For an abelian lattice ordered group $G$ let $G$ be the system of all compatible convergences on $G$; this system is a meet semilattice but in general it fails to be a lattice. Let ${\alpha }_{nd}$ be the convergence on $G$ which is generated by the set of all nearly disjoint sequences in $G$, and let $\alpha$ be any element of $G$. In the present paper we prove that the join ${\alpha }_{nd}\vee \alpha$ does exist in $G$.

We consider algebras determined by all normal identities of $MV$-algebras, i.e. algebras of many-valued logics. For such algebras, we present a representation based on a normalization of a sectionally involutioned lattice, i.e. a $q$-lattice, and another one based on a normalization of a lattice-ordered group.