X q h pY x , G x q xPX be a resolution of convex geometries.
If the convex geometry pZ, G Z q is ordinal, equivalently G Z is closed under union (Theorem 2.5), Remark 3.4 implies that all geometries pX, G X q and pY x , G x q, for x P X, are also ordinal (because a subgeometry of an ordinal geometry is itself ordinal).
To prove the converse, suppose now that pX, G X q and pY x , G x q, for x P X, are all ordinal convex geometries. By Theorem 2.5, it suffices to show that G Z is closed under union. Let B, C P G Z . We shall prove that D " B Y C satisfies Requirements (R1)-(R3) in Definition 3.1.
(R1) Requirement (R1) applied to B and C yields πpBq, πpCq P G X , hence πpDq " πpBq Y πpCq P G X , because G X is closed under union.
(R2) Let x P πpDq. Without loss of generality, assume that x P πpBq X πpDq (indeed, if x belongs to exactly one between πpBq and πpCq, the result is trivial). Now Requirement (R2) applied to B and C yields B X Y x , C X Y x P G x . Since D X Y x " pB X Y x q Y pC X Y x q and G x is closed under union by assumption, we derive D X Y x P G x , as claimed.
(R3) Let x P πpDqz ex G X pπpDqq " pπpBq Y πpCqqz ex G X pπpBq Y πpCqq. It follows that x P πpBqz ex G X pπpBqq or x P πpCqz ex G X pπpCqq holds. By Requirement (R3) applied to B or C, we get Y Thus, D satisfies (R3), too.
Next, we prove the second assertion. Let ď X , ď x , and ď Z be the partial orders associated to the ordinal convex geometries pX, G X q, pY x , G x q, and pZ, G Z q, respectively. We show that pZ, ď Z q is the resolution (in terms of relations) of pX, ď X q into the family pY x , ď x q xPX , that is, Indeed, in view of the equation (6) in Lemma 3.5, the equivalence (10), and the equivalence (11) in Definition 4.12, we have, for all z, z 1 P Z, We conclude that z ă Z z 1 does not hold if and only if the pair pz, z 1 q does not belong to the resolution of the partial order ď X into the family tď x x P Xu. This completes the proof.
The following consequence of Theorem 4.14 is immediate: Corollary 4.15. An ordinal convex geometry is primitive if and only its associated partial order is primitive.
For information on primitive posets, we refer the reader to Schmerl and Trotter (1993) or Boudabbous, Zaguia, and Zaguia (2010). The concept of primitivity applies to more general relational structures: see Ille (2005) for a survey.
The shrinkable sets of an ordinal convex geometry pZ, Gq are exactly the autonomous sets of the associated partial order ď, where (see Schröder, 2016) S Ď Z is autonomous if for all s, s 1 P S and z P ZzS, s ď z ùñ s 1 ď z and z ď s ùñ z ď s 1 .
Finally, observe that Theorem 4.14 does not hold for compositions: indeed, Example 3.2 exhibits a composition of ordinal convex geometries that fails to be a convex geometry. 8

Primitivity of Small Convex Geometries
Here we determine all primitive convex geometries on at most four elements. Observe preliminarily that our classification task is simple for the special case of ordinal convex geometries. In fact, by Corollary 4.15, to test whether an ordinal convex geometry is primitive, it suffices to check whether its associated poset is primitive (as a poset), which in turn amounts to investigate whether the poset has an autonomous subset.
To start, note that all convex geometries on one or two elements are primitive. The next proposition inspects which convex geometries on three and four elements are primitive, also determining whether they are ordinal or affine. A list of all 34 convex geometries on four elements appear in Merckx (2013), and their number is confirmed in Uznanski (2013).
Proposition 5.1. Up to isomorphisms, there are: (i) 6 convex geometries on three elements, of which 1 is primitive and nonordinal, and 5 are resolvable and ordinal; (ii) 34 convex geometries on four elements, 12 of which are primitive; among the primitive ones, 1 is ordinal and 2 are affine.
Proof. (i) On three elements there are, up to isomorphisms, 6 geometries, which are listed in Example 2.3. By using Theorem 2.5, one can readily check that exactly 5 of them are ordinal: in fact, there is only one convex geometry that is not closed under union, namely G 5 . All 5 corresponding posets on three elements are resolvable, hence also the associated ordinal convex geometries are resolvable. Furthermore, the unique non-ordinal geometry G 5 is primitive, since otherwise its base and fibers would be ordinal, and so G 5 itself would also be ordinal by Theorem 4.14.
(ii) On four elements, there are 34 convex geometries, 16 of which are ordinal. The 16 posets on four elements are listed, for instance, in Monteiro, Savini and Viglizzo (2017) and Steinbach (1990); only one of these posets is primitive (it is the 'Nposet'). To find out how many of the 18 non-ordinal convex geometries on four elements are primitive, we rather look for the number of resolvable ones.
By Theorem 4.14, non-ordinal resolutions have either a non-ordinal base or a non-ordinal fiber (or both). Furthermore, all fibers of a nontrivial resolution on four elements have size at most three. Thus there are only two cases: (a) the base is the unique non-ordinal geometry G 5 on three elements, and the three fibers have one, one, and two elements, respectively; (b) the base has two elements and one fiber is G 5 .
Taking into account the automorphisms of small convex geometries, we are left with 7 possible resolutions, of which 4 are of type (a), and 3 of type (b). It is simple to construct these 7 resolvable geometries, and check that they are pairwise non-isomorphic. We conclude that among the 18 non-ordinal convex geometries on four elements, 11 are primitive. Moreover, exactly 2 of these 11 primitive geometries are affine. In fact, there are exactly 4 affine convex geometries on four points, which are those induced on the subsets of the real affine plane shown in Figure 4; only the first and the fourth produce a primitive convex geometry.

Future Work and Open Problems
Here we list a few natural problems (they might be easy or difficult). By a class of convex geometries, we mean a class closed under taking isomorphic images.
1. For a property (P) shared by some convex geometries, consider the following two assertions: (i) if the base and the fibers of a resolution all satisfy (P), then the resolution also satisfies (P); (ii) if a resolution satisfies (P), then its base and its fibers satisfy (P).
We say that the property (P) is forward stable when (i) is true, backward stable when (ii) is true, and stable when both (i) and (ii) are true. For instance, ordinality of a convex geometry is a stable property (Theorem 4.14), whereas affineness is a backward stable property (Lemma 4.1) that fails to be forward stable (Example 4.2). An interesting problem consists of determining which (additional) properties of convex geometries considered in the literature are preserved by resolutions, in particular which of the known families of convex geometries are stable under resolutions (see Goecke et al., 1989 for several types of such families). Carpentiere (2019) shows that neither monophonic convex geometries nor geodetic convexity geometries form a stable family (for monophonic vs geodetic convex sets in graphs, see Farber and Jamison, 1986).
2. Any class C of convex geometries is included in a smallest class S of convex geometries forward stable under resolutions. When C itself is not forward stable, S differs from C. Characterize S when C is the class of affine convex geometries, and also for other nonstable classes C.
3. Design a non-naïve algorithm to test whether a given convex geometry is primitive, and another one to generate the primitive convex geometries on small numbers of elements. Enright (2001) investigates various encodings of convex geometries. Uznanski (2013) discusses a code generating all convex geometries up to seven elements (reporting ingegneous programming efforts).
4. As it is the case for many classes of structures with respect to compositions (see Möhring and Radermacher, 1984), does the fraction of primitive convex geometries on n elements tend to one? (Notice that there are two questions here, one for convex geometries on labeled sets and one for convex geometries up to isomorphisms.) About the asymptotic number of labelled convex geometries, see Echenique (2007) and Monjardet (2008). For examples of recent results concerning prime structures, see Boudabbous and Ille (2011) for binary relations, Guillet, Leblet, and Rampon (2017) for posets, Ille and Villemaire (2014) as well as Chudnovsky et al. (2016) for graphs.