If element $z$ of a lattice effect algebra $(E,\oplus ,\mathbf{0},\mathbf{1})$ is central, then the interval $[\mathbf{0},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus$. It is a known fact that if the set $C\left(E\right)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C\left(E\right)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion $\widehat{E}$ of $E$ which is its extension...

In this paper we present representation of finite effect algebras by matrices. For each non-trivial finite effect algebra $E$ we construct set of matrices $M\left(E\right)$ in such a way that effect algebras $E_1$ and $E_2$ are isomorphic if and only if $M\left(E_1\right)=M\left(E_2\right)$. The paper also contains the full list of matrices representing all nontrivial finite effect algebras of cardinality at most $8$.

Lattice effect algebras generalize orthomodular lattices and $MV$-algebras. We describe all complete modular atomic effect algebras. This allows us to prove the existence of ordercontinuous subadditive states (probabilities) on them. For the separable noncomplete ones we show that the existence of a faithful probability is equivalent to the condition that their MacNeille complete modular effect algebra.