Large rigid sets of algebras with respect to embeddability

Czédli, Gábor; Jakubíková-Studenovská, Danica

doi:10.1515/ms-2015-0145

Cited by 1 publication

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“…Note that rigidity (with respect to a given family of maps) is a frequently studied concept in various fields of mathematics; we mention only [7] and [8] from 2016, when the present paper was submitted. As usual, N 0 stands for the set {0, 1, 2, .…”

Section: Affine-rigid Functionsmentioning

confidence: 99%

Representing convex geometries by almost-circles

Czédli¹,

Kincses²

2017

Acta Sci. Math.

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Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set T RR of planar convex polygons such that T RR with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of T RR to a finite subset in a natural way. An almost-circle of accuracy 1 − ǫ is a differentiable convex simple closed curve S in the plane having an inscribed circle of radius r 1 > 0 and a circumscribed circle of radius r 2 such that the ratio r 1 /r 2 is at least 1−ǫ. Motivated by Richter and Rogers' result, we construct a set Tnew such that (1) Tnew contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves;(2) Tnew with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to T RR , Tnew is closed with respect to nondegenerate affine transformations; and (4) for every (small) positive ǫ ∈ R and for every finite convex geometry, there are continuum many pairwise affinedisjoint finite subsets E of Tnew such that each E consists of almost-circles of accuracy 1−ǫ and the convex geometry in question is represented by restricting the convex hull operator to E. The affine-disjointness of E 1 and E 2 means that, in addition to E 1 ∩ E 2 = ∅, even ψ(E 1 ) is disjoint from E 2 for every non-degenerate affine transformation ψ. Date: August 23, 2016. 1991 Mathematics Subject Classification. Primary 05B25; Secondary 06C10, 52A01. Key words and phrases. Abstract convex geometry, anti-exchange system, differentiable curve, almost-circle. This research was supported by NFSR of Hungary (OTKA), grant number K 115518. 1 (1.3) Points(X) = C∈X C, and Conv T (X) : = {D ∈ T : D ⊆ Conv R 2 (Points(X))}.

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Section: Affine-rigid Functionsmentioning

confidence: 99%