Novita Dahoklory 

Program Studi Matematika, Universitas Pattimura

Henry W. M. Patty

Program Studi Matematika, Universitas Pattimura

DOI: https://doi.org/10.19184/mims.v24i2.45887

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