A Diamond Embedding Theorem in the Quotient Structure \(\textbf{R}/{NCup}\)

Li, Wu, and Yang constructed two incomplete cuppable c.e. degrees \(\textbf{a},\textbf{b}\) such that there is no incomplete c.e. degree which cups both \(\textbf{a}, \textbf{b}\) to \(\textbf{0}'\) . This result can be interpreted as the existence of a minimal pair in \(\textbf{R}/{NCup}\) , the quotient structure of upper semilattice of computably enumerable degrees modulo the ideal of noncuppable degrees. In this paper, we will prove a stronger result by constructing \(\textbf{a}\) and \( \textbf{b}\) such that \(\textbf{a}\cup \textbf{b}=\textbf{0}'\) and no incomplete c.e. degree can cup both of them to \(\textbf{0}'\) . In other words, the diamond lattice can be embedded into \(\textbf{R}/{NCup}\) preserving 0 and 1.