Joins and meets in effect algebras

TL;DR

This paper investigates the conditions under which joins and meets exist in effect algebras represented by test spaces, providing formulas and examples that highlight structural properties and limitations.

Contribution

It characterizes when joins and meets exist in effect algebras via test spaces and offers explicit formulas, including an example of a non-homogeneous effect algebra with lattice sharp elements.

Findings

Formulas for joins and meets using test space elements

Conditions for existence of joins and meets in effect algebras

Example of a finite effect algebra with lattice sharp elements but not a lattice

Abstract

We know that each effect algebra $E$ is isomorphic to $\pi(X)$ for some $E$-test spaces $(X,{\cal T})$.We describe when $\pi(x)\lor \pi(y)$ and $\pi(x)\land\pi(y)$ exists for $x,y\in{\cal E}(X,{\cal T})$. Moreover we give the formula for $\pi(x)\lor\pi(x)$ and $\pi(x)\land\pi(y)$ using only $x,y$ and tests which are elements of ${\cal T}$. 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.