AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient setω/E(or equivalently, the relationE)realizesa linearly ordered setLif there exists a c.e. relation ⊴ respectingEsuch that the induced structure (ω/E; ⊴) is isomorphic toL. Thus, one can consider the class of all linearly ordered sets that are realized byω/E; formally,K(E)={L∣theorder−typeLisrealizedbyE}. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order≤loon the class of all c.e. equivalence relations:E1≤loE2if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order≤lo. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.</jats:p