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Geometry of Autometrized Lattice Ordered Monoids

Tekalign Regasa Ashale, Girum Aklilu Abebe, Kolluru Venkateswarlu

Abstract

In this paper, we study the geometry of AL-monoids by introducing the concept of metric betweeness and its properties t1, t2,, B-linearity, D-linearity, lattice betweeness, B-linearity, and Dlinearity, segments and equilateral triangles. It is proved that there do not exist equilateral triangles in AL-monoids. It is also proved that any AL-monoid is ptolemaic.

Geometry of Autometrized Lattice Ordered Monoids

Abstract

In this paper, we study the geometry of AL-monoids by introducing the concept of metric betweeness and its properties t1, t2,, B-linearity, D-linearity, lattice betweeness, B-linearity, and Dlinearity, segments and equilateral triangles. It is proved that there do not exist equilateral triangles in AL-monoids. It is also proved that any AL-monoid is ptolemaic.
Paper Structure (3 sections, 17 theorems, 28 equations)

This paper contains 3 sections, 17 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Result

Key Result

lemma 1

$a\leq b$ implies $a\ast c\leq b\ast c,$ for any $c\in A.$

Theorems & Definitions (44)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • remark 1
  • remark 2
  • lemma 1
  • proof
  • definition 5
  • theorem 1
  • ...and 34 more