A single-element extension of antimatroids

An antimatroid is a family of sets which is accessible, closed under union, and includes an empty set. A number of examples of antimatroids arise from various kinds of shellings and searches on combinatorial objects, such as, edge=node shelling of trees, poset shelling, node-search on graphs, etc. (Discrete Math. 78 (1989) 223; Geom. Dedicata 19 (1985) 247; Greedoids, Springer, Berlin, 1980) [1–3]. We introduce a one-element extension of antimatroids, called a lifting, and the converse operation, called a reduction. It is shown that a family of sets is an antimatroid if and only if it is constructed by applying lifting repeatedly to a trivial lattice. Furthermore, we introduce two speci9c types of liftings, 1-lifting and 2-lifting, and show that a family of sets is an antimatroid of poset shelling if and only if it is constructed from a trivial lattice by repeating 1-lifting. Similarly, an antimatroid of edge-shelling of a tree is shown to be constructed by repeating 2-lifting, and vice versa. ? 2002 Elsevier Science B.V. All rights reserved. 1. Posets, lattices and antimatroids We 9rst present the de9nition of terminology. For a partially ordered set P=(S;4), an ideal of P is a subset K of S such that if x∈K and y 4 x for y∈ S, then y∈K . A (lter is the complement set of an ideal. [x; y] = {z ∈ S: x 4 z; z 4 y} is the interval between x and y. The lattice consisting only of an empty set is called a trivial lattice, and 2[n] denotes the Boolean algebra of all the subsets of an n-element set. For distinct elements x; y∈ S with x 4 y, if x 4 z 4 y necessarily implies x= z or z=y, then x is covered by y. A poset is called a forest if every element is covered by at most one element. In a forest, we call a maximal element a treetop. For the treetops t1; : : : ; tk of a forest Q=(S;4), clearly their principal ideals Ti = {x∈Q: x 4 ti} for i=1; : : : ; k form a partition of S. E-mail address: [email protected] (M. Nakamura). 0166-218X/02/$ see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S0166 -218X(01)00288 -8 160 M. Nakamura /Discrete Applied Mathematics 120 (2002) 159–164 Let E denote a non-empty 9nite set, and L a family of subsets of E. For a set X and an element p, we write X \p, X ∪p instead of X \ {p}, X ∪{p} for the sake of simplicity. Also, We let L − p= {X \ p: X ∈L} and for a new element q not in E, L+ q= {X ∪ q: X ∈L}. L is called an antimatroid on E if it satis9es the following: (L1) ∅∈L, (L2) if X = ∅ and X ∈L, then X \ x∈L for some x∈X , (L3) if X; Y ∈L and X * Y , then Y ∪ x∈L for some x∈X \ Y . The element of L is called a feasible set. Under the assumption (L2), (L3) is equivalent to (L3′). (L3′) if X; Y ∈L, then X ∪ Y ∈L. The family of all the ideals of a poset is an antimatroid, which we call a poset-shelling antimatroid. For a tree T =(V; E), L= {X ⊆ E: T − X is connected} (1) is an antimatroid called an edge-shelling antimatroid of T . 2. Lifting and reduction of antimatroids We shall de9ne a one-element extension of antimatroids. Let L1, L2 be the subfamilies of an antimatroid L. Suppose that they satisfy the following: (E0) L1 ∪ L2 =L, (E1) L1 is an antimatroid, (E2) L2 is a 9lter in L, (E3) L2 = {Y ∈L: X ⊆ Y for some X ∈L1 ∩ L2}. Let p be a new element not in E. Then we can de9ne a one-rank higher lattice by (L ↑ p)(L1 ;L2) =L1 ∪ (L2 + p)=L1 ∪ {Y ∪ p: Y ∈L2} (2) which we call a lifting of L at (L1;L2) by p. We write L ↑ p to denote (L ↑ p)(L1 ;L2) when no confusion may occur. Then we have the following theorem. Theorem 2.1. A lifting (L ↑ p)(L1 ;L2) is an antimatroid on a set E ∪ p. Proof. ∅∈L ↑ p is obvious. To see (L2), take any X ∈L ↑ p. If X ∈L1, (L2) is clear. Otherwise suppose X =X ′ ∪ p and X ′ ∈L. If X ′ is not minimal in L2, there exists an element x∈X ′ such that X ′ \x∈L2, and we have X \x=(X ′ \x)∪p∈L ↑ p. If X ′ is minimal in L2, X ′ ∈L1 follows from (E3). Hence X \ p=X ′ ∈L1 ⊆ L ↑ p. So (L2) holds. Finally we shall show (L3′). The only interesting case is that X ∈L1 and Y =Y ′ + p∈L2 + p. By (E2) L2 is a 9lter, and we have X ∪ Y ′ ∈L2. Hence X ∪ Y =(X ∪ Y ′) + p∈L ↑ p. M. Nakamura /Discrete Applied Mathematics 120 (2002) 159–164 161 Next we introduce the converse operation of lifting. Take an element p∈E. Then we have a one-rank lower lattice L ↓ p=L− p= {X \ p: X ∈L}: (3) As is easy to observe, L ↓ p is an antimatroid on E \ p. We call it a reduction of L at p. The reduction and the lifting are the converse of each other. Theorem 2.2. (a) For any p∈E, we have ((L ↓ p) ↑ p)(L1 ;L2) =L; (4) where L1 = {X : X ∈L; p ∈ X }; L2 = {X − p: X ∈L; p∈X }. (b) Conversely; take a new element q not in E; and suppose L1 and L2 satisfy (E0)–(E3). Then ((L ↑ q)(L1 ;L2)) ↓ q=L: (5) Proof. We shall 9rst show (a). Obviously, (E0) and (E1) hold for L1;L2 in L ↓ p. To see that (E2) holds, take any X ′ ∈L ↓ p=L1 ∪ L2 such that X ⊆ X ′ for some X ∈L2, and we shall show that X ′ ∈L2. Suppose contrarily X ′ ∈ L2. Then X ′ ∈L1. So we have X ′ ∈L, while X ∪p∈L holds from the assumption. It follows from (L3) that X ′ ∪ p∈L. Hence X ′ ∈L2, a contradiction. Accordingly, L2 is a 9lter in L ↓ p. To show (E3), take any X ∈L2. Let Z be a minimal element of L2 such that Z ⊆ X . We shall show that Z belongs to L1. By assumption, Z ′ =Z ∪ p∈L. By (L2), there exists a∈Z ′ =Z ∪p such that Z ′ \ a∈L. If a=p, we have Z =Z ′ \ a∈L and Z ∈L1 follows. If a =p, then Z ′ \ a=(Z \ a) ∪ p∈L. Hence, we have Z \ a∈L2, which contradicts the minimality of Z . Hence we have Z ∈L1∩L2 and (E3) follows. Since it is easy to check that the lifting of L ↓ p at (L1;L2) is equal to L, (a) readily follows. Similarly (b) can be shown. From Theorems 2.1 and 2.2, we have the following. Corollary 2.1. Let L be a family of subsets of E. Then L is an antimatroid if and only if it can be constructed from a trivial lattice by applying lifting repeatedly. Proof. Order arbitrarily the elements of E as p1; p2; : : : ; pn. Then (· · · ((L ↓ p1) ↓ p2) · · ·) ↓ pn is a trivial lattice, and repeating the reverse lifting n times gives L. 3. Characterizations of poset-shelling antimatroids and edge-shelling antimatroids of trees In this section, we shall present the characterizations of poset-shelling and tree edge-shelling antimatroids in terms of certain special liftings. 162 M. Nakamura /Discrete Applied Mathematics 120 (2002) 159–164 Let L be an antimatroid on E, and A a feasible set of L. When we de9ne L1 and L2 by L1 =L; L2 = [A; E]; (6) then (E0)–(E3) are trivially satis9ed, and the resultant lifting is a 1-lifting. If E\A∈L is further satis9ed, we call it a self-dual 1-lifting. The poset-shelling antimatroids are characterized by 1-lifting. Theorem 3.1. Let L be a family of subsets of E. Then L is a poset-shelling antimatroid if and only if it can be constructed from a trivial lattice by repeating 1-lifting. Proof. First, suppose L is a poset-shelling antimatroid on E, we shall prove that L can be constructed by 1-lifting. We use induction on n= |E|. If n=0, the assertion is trivial. Suppose the assertion holds until n= k, and let L′ be a poset-shelling antimatroid on the underlying set E′ with |E′|= k + 1. Take a maximal element p of E′ and set A′ = {x∈E′: x 4 p}. Then the reduction L=L′ ↓ p is easily seen to be equal to the shelling antimatroid of the poset on E=E′ \p. Obviously A=A′ \p is an ideal in E. Hence, we can de9ne a 1-lifting L′′ =(L ↑ p)(L; [A;E]) of L and it is easy to check that L′′ is equal to L. This completes the induction step. Conversely, suppose L is constructed from a trivial lattice by applying 1-lifting n times. We shall show L is a poset-shelling antimatroid. We use induction on n. If n=0 then the assertion is trivial. Let p be a new element not in E. Take a feasible set A∈L, and consider 1-lifting L′ =(L ↑ p)(L; [A;E]). We extend the partial order to that on E′ =E ∪ p by { x 4 p for x∈A; x and p are incomparable in E′ for x∈E \ A; (7) (In E′, the other relations of elements are the same as those in E.) Now it is an easy routine to check that L′ is the poset-shelling antimatroid of (E′;4). This completes the proof. An antimatroid of shelling of a forest, which is a special case of posets, can be characterized by self-dual 1-lifting. Corollary 3.1. L is a poset-shelling antimatroid of a forest if and only if it is constructed from a trivial lattice by repeating self-dual 1-lifting. Proof. We shall show the suKciency part 9rst. Let L be an antimatroid on E obtained by repeating self-dual 1-lifting. We use induction on n= |E|. The case of n=0 is trivial. Take a feasible set A of L such that E \ A is also feasible. Let L′ =(L ↑ q)(L; [A;E]) be the associated self-dual 1-lifting. By induction hypothesis, L is a shelling antimatroid of a certain forest F =(E;4). Let S be the set of the treetops of F . Since A and E \A are both feasible sets, they are ideals of F . So if x 4 y in F , then either x; y∈A or M. Nakamura /Discrete Applied Mathematics 120 (2002) 159–164 163 x; y∈E \ A holds. It follows from this that there exists a partition of treetops S into two sets S1; S2 such that