A note on joins and meets for positive linear preorders

Abstract

A positive preorder R is linear if the corresponding quotient poset is linearly ordered. Following the recent advances in the studies of positive preorders, the paper investigates the degree structure Celps of positive linear preorders under computable reducibility. We prove that if a positive linear preorder L is non-universal and the quotient poset of L is infinite, then L is a part of an infinite antichain inside Celps.

For a pair L,R from Celps, we obtain sufficient conditions for when the pair has neither join, nor meet (with respect to computable reducibility). We give an example of a pair from Celps that has a meet. Inside the substructure \Omega of Celps containing only computable linear orders of order-type \omega, we build a pair that has a join inside \Omega.