Joins and meets in effect algebras

Abstract

We know that each effect algebra $E$ is isomorphic to $\pi (X)$ for some $E$-test spaces $(X,\mathcal{T})$.We describe when $\pi (x)\vee \pi (y)$ and $\pi (x)\wedge \pi (y)$ exists for $x,y\in \mathcal{E}(X,\mathcal{T})$. Moreover we give the formula for $\pi (x)\vee \pi (x)$ and $\pi (x)\wedge \pi (y)$ using only $x,y$ and tests which are elements of $\mathcal{T}$. We obtain an example of finite, not homogeneous effect algebra $E$ such that sharp elements of $E$ form a lattice, whereas $E$ is not a lattice.