#### 54 Citations

Quasiplanar Diagrams and Slim Semimodular Lattices

- Mathematics, Computer Science
- Order
- 2016

A bijection allows one to describe finite posets of order dimension at most 2 by finite slim semimodular lattices, and it is obtained that there are exactly (n−2)! quasiplanar diagrams of size n. Expand

Finite convex geometries of circles

- Computer Science, Mathematics
- Discret. Math.
- 2014

It is proved that if the circles are collinear and they are arranged in a “concave way”, then they determine a convex geometry of convex dimension at most 2, and each finite convex geometries can be represented this way. Expand

Convexities on ordered structures have their Krein--Milman theorem

- Mathematics
- 2013

We show analogues of the classical Krein-Milman theorem for several ordered algebraic structures, especially in a semilattice (non-linear) framework. In that case, subsemilattices are seen as convex… Expand

Length-preserving extensions of semimodular lattices by lowering join-irreducible elements

- Mathematics
- 2021

We prove that if e is a join-irreducible element of a semimodular lattice L of finite length and h < e in L such that e does not cover h, then e can be “lowered” to a covering of h by taking a… Expand

Resolutions of Convex Geometries

- Mathematics
- 2021

Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries,… Expand

A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries

- Computer Science, Mathematics
- Order
- 2020

A duality between nonmonotonic consequence relations and well-founded convex geometries is presented, which is an extension of an existingDuality between path independent choice functions and convex Geometries that has been developed independently by Koshevoy and by Johnson and Dean. Expand

On the number of atoms in three-generated lattices

- Mathematics
- 2020

As the main achievement of the paper, we construct a three-generated, 2-distributive, atomless lattice that is not finitely presented. Also, the paper contains the following three observations.… Expand

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

- Physics, Mathematics
- Categories and General Algebraic Structures with Application
- 2019

K. Adaricheva and M. Bolat have recently proved that if $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1-k}$… Expand

Circles and crossing planar compact convex sets

- Mathematics
- Acta Scientiarum Mathematicarum
- 2019

Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that whenever $K_1\subseteq \mathbb R^2$ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense… Expand

Description of closure operators in convex geometries of segments on the line

- Mathematics
- Algebra universalis
- 2019

Convex geometry is a closure space $(G,\phi)$ with the anti-exchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear… Expand

#### References

SHOWING 1-10 OF 24 REFERENCES

Ordered Sets and Social Sciences

- Computer Science
- 1982

Problems of modeling, of analysis and of aggregation of preferences provide important examples to illustrate the connections between ordered sets and social sciences. We review these problems… Expand

The Subposet Lattice and the Order Polynomial

- Computer Science, Mathematics
- Eur. J. Comb.
- 1982

The structure of the lattice of all subposets of a fixed poset is explored. This lattice is then used to prove some identities for the order polynomial of that poset.

The lattice of convex sets of an oriented matroid

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 1982

Abstract Using the theory of the anti-exchange closure the structure of the lattice of convex sets of an acyclic oriented matroid is described. A new expression for the characteristic polynomial of… Expand

Meet-distributive lattices and the anti-exchange closure

- Mathematics
- 1980

This paper defines the anti-exchange closure, a generalization of the order ideals of a partially ordered set. Various theorems are proved about this closure. The main theorem presented is that a… Expand

Convexity in directed graphs

- Mathematics
- 1971

Abstract In this paper the concept of convexity in directed graphs is described. It is shown that the set of convex subgraphs of a directed graph G partially ordered by inclusion forms a complete,… Expand