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What Convex Geometries Tell About Shattering-Extremal Systems Bogdan Chornomaz

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What Convex Geometries Tell About Shattering-Extremal Systems Bogdan Chornomaz What convex geometries tell about shattering-extremal systems Bogdan Chornomaz To cite this version: Bogdan Chornomaz. What convex geometries tell about shattering-extremal systems. 2020. hal- 02869292 HAL Id: hal-02869292 https://hal.archives-ouvertes.fr/hal-02869292 Preprint submitted on 15 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. What convex geometries tell about shattering-extremal systems Bogdan Chornomaz [email protected] Vanderbilt University 1 Introduction Convex geometries admit many seemingly distinct yet equivalent characteriza- tions. Among other things, they are known to be exactly shattering-extremal closure systems [2]. In this paper we exploit this connection and generalize some known characterizations of convex geometries to shattering-extremal set families, which are a subject of intensive study in their own right. Our first main result is Theorem 2 in Section 4, which characterizes shattering- extremal set families in terms of forbidden projections, similar to the character- ization of convex geometries by Dietrich [3], discussed in Section 3. Another known characterization of convex geometries, given in Theorem 3, is that they are exactly closure systems in which any non-maximal set can be extended by one element. In Section 5 we prove a similar property for shattering-extremal systems, together with another topological property that every loop can be con- tracted. However, even together these two properties do not give a sufficient condition. Finally, in Section 6 we outline some open problems and related questions. 2 Preliminaries If not mentioned otherwise, all objects that we consider in this paper are finite. In a poset P, an antichain is a subset A ⊆ P such that no two distinct elements from A are comparable in P; moreover, A is maximal if A [ fxg is not an antichain for any x 2 P − A. A subset S ⊆ P is hereditary (upward closed), if for any x 2 S and any y ≤ x (y ≥ x) it follows that y 2 S. There is a natural bijection between antichains and upward closed sets in P: for an antichain A its upward closure Au = fx 2 P j x ≥ a for some a 2 Ag is, as expected, upward closed. Conversely, for an upward closed set I, the set of its minimal elements Im = fx 2 I j y 6< x for every y 2 Ig is an antichain. Moreover, these two operations are mutually inverse. We denote the operation of taking minimal antichain by Min: 2P ! 2P . For two subsets Q; R ⊆ P we say that Q refines R, denoted Q R, if for any r 2 R there is q 2 Q such that q ≤ r. Refinement relation is a preorder on 2P , however, when restricted to antichains in P, it becomes a partial order. It is easy to see that Q R iff Qu ⊇ Ru, in particular, an antichain of minimal elements of a set refines this set. Our primary object of study will be a set family F (which we also call system) over a base set U, that is, F ⊆ 2U . We will denote the powerset of U by P(U) when we consider it as a poset ordered by inclusion, and by 2U when we do not care about the order. A system F shatters a set X ⊆ U (alternatively, X is shattered by F) if for any Y ⊆ X there is F 2 F such that F \ X = Y . We denote the family of sets shattered by F by Str(F). Trivially, for any F, Str(F) is hereditary. Also, by the Sauer-Shelah-Perles (SSP) lemma, it holds: jFj ≤ j Str(F)j; (1) and we say that F is shattering-extremal if it attains equality in (1), that is, if jFj = j Str(F)j. Every hereditary system H is shattering-extremal with Str(H) = H. For A; B ⊆ U, we denote the set difference of A and B by A − B and the symmetric difference by A4B. For A ⊆ U and x 2 U we write A−x for A−fxg and A + x for A [ fxg. For ': X ! Y and A ⊆ X, '[A] denotes the image of X under '. We call a tuple (P; hP ), where P ⊆ U and hP : P ! f0; 1g a projection, where P is called the support and hP the pattern of the projection. Alternatively, a projection can be given by a tuple (P; HP ), H ⊆ P ⊆ U. Two definitions are ob- tained from each other by identifying HP with its characteristic function hP (x). If no confusion arises, we will drop hP (or HP ) when mentioning a projection. For a function h: P ! f0; 1g and Q ⊆ P we define a function hjQ : Q ! 0; 1 as hjQ(x) = h(x), for x 2 Q. We define PRJ to be a poset of all projections with the following partial order: for projections (P; hP ) and (Q; hQ), (P; hP ) ≤ (Q; hQ) if P ⊆ Q and hP = hQjP ; empty projection (;;"), ": ; ! f0; 1g, is a minimal element of PRJ. Almost exclusively we talk about projections it in the context of them being forbidden, in particular, we say that F ⊆ U satisfies a projection P if HP 6= P \ F , otherwise P invalidates F . Similarly, system F satisfies projection set P if F satisfies P , for all F 2 F, P 2 P. Note that for all P; Q 2 PRJ, P ≤ Q, if F satisfies P then it satisfies Q; alternatively, if Q invalidates F then so does P . u For an arbitrary system F we define PF as a set of all projections satisfied u by F, that is, PF = fP 2 PRJ j HP 6= P \ F for all F 2 Fg, and define PF as u PF = Min(PF ). The latter is called the set of forbidden projections of F. Note that in general P can have projections with same supports (but with different u patterns). In fact, save for some degenerate cases, PF will have a lot of them. The situation with PF is similar: although in some cases, which are of particular interest to us, supports of PF will all be different and form an antichain (in P(U)), in general supports of PF can coincide or be comparable to each other. In the opposite direction, we define PRJ∗ to be the poset of antichains in PRJ ordered by refinement; its maximal element is an empty set of projections and its minimal element is a one-element antichain f(;;")g. For P 2 PRJ∗ we define FP as a system of sets satisfying P, that is, FP = fF ⊆ U j HP 6= P \ F for all (P; HP ) 2 Pg. Sometimes we will write F(P) and P(F) instead of FP and PF . Note that the definition of FP does not require P to be an antichain. However, including other projection sets does not add to the expressiveness of this definition. We justify this in Proposition 1 below, together with some other basic properties of operations P and F. Proposition 1. 1. For any set of projections P, F(P) = F(Min(P)) = F(Pu); 2. For any system F, F(P(F)) = F; 3. For any P 2 PRJ∗, P(F(P)) P, and there are antichains of projections for which this inequality is strict; 4. For any P 2 P RJ ∗, P(F(P)) = P(F(P(F(P)))). Proof. (1). As Min(P) ⊆ P, F(Min(P)) ⊇ F(P), because the latter is the family of sets satisfying smaller set of forbidden projections than the latter. And if F2 = F(P), then there is P 2 P, invalidating F . Take Q 2 Min(P), Q ≤ P . Then Q also invalidates F , hence F2 = F(Min(P)). So, F(P) = F(Min(P)); the second equality follows from the fact that Min(P) = Min(Pu). (2). Let us define F 0 = F(P(F)). Thus, F 0 is a family of all subsets satisfying 0 PF . As all elements of F satisfy PF , F ⊇ F. On the other hand, for any N2 = F, u (U; N) is satisfied by F and hence (U; N) 2 PF . Then there is P 2 PF such that P ≤ (U; N). As (U; N) invalidates N then so does P . But then N2 = F 0. (3). Let F 0 = F(P), P0 = P(F 0), and P0u and Pu be upward closures of P0 0u u 0 u u 0 and P. Then P = PF 0 and F = F(P ). As in (2), P is satisfied by F and P0u is the set of all projections satisfying F 0, hence P0u ⊇ Pu which holds iff P0 P. For an example of the strict inequality, let U be a one element set, U = f1g, and let P be a two-element antichain of projections, P = f(U; ;); (U; U)g. Then F(P) is empty and hence P(F(P)) = f(;;")g 6= P. (4). A consequence of (2). ∗ We define PRJ ( PRJ as an image of operation P, that is, PRJ = P[F]. Note that by Proposition 1(4), P 2 PRJ iff P = P(F(P)). The family Str(F) of shattered sets of F can be trivially defined via PF .
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