Medians are Below Joins in Semimodular Lattices of Breadth 2
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Medians are Below Joins in Semimodular Lattices of Breadth 2 ´ ´ 1 Gabor Czedli
· Robert C. Powers2 · Jeremy M. White3
Received: 5 November 2019 / Accepted: 20 October 2020 / © The Author(s) 2020
Abstract Let L be a lattice of finite length and let d denote the minimum path length metric on the covering graph of L. For any ξ = (x1 , . . . , xk ) ∈ Lk , an element y belonging to L is called a median of ξ if the sum d(y, x1 ) + · · · + d(y, xk ) is minimal. The lattice L satisfies the c1 -median property if, for any ξ = (x1 , . . . , xk ) ∈ Lk and for any median y of ξ , y ≤ x1 ∨· · ·∨xk . Our main theorem asserts that if L is an upper semimodular lattice of finite length and the breadth of L is less than or equal to 2, then L satisfies the c1 -median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that 2 cannot be replaced by 3. Keywords Semimodular lattice · Breadth · c1 -median property · Covering path · Join-prime element
1 Introduction Given a lattice L of finite length and ξ = (x1 , . . . , xk ) ∈ Lk , an element y ∈ L is called a median of ξ if the sum d(y, x1 ) + · · · + d(y, xk ) is minimal, where
This research was supported by the Hungarian Research, Development and Innovation Office under grant number KH 126581. G´abor Cz´edli
[email protected] Robert C. Powers [email protected] Jeremy M. White [email protected] 1
Bolyai Institute, University of Szeged, Szeged, 6720, Hungary
2
Department of Mathematics, University of Louisville, Louisville, Kentucky 40292 USA
3
School of Natural Science, Spalding University, Louisville, Kentucky 40203 USA
Order
d(y, xi ) stands for the path distance in the Hasse diagram of L. Our goal is to prove that ⎫ whenever L is, in addition, upper semimodular and of breadth ⎪ ⎬ at most 2, to be defined in Eq. 1.10, then y ≤ x1 ∨ · · · ∨ xk (1.1) holds for every k ≥ 2 and for any median y of every ξ = ⎪ ⎭ (x1 , . . . , xk ) ∈ Lk ; see our main result, Theorem 4.1, for more details.
1.1 Outline The paper is structured as follows. In Section 1.2, we survey some earlier results on medians in lattices. Section 1.3 recalls some definitions, whereby the paper is readable with minimal knowledge of Lattice Theory. In Section 2, we give a new way of constructing semimodular lattices; see Proposition 2.1, which can be of separate interest. As a particular case of our construction, we present a semimodular lattice L(n, k) with breadth k and size |L(n, k)| = 2nk − (n − 1)k for any integers k ≥ 3 and n ≥ 4 such that L(n, k) fails to satisfy the c1 -median property. Section 3 is devoted to two technical lemmas that will be used later. Finally, Section 4 presents our main result, Theorem 4.1, which asserts somewhat more than Eq. 1.1. Using the auxiliary statements proved in Sections 2 and 3, 4 concludes with the proof of Theorem 4.1. Note that the survey given in Section 1.2 is mainly for lattice theorists; this is why some well-known lattice the
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