Joins and meets in the structure of ceers

The theory of ceers computes true arithmetic

“…The classes I, Light, and Dark. We recall the following partition of ceers, introduced and studied in [3]. Let R be a ceer:…”

“…These classes partition the ceers, and give rise to a corresponding partition of the degrees of ceers into three classes of degrees (still denoted by I, Light, Dark): I is an initial segment of Ceers having order type ω. In Ceers (in the language of posets), the degree of Id, and thus each of these three classes are first order definable [3]. Ceers, Light and Dark are neither upper nor lower semilattices: in this regard, the most spectacular case is provided by dark degrees, as no pair of incomparable dark degrees has either meet or join in Ceers or in Dark.…”

“…The first item is essentially [3, Lemma 2.1]. The second item comes from Lemma 2.8 in [3]; the third item follows from the previous one and the fact that under the assumptions the inclusion reduction RaeW ď R misses exactly k equivalence classes.…”

“…Proof. In this proof we use that in Ceers {I every non-universal element has infinitely many distinct self-full strong minimal covers, see [3,Theorem 7.9]. For each pair ta i , b i u P F , we let c i be a self-full I-strongly minimal cover of a i ' b i .…”

“…More explicitly oriented toward a degree-theoretic approach are the papers on ceers by Gao and Gerdes [15], Andrews et al [1], [2], and finally Andrews and Sorbi [3]: the last paper provides a thorough investigation of the structure Ceers, with emphasis on existence and non-existence of meets and joins, minimal covers, definable classes of degrees, and automorphisms.…”