We construct a high c.e. degree which is not the join of two minimal degrees and so refute Posner's conjecture that every high c.e. degree is the join of two minimal degrees. Additionally, the proof shows that there is a high c.e. degree a such that for any splitting of a into degrees b and c one of these degrees bounds a 1-generic degree.
"A high c.e. degree which is not the join of two minimal degrees." J. Symbolic Logic 75 (4) 1339 - 1358, December 2010. https://doi.org/10.2178/jsl/1286198150