A preorder $ R $ is linearwhenever the corresponding quotient poset is linearly ordered.This article discusses computable reducibility on binary relations.We study the degree structure Celpsof computably enumerable linear preordersunder computable reducibility.Concatenation yields the ordered sum of two given linear preorders.We show thatthe elementary theory of Celps with concatenationis recursively isomorphic to first-order arithmetic.We also show thatthe theory of all countable linear preorders(under computable reducibility)with concatenationis recursively isomorphic to second-order arithmetic.
