Base Axioms of Modular Supermatroids
This paper studies axiom systems of supermatroids. Barnabei et al.'s base axioms concerning poset matroids (i.e., distributive supermatroids) are generalized to modular supermatroids, and a mistake in the proof of base axioms of poset matroids is pointed out.
Matroids, as an important combinatorial structure, have been generalized by many authors, such as polymatroids (Edmonds ), supermatroids (Dunstan et al. ), greedoids (Korte et al. ), and fuzzy matroids (Goetschel and Voxman ).
One of the most natural extensions may be supermatroids, which generalize the underlying sets of matroids to arbitrary finite partially ordered sets. Supermatroids connect closely with other extensions; for example, integral polymatroids are essentially supermatroids on a special class of finite distributive sublattices of and greedoids are strong supermatroids on graded posets. As supermatroids, Faigle’s ordered geometries  combine the exchange properties with finite posets, meanwhile, ordered geometries generalize a particular interest case of supermatroids and distributive supermatroids (i.e., supermatroids on distributive lattices). Tardos  showed an intersection theorem for distributive supermatroids. Moreover, Barnabei et al.  studied distributive supermatroids in more detail (in the name of poset matroids). Another approach to generalizing the concept of distributive supermatroids was proposed by Fujishige et al. ; they studied supermatroids on lattices of closed sets of convex geometries (cg-matroids). For a related general framework, see [3, 9].
To study axiom systems of generalized matroidal structures are fundamental problems, no matter finite or infinite extensions of matroids (see, e.g., [8, 10–12]). Barnabei et al.  gave many equivalent axiomatizations of distributive supermatroids. The main missing axiom in , that is, flat lattice axioms of distributive supermatroids, was completed by Wild . The present work studies mainly bases axioms of supermatroids on modular lattice (named modular supermatroids). We give two equivalent characterizations of modular supermatroids in terms of bases and the importance of the condition of modular lattices reflected in the proofs. We note here that, without further mention, all sets and structures in this paper are finite.
In this section, we introduce some basic concepts and results on matroids, lattices, and supermatroids. For a more detailed exposition of these topics, refer to [3, 13, 14], respectively.
Definition 1 (Welsh ). A matroid is an ordered pair consisting of a finite set and a nonempty collection of subsets of having the following two properties.(I1)If , , then .(I2*)If and , then there is an element such that .
Let be a matroid; then the members of are the independent sets of , and we call a maximal independent set in a base of . Definition 1 defines a matroid via independent sets; we now present base axioms of matroids.
Theorem 2 (Welsh ). Let be a set of subsets of a finite set . Then is a collection of bases of a matroid on if and only if it has the following properties.(B1) is nonempty.(B2*)If and , then there is an element of such that (base exchange axiom).
Axiom (B2*) can be replaced by the following middle base axiom.
Theorem 3 (White ). Let be a set of subsets of a finite set . Then is a collection of bases of a matroid on if and only if satisfies (B1) and the following conditions.(B2)For , if has members , such that , , then has a member such that (middle base axiom).(B3)If and , then .
Remark 4. Matroids have several equivalent descriptions. Extensions of matroids are used to generalize different axiom systems of matroids. For example, Goetschel and Voxman fuzzy matroids generalize (I2*) to fuzzy sets. It can be verified that (I2) in Definition 1 can be replaced by the following condition.
(I2) If , , are maximal members of , then . Supermatroids use an extension of (I2) in posets. As mentioned above, poset matroids are equivalent to distributive supermatroids, but the original definition of poset matroids extends middle base axiom to posets.
Now we summarize some facts of posets and lattices needed in this paper.
Let be a poset, . If and implies or , we say that covers (denoted by or ). An ideal of a poset is a subset such that , implies . Down() denotes the set of all ideals of . Dually, a filter is a subset such that , implies . Let ); this ideal is called the principal ideal generated by , and principal filters are defined analogously. The intersection of a principal ideal and a principal filter is an interval ; that is, . A subset such that any two elements of are comparable (incomparable) is a (an) chain (antichain). The length of the chain is . The height of an element is the length of the longest chain from 0 to .
An element is the meet of and (denoted by ) if for any we have that if and only if and . An element is the join of and (denoted by ) if for any we have that if and only if and . A poset is a lattice if any two elements have and . A lattice is said to satisfy the upper covering condition if implies or for all . The lower covering condition is the dual. A lattice is a modular lattice if satisfies both the lower covering condition and the upper covering condition. The following theorem gives some equivalent forms of modular lattice.
Theorem 5 (Roman ). For a lattice , the following conditions are equivalent.(1) is modular.(2) does not contain (also called a pentagon).(3)For all , .
In 1972, Dunstan et al.  introduced a generalization of matroids to partial sets as follows.
Definition 6 (Dunstan et al. ). Let be a poset with a minimal element 0 and height function . is a supermatroid if satisfies the following three conditions.(LI0).(LI1)If , , then .(LI2)For any , all maximal elements of have the same height.
The elements of are called independent elements of the supermatroids; otherwise an element is dependent. A base of is a maximal independent element.
We now introduce another matroidal structure on poset sets proposed by Barnabei et al. in 1998.
Definition 7 (Barnabei et al. ). A poset matroid on the partial set is a family of filters of , called bases, and satisfies the following axioms.(DLB1).(DLB2)For , if has members , such that , , then has a member such that .(DLB3)If and , then .
Remark 8. Poset matroids deal with filters of posets, which seems different from supermatroids that concern mainly elements of partial sets. However, by a fundamental theorem of Birkhoff (every finite distributive lattice is isomorphic to the lattice of all filters of a finite partially ordered set. Conversely, every finite partially ordered set is isomorphic to the partially ordered set of the meet-irreducible elements of a distributive lattice), poset matroids are just supermatroids on distributive lattices. Though supermatroids were introduced before poset matroids, unfortunately, Barnabei et al.’s paper  never mentioned the name “supermatroids."
3. Base Axioms of Modular Supermatroids
In this section, we will characterize modular supermatroids in terms of bases. First, we will prove some properties of bases of modular supermatroids.
Theorem 9. Let be a supermatroid on a lattice and let be the set of bases of ; then satisfies the following condition.(LB2)Suppose ; then for every pair satisfying , , and , there exists such that .
Proof. (LB2) obviously holds if or . In the following, we assume and . It follows from (LI2) that ; thus, . Since maximal elements of have the same height, there is such that . If , similarly, it follows from (LI2) that there exists satisfying . If , then there exists satisfying . If , we continue the above process. After finite steps, we have , , and . Note that maximal elements of have the same height; thus, ; this concludes the proof.
Remark 10. (B2) characterizes matroids (Theorem 3), and this fact can be extended to distributive supermatroids (poset matroids are actually defined by generalized form of (B2), and then independence axioms of poset matroids, i.e., generalized forms of (I2*), and equivalent descriptions of independence axioms of poset matroids were proved in ). However, an antichain with (LB2) cannot define a supermatroid on a lattice, as the following example shows.
Example 11. Consider the lattice in Figure 1(a). is the set of black-filled points. Let ; then satisfies (LB2), but is not a supermatroid. Note that the two elements in have different heights, and the following result tells us that this cannot happen when is a modular lattice.
Lemma 12. Suppose that is a modular lattice; . Let ; if satisfies (LB2), then all elements in have the same height.
Proof. Suppose ; we apply induction on . Obviously, and for the assertion is true. Assume that and have the same height when . Let . Choose and ; then . Consider the following two cases.
Case 1 ( (thus )). By the upper covering condition, ( contradicts the fact that , are maximal elements of ). Choose and . If , then . We now consider the case .
Since , it follows from the upper covering condition that or . If , it is easy to verify that the sublattice of is just a pentagon, contrary to the fact that is a modular lattice. Thus, .
Since , and , it follows from (LB2) that there exists such that . Because , we have By the induction hypothesis, we have . and imply . Note , so Thus holds by the induction hypothesis. As a result, we get (cf. Figure 1(b)).
Case 2 (). Since , it holds by the upper covering condition that or . If , then forms a sublattice of , contradicting the fact that is a modular lattice. Thus, . Consider that , , and ; it follows from (LB2) that there exists satisfying . The subsequent proof is similar to Case 1; we can verify , as desired (cf. Figure 1(c)).
Now we prove the first base axioms of modular supermatroids.
Theorem 13 (base axioms of modular supermatroids). Let be a modular lattice and . Then is the set of bases of supermatroids on if and only if it has the following property.(LB2)Suppose ; then for every pair satisfying , , and , there exists such that .(LB3)If and , then .
Proof. We have already proved that the collection of bases of a supermatroid satisfies (LB2) (i.e., Theorem 9) and obviously satisfies (LB3). To show the converse, let . It follows from (LB3) that Max . Assume and . We will prove the following.
(LI2*) If , , then there exists such that holds. For (LI2*) (LI2) is trivial, we conclude the proof.
The definition of implies that there exists such that and . Since , , and , it holds from (LB2) that there exists satisfying . If , then (LI2*) holds. We now consider the case . By , we have . From Lemma 12, holds; thus, . Clearly, . To prove , we assume and derive a contradiction. is a modular lattice; thus,
By , so
It follows from (3) and (4) that
And since with (5), we have
Note the assumption ; (7) implies
It holds from and (8) that
Let . By the upper covering condition, forms a maximal chain from to . Therefore, , contrary to (9) (note ). Thus, is proved.
Choose and . It is trivial that . Since we have ; thus, . implies , and obviously , so (LI2*) holds and this completes the proof.
Remark 14. Barnabei et al. defined poset matroids in terms of base axioms. They proved the equivalence of base axioms and independence axioms for distributive supermatroids (see Theorem 5.1 in ). Our theorem (Theorem 13) actually generalizes their results about base axioms to modular supermatroids.
Remark 15. In the proof of Theorem 5.1 in , (LB2*) (LI2*) was verified by induction ((LB2*) is equivalent to (LB2) for distributive supermatroids; see Theorem 4.3 of . We will prove the equivalence of (LB2*) and (LB2) for modular supermatroids in the following). Unfortunately, there is something wrong with the induction. We take a matroid, that is, a special case of poset matroids, as an example to verify our assertion. Consider the cycle matroid , where is shown in Figure 2(b). is an independent set and is a base. Though for every , obviously, is an independent set (i.e., we can augment by an element in ), it is easy to check that we cannot augment to a larger independent set by the base exchange axiom. This example shows that the assertion “if , the thesis follows immediately by the exchange property and by Theorem 4.1” in the proof of [7, Theorem 5.1] is not right. We give a direct proof of a more generalized result of [7, Theorem 5.1] in Theorem 13 instead of the induction approach and our proof presents the important role of modular properties.
We now give another base axiom of modular supermatroids.
Theorem 16. Let be a modular lattice, , and . Then the following are equivalent.(LB2)Suppose ; then for every pair satisfying , , and , there exists such that .(LB2*)Suppose ; then for every , there exists such that .
Proof. (LB2) (LB2*). Let , , satisfying . Since , , and , it holds from (LB2) that there exists such that . By Lemma 12, ; thus . Then the lower covering condition implies
If , observe that implies ; then is the sublattice of , contradicting the fact that is a modular lattice. Hence, .
It is easy to check . Since and are incomparable, we have Thus, . Then it follows from that and . Because and , as desired.
(LB2*) (LB2). Let and , satisfying , , and . If or , then (LB2) holds. We now consider the case and . In the subset of that all elements are equal to or greater than , we choose a such that the height of is maximal (i.e., and for all , ). Note that or they are incomparable. We assert ; then this finishes our proof. To prove our assertion, we assume that and are incomparable and derive a contradiction.
Since , choose , satisfying . By (LB2), there exists such that . Let ; then . Obviously, . We consider the following two cases.
Case 1 (). Note , and it is easy to verify ; hence, forms a sublattice of , contradicting the fact that is a modular lattice.
Case 2 (). In this case, holds. Since is a modular lattice, we have (13) implies It is easy to check ((LB2*) implies (LB2); then it holds from Lemma 12 that . We certainly can prove directly that (LB2*) implies that all elements in have the same height (but not needed here), though the proof is not trivial); thus it follows from (14) that , contrary to the definition of .
This paper establishes the middle base axiom and base exchange axiom of modular supermatroids. For independence axioms, we can prove that (I2*) and its other equivalent characterizations can also be generalized to modular supermatroids, and most of these extensions do not need the restriction of modular lattices. However, we cannot extend directly circuit axioms of distributive supermatroids (i.e., Theorem 9.1 of ) to modular supermatroids. Note that there are many equivalent descriptions of circuits for matroids; thus, choosing one proper definition and constructing circuit axioms of modular supermatroids are our future work. Apart from axiom systems, the second reason for us to consider properties of circuits is to study the connectedness of supermatroids. Connectedness is an important topic in matroid theory, while few papers concerning extensions of matroids refer to connectedness. Our work of this paper, meanwhile, proposes preliminaries for subsequent study about circuits.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors wish to thank the anonymous reviewers for their careful reading and corrections. This work is supported by the National Natural Science Foundation of China (Grant nos. 61202178 and 61203372) and the Fundamental Research Funds for the Central Universities (Grant no. K50510700006).
J. Edmonds, “Submodular functions, matroids and certain polyhedra,” in Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications, pp. 69–87, Gordon and Breach, New York, NY, USA, 1969.View at: Google Scholar
F. D. J. Dunstan, A. W. Ingleton, and D. J. A. Welsh, “Supermatroids,” in Proceedings of the conference on Combinatorial Mathematics, pp. 72–122, Mathematical Institute, Oxford, UK, 1972.View at: Google Scholar
B. Korte, L. Lovsz, and R. Schrader, Greedords, Springer, Berlin, Germany, 1991.
R. Goetschel, and W. Voxman, “Fuzzy matroids,” Fuzzy Sets and Systems, vol. 27, no. 3, pp. 291–302, 1988.View at: Publisher Site | Google Scholar | MathSciNet
U. Faigle, “Geometries on partially ordered sets,” Journal of Combinatorial Theory B, vol. 28, no. 1, pp. 26–51, 1980.View at: Publisher Site | Google Scholar | MathSciNet
E. Tardos, “An intersection theorem for supermatroids,” Journal of Combinatorial Theory B, vol. 50, no. 2, pp. 150–159, 1990.View at: Publisher Site | Google Scholar | MathSciNet
M. Barnabei, G. Nicoletti, and L. Pezzoli, “Matroids on partially ordered sets,” Advances in Applied Mathematics, vol. 21, no. 1, pp. 78–112, 1998.View at: Publisher Site | Google Scholar | MathSciNet
S. Fujishige, G. A. Koshevoy, and Y. Sano, “Matroids on convex geometries,” Discrete Mathematics, vol. 307, no. 15, pp. 1936–1950, 2007.View at: Publisher Site | Google Scholar | MathSciNet
M. Wild, “Weakly submodular rank functions, supermatroids, and the flat lattice of a distributive supermatroid,” Discrete Mathematics, vol. 308, no. 7, pp. 999–1017, 2008.View at: Publisher Site | Google Scholar | MathSciNet
S.-G. Li, X. Xin, and Y.-L. Li, “Closure axioms for a class of fuzzy matroids and co-towers of matroids,” Fuzzy Sets and Systems, vol. 158, no. 11, pp. 1246–1257, 2007.View at: Publisher Site | Google Scholar | MathSciNet
Y. Sano, “Rank functions of strict cg-matroids,” Discrete Mathematics, vol. 308, no. 20, pp. 4734–4744, 2008.View at: Publisher Site | Google Scholar | MathSciNet
W. Yao and F.-G. Shi, “Bases axioms and circuits axioms for fuzzifying matroids,” Fuzzy Sets and Systems, vol. 161, no. 24, pp. 3155–3165, 2010.View at: Publisher Site | Google Scholar | MathSciNet
D. J. A. Welsh, Matroid Theory, Academic Press, London, UK, 1976.View at: MathSciNet
S. Roman, Lattice and Ordered, Springer, New York, NY, USA, 2008.
N. White, Ed., Theory of Matroids, Cambridge University Press, Cambridge, UK, 1986.