The density of infima in the recursively enumerable degrees (Q809074)

The author shows that, for any recursively enumerable degrees \(\underset{\tilde{}} a\), \(\underset{\tilde{}} e\) with \(\underset{\tilde{}} e<\underset{\tilde{}} a\), there exist incomparable recursively enumerable degrees \(\underset{\tilde{}} b\), \(\underset{\tilde{}} c\) with infimum \(\underset{\tilde{}} d\) such that \(\underset{\tilde{}} e<\underset{\tilde{}} d,\underset{\tilde{}} b,\underset{\tilde{}} c<\underset{\tilde{}} a\). Then, combining \textit{K. Ambos-Spies}' result of ``On pairs of recursively enumerable degrees'' [Trans. Am. Math. Soc. 283, 507-531 (1984; Zbl 0541.03023)], any nontrivial interval in the recursively enumerable degrees contains both incomparable pairs with and without infima in recursively enumerable degrees.