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.

On the measure of fuzziness and negation. II. Lattices

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.

Median algebras

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.

https://www.ams.org/proc/1961-012-03/S0002-9939-1961-0125807-5/S0002-9939-1961-0125807-5.pdf

Metric ternary distributive semi-lattices

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.

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Coarse median spaces and groups

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.

/pdf/medians-lattices-and-trees-4q57l1b1x2.pdf

Medians, lattices, and trees

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.

/pdf/trees-lattices-order-and-betweenness-1ajqqmwefm.pdf

Marlow Sholander

Papers

Trees, lattices, order, and betweenness