Trees, lattices, order, and betweenness
01 Mar 1952-Vol. 3, Iss: 3, pp 369-381
TL;DR: This paper considers postulates expressed in terms of "segments," "medians," and "betweenness" for trees, lattices, and partially ordered sets.
Abstract: In this paper we consider postulates expressed in terms of "segments," "medians," and "betweenness." Characterizations are obtained for trees, lattices, and partially ordered sets. In general a characterization is given by a system of three postulates. These systems fall in pairs; systems of a pair have two postulates in common. An algebra which has both lattices and trees as special cases is given in the final section.
Citations
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TL;DR: The lattice concept of betweenness is introduced to operationalize the measure of fuzziness of a concept and properties of this concept are investigated.
Abstract: We propose that the measure of fuzziness of a concept is related to the distinction between the concept and its negation. In this paper we are concerned with the situation in which our objects lie in a lattice. We introduce the lattice concept of betweenness to operationalize our definition. We investigate properties of this concept of fuzziness. We also discuss the related notion of negation in lattices in detail.
226 citations
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TL;DR: A median algebra is a ternary algebra which satisfies all the identities true for the median operation in distributive lattices and all intrinsic semilattice orders on a median algebra are exhibited and characterized in several ways.
Abstract: A median algebra is a ternary algebra which satisfies all the identities true for the median operation in distributive lattices. The relationship of median algebras to (abstract) geometries, graphs, and semilattices is discussed in detail. In particular, all intrinsic semilattice orders on a median algebra are exhibited and characterized in several ways.
206 citations
01 Mar 1961
TL;DR: In this article, Lillo et al. discuss the stability of continuous matrices and the stability theory of differential systems with almost periodic coefficients, and present a matrix transformation for linear differential equations.
Abstract: 1. J. Lillo, Continuous matrices and the stability theory of differential systems, Math. Z. vol. 73 (1960) pp. 45-58. 2. -, Perturbations of nonlinear systems, Acta Math. vol. 103 (1960) pp. 123138. 3. -, Linear differential equations with almost periodic coefficients, Amer. J. Math. vol. 81 (1959) pp. 37-45. 4. L. Markus, Continuous matrices and the stability of differential systems, Math. Z. vol. 62 (1955) pp. 310-319. 5. W. T. Reid, Remarks on a matrix transformation for linear differential equations, Proc. Amer. Math. Soc. vol. 8 (1957) pp. 708-712.
116 citations
Cites background from "Trees, lattices, order, and between..."
...If a TDSL 3 satisfies (TF), then the metric space ¿1(3), as defined and metrized above, satisfies (U)....
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...The corresponding sequence of ¿J(3) of Lemma 7: aRaiR ■ ■ ■ Ram = b is thus minimal so that ab = Sa[b], the dimension of b in <P(a, 3)....
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...(3) (P(a, 3) is closed with respect to meet given by br\ac—(b, a, c)....
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...In a TDSL 3 satisfying (TF), bRc in ¿J(3) iff bRaC in g(9(a, 3)), where Ra is $a....
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...In ¿J(3) bRc iff (b, x, c) =b or c for all x including (b, a,c)=b or c....
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TL;DR: The notion of a coarse median on a metric space was introduced in this paper, which satisfies the axioms of a median algebra up to bounded distance, and the existence of such a median is quasi-isometry invariant, and so it applies to finitely generated groups via Cayley graphs.
Abstract: We introduce the notion of a coarse median on a metric space. This satisfies the axioms of a median algebra up to bounded distance. The existence of such a median on a geodesic space is quasi-isometry invariant, and so it applies to finitely generated groups via their Cayley graphs. We show that asymptotic cones of such spaces are topological median algebras. We define a notion of rank for a coarse median and show that this bounds the dimension of a quasi-isometrically embedded euclidean plane in the space. Using the centroid construction of Behrstock and Minsky, we show that the mapping class group has this property, and recover the rank theorem of Behrstock and Minsky and of Hamenstadt. We explore various other properties of such spaces, and develop some of the background material regarding median algebras.
85 citations
Cites background from "Trees, lattices, order, and between..."
...They appear in [Sholander 1952] and as “tree algebras” in [Bandelt and Hedlíková 1983]....
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01 May 1954
TL;DR: These systems are characterized in terms of a binary operation and it is further shown that these, along with more general systems, can be imbedded in distributive lattices.
Abstract: In other papers [3; 4]2 the author discussed the behavior of medians, segments, and betweenness in systems called median semilattices. A tree is a type of median semilattice. In this paper these systems are characterized in terms of a binary operation. It is further shown that these, along with more general systems, can be imbedded in distributive lattices. The product notation in this paper is not to be confused with the betweenness notation of [4], nor is the ordered pair notation here to be confused with the segment notation there.
77 citations
Cites background from "Trees, lattices, order, and between..."
...This is Postulate M of [3]....
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...We have shown (K, L1) implies (M, N, O1) of [3]....
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...7) of [3], x(a, b, c) = (xa, xb, c) =(ax, bx, cx) and medians are preserved under multiplication....
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References
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TL;DR: In this paper, the first part of this paper continues the analysis of betweenness in metric lattices into an examination of postulates involving five points, and the second part deals mainly with a definition of the betweenness relation in lattices.
Abstract: Introduction. The examination of the foundations of geometry which interested many prominent mathematicians about the turn of the century brought to light the importance of the fundamental notion of betweenness (see, for example('), [10, 11]). This notion has suffered the treatment which modern mathematics metes out to all its concepts, namely, first an examination of the concept in a particular instance followed by wider and wider generalizations. The first part of this program for the concept of betweenness was carried through by Pasch, Huntington and Kline [8, 10]. The simplicity of the concept permitted them to give an elegant and complete theory for the case of linear order. In the direction of generalizations(2), K. Menger and his students have been one of the most important influences in the study of betweenness in metric spaces [9, 3 ]. We purpose here to add to both phases of this program. The first part of our paper continues the analysis of Huntington and Kline into an examination of postulates involving five points; the second part deals mainly with a definition of betweenness in lattices which generalizes metric betweenness in metric lattices (see [5, 6]). It is hoped that the five point transitivities may prove interesting and their analysis valuable. If we restrict our attention to the relation of betweenness in linear order such cannot be the case since four point properties are then sufficient to describe completely the betweenness relation. We feel that the results of the second part exhibit the properties of the betweenness relation as reflections of properties of the underlying space(3). We shall use the notations of set theory which have become standard. In the second part we shall assume a knowledge of the fundamentals of both lattice theory and metric geometry. We refer the reader to the recent books Distance Geometry by L. M. Blumenthal [3] and Lattice Theory by Garrett Birkhoff [1]. We shall use the terminology and notation of these books in the second part.
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TL;DR: In this paper, a set P of elements a, b, c, in which a binary relation is defined, having the properties (1) a?a for all aCP, (2) b?b and b?a imply a=b, (3) b_c imply a_c, will be called an ordered set.
Abstract: A set P of elements a, b, c, in which is defined a binary relation " " having the properties (1) a?a for all aCP, (2) a?b and b?a imply a=b, (3) a?b, b_c imply a_c, will be called an ordered set. If for any pair of elements a, b CP one of the three relations a a, b and c > a, b implies c > a + b) and their greatest lower bound ab (an element such that ab a for all a CL is the unit of L. Two elements a and b such that ab = 0 are called ,-independent; if a+ b = 1, they are a-independent(3); and if they are both a-independent, and ,u-independent, they are
12 citations