We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.
The extension of a lattice ordered group $A$ by a generalized Boolean algebra $B$ will be denoted by $A_B$. In this paper we apply subdirect decompositions of $A_B$ for dealing with a question proposed by Conrad and Darnel. Further, in the case when $A$ is linearly ordered we investigate (i) the completely subdirect decompositions of $A_B$ and those of $B$, and (ii) the values of elements of $A_B$ and the radical $R(A_B)$.