Hubungan antara latis distributif dan aljabar median

Abstract

Let M be a non-empty set equipped by a ternary operation m:M×M×M→M. The set M is called a median algebra if (M,m) satisfies these properties (1) majority: m(a,a,b)=a, associativity: m(a,b,m(c,b,d) = m(m(a,b,c),b,d), and commutativity: m(a,b,c) = m(a,c,b) = m(b,a,c) for every a,b,c,d∈M. In this paper, we will relate a median algebra and a distributive lattice; every distributive lattice is a median algebra. Moreover, we will study an interval [a,b] in a median algebra (M,m) motivated by closed intervals in R. We will also investigate the basic properties of the interval [a,b] in a median algebra. Furthermore, using these properties, we will show that every interval in a median algebra is conversely a distributive lattice.

Keywords: median algebra, distributive lattices, interval.
MSC2020: 06D99.