A generalized $MV$-algebra $\mathcal A$ is called representable if it is a subdirect product of linearly ordered generalized $MV$-algebras. Let $S$ be the system of all congruence relations $\rho $ on $\mathcal A$ such that the quotient algebra $\mathcal A/\rho $ is representable. In the present paper we prove that the system $S$ has a least element.
Let $\mathcal A$ and $\mathcal B$ be graph algebras. In this paper we present the notion of an ideal in a graph algebra and prove that an ideal extension of $\mathcal A$ by $\mathcal B$ always exists. We describe (up to isomorphism) all such extensions.
Some results concerning congruence relations on partially ordered quasigroups (especially, Riesz quasigroups) and ideals of partially ordered loops are presented. These results generalize the assertions which were proved by Fuchs in [5] for partially ordered groups and Riesz groups.
In the paper the notion of an ideal of a lattice ordered monoid A is introduced and relations between ideals of A and congruence relations on A are investigated. Further, it is shown that the set of all ideals of a soft lattice ordered monoid or a negatively ordered monoid partially ordered by inclusion is an algebraic Brouwerian lattice.