Semimodular Lattices Research Articles

Relations Between Some Dimensions of Semimodular Lattices

The aim of this paper is to present relations between Goldie, hollow and Kurosh-Ore dimensions of semimodular lattices. Relations between Goldie and Kurosh-Ore dimensions of modular lattices were studied by Grzeszczuk, Okiński and Puczyłowski.

Excluded-minor characterizations of antimatroids arisen from posets and graph searches

An antimatroid is a family of sets such that it contains an empty set, and it is accessible and closed under union of sets. An antimatroid is an ‘antipodal’ concept of matroid. We shall show that an antimatroid is derived from shelling of a poset if and only if it does not contain a minor isomorphic to S 7 where S 7 is the smallest semimodular lattice that is not modular. It is also shown that an antimatroid is a node-search antimatroid of a rooted digraph if and only if it does not contain a minor isomorphic to D 5 where D 5 is a lattice consisting of five elements ∅,{ x},{ y},{ x, y} and { x, y, z}. Furthermore, it is shown that an antimatroid is a node-search antimatroid of a rooted undirected graph if and only if it does not contain D 5 nor S 10 as minor: S 10 is a locally free lattice consisting of 10 elements.

Reconstruction of Finite Truncated Semi-Modular Lattices

In this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general.

Excluded–Minor Characterizations of Antimatroids arisen from Posets and Graph Searches

Abstract An antimatroid is a family of sets such that it contains an empty set, and it is accessible and closed under union of sets. An antimatroid is a ’dual’ or ‘antipodal’ concept of matroid. We shall show that an antimatroid is derived from shelling of a poset if and only if it does not contain a minor isomorphic to S7 where S7 is the smallest semimodular lattice that is not modular (See Fig. 1). It is also shown that an antimatroid is a node-search antimatroid of a digraph if and only if it does not contain a minor isomorphic to D5 where D5 is a lattice consisting of five elements O {x},{y}, {x, y} and {x, y, z}. Furthermore, an antimatroid is shown to be a node-search antimatroid of an undirected graph if and only if it does not contain D5 nor S10 as a minor: S10 is a locally free lattice consisting of ten elements shown in Fig. 2.

Representations of Matroids in Semimodular Lattices

Representations of matroids in semimodular lattices and Coxeter matroids in chamber systems are considered in this paper.

Reducibility in Finite Posets

A notion of reducibility in finite posets is studied. Deletable elements in upper semimodular posets are characterized. Though it is known that the class of upper semimodular lattices is reducible, we construct an example of an upper semimodular poset that is not reducible. Reducibility of pseudocomplemented posets is studied.

Strong semimodular lattices and Frankl's conjecture

In this paper, we show that Frankl's conjecture holds for strong semimodular lattices. The result is the first step to deal with the case of upper semimodular lattices.

Frankl's Conjecture Is True for Lower Semimodular Lattices

It is shown that every finite lower semimodular lattice L with |L|≥2 contains a join-irreducible element x such that at most |L|/2 elements y∈L satisfy y≥x.

Graph automorphisms of semimodular lattices

Graph automorphisms of semimodular lattices

On quasi-invariant elements of a lattice

Let L be a lattice and G a group acting on L. An element x of L is said to be quasi-G-invariant if for every $ g \in G $ either $ x \le g(x) $ or x covers $ x \wedge g(x)$ . We prove that if L is an upper semimodular lattice and $ x \in L $ is quasi-G-invariant then there is a $ g \in G $ such that x covers $ x \wedge g(x)$ and $ x \wedge g(x) $ is G-invariant, or there is a $ g \in G $ such that $x \vee g(x) $ covers x and $ x \vee g(x) $ is G-invariant.

Congruence Lattices of Finite Semimodular Lattices

AbstractWe prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

Congruences on ideal nil-extension of completely regular semigroups

Let S be an ideal nil-extension of a completely regular semigroupK by a nil semigroupQ with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a congruence onK respectively. It is proved that every congruence on S can be uniquely respresented by an admissible congruence pair (δ,ω) ofS. Suppose that ϱk denotes the Rees congruence induced by the idealK ofS. Then it is shown that for any congruenceσ on S, a mapping Γ: σ ¦→ (σQ, σK) is an order-preserving bijection from the set of all congruences on S onto the set of all admissible congruence pairs of S, where σK is the restriction of σ toK and σQ = (σ V ϱK) / ϱK. Moreover, the lattice of congruences of S is also discussed. As a special case, every congruence on completely Archimedean semigroups S is described by an admissible quarterple of S. The following question is asked: Is the lattice of congruences of the completely Archimedean semigroup a semimodular lattice?

Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices

We study paths between maximal chains, or “flags,” in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the subposet is equal to the sublattice generated by the flags. It is a distributive lattice which is determined by the “Jordan-Hölder permutation” between the flags. The minimal paths correspond to all reduced decompositions of this permutation. In a semimodular lattice, the subposet is not uniquely determined by the Jordan-Hölder permutation for the flags. However, it is a join sublattice of the distributive lattice corresponding to this permutation. It is semimodular, unlike the lattice generated by the two flags, which may not be ranked. The minimal paths correspond to some reduced decompositions of the permutation, though not necessarily all. We classify the possible lattices which can arise in this way, and characterize all possibilities for the set of shortest paths between two flags in a semimodular lattice.

Semimodular Lattices and Semibuildings

In a ranked lattice, we consider two maximal chains, or “flags” to be i-adjacent if they are equal except possibly on rank i. Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a “Jordan-Holder permutation” between any two flags. This permutation has the properties of an S_n-distance function on the chamber system of flags. Using these notions, we define a W-semibuilding as a chamber system with certain additional properties similar to properties Tits used to characterize buildings. We show that finite rank semimodular lattices form an S_n-semibuilding, and develop a flag-based axiomatization of semimodular lattices. We refine these properties to axiomatize geometric, modular and distributive lattices as well, and to reprove Tits‘ result that S_n-buildings correspond to relatively complemented modular lattices (see [16], Section 6.1.5).

The Lattice of Completions of an Ordered Set

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind–MacNeille completion P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of closure operators on any complete lattice. In particular, if K(P) is finite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive.

Order Analogues and Betti Polynomials

We exhibit an order-preserving surjection from the lattice of subgroups of a finite abelianp-group of typeλonto the product of chains of lengths the parts of the partitionλ. Thereby, we establish the subgroup lattice as an order-theoretic, not just enumerative,p-analogue of the chain product. This insight underlies our study of the simplicial complexesΔS(p), whose simplices are chains of subgroups of orderspk, somek∈S. Each of these subgroup complexes is homotopy equivalent to a wedge of spheres of dimension |S|−1. The number of spheres in the wedge,βS(p), is known to have nonnegative coefficients as a polynomial inp. Our main result provides a topological explanation of this enumerative result. We use our order-preserving surjection to findβS(pmaximal simplices inΔS(pwhose deletion leaves a contractible subcomplex. This work suggests a definition of order analogue; our main result holds for any semimodular lattices that are order analogues of a semimodular lattice.

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.

A generalization of semiconductor supersolvable lattices

Stanley ( Algebra Universalis 2, 1972, 197–217) introduced the notion of a supersolvable lattice, L, in part to combinatorially explain the factorization of its characteristic polynomial over the integers when L is also semimodular. He did this by showing that the roots of the polynomial count certain sets of atoms of the lattice. In the present work we define an object called an atom decision tree. The class of semimodular lattices with atom decision trees strictly contains the class of supersolvable lattices, but their characteristic polynomials still factor for combinatorial reasons. We then apply this notion to prove the factorization of polynomials associated with various hyperplane arrangements having non-supersolvable lattices.

Extreme k-families

Let P be a poset. A subset A of P is a k-family iff A contains no ( k + 1)-element chain. For i ⩾ 1, let A i be the set of elements of A at depth i − 1 in A. The k-families of P can be ordered by defining A ⩽ B iff, for all i, A i is included in the order ideal generated by B i . This paper examines minimal r-element k-families, defined as k-families A such that | A| = r and for every B < A, | B| < r. Minimal k-families are related to maximal r-antichains and an operation called Sperner closure, which have been used to obtain extremal results for families of sets with width restrictions. Let M k,r be the set of minimal r-element k-families and let M k = ∪ r ≥ 0 M k, r . It is shown that M k is a join-subsemilattice by the lattice A k of k-families. M k is a lower semimodular lattice, where the rth rank is given by M k,r . If w k is the maximum size of a k-family, then | M k, r | ⩽ ( w r k)and |∪ M k | ⩽ Σ i = 1 w k ⌈ i/ k⌉. Let D(A) = max{| B| − | A| | B is a k-family and B ⩽ A}. For k-families A and B, D(A v B) ⩽ D(A) + D(B). This result shows that { A | D( A) = 0} is also a join-subsemilattice of A k .

Hereditarily strong lattices

Lower (or: dually) semimodular lattices behave in certain situations like consistent upper semimodular lattices. A common property of both classes of lattices is that they are hereditarily strong in the sense that every interval is a strong sublattice (cf. Section 2). We relate the properties “hereditarily strong” and “consistent” as well as their duals to lower semimodularity and to modularity (Section 3). Finally we characterize modularity for consistent lower semimodular lattices by means of forbidden sublattices (Section 4).