Let \(X\) be a set with an arbitrary binary relation (not necessarily a partial order). Then for \(A\subset X\) the authors define \(lb(A)=\{x\in X:\forall \in A:x\leq a\}\); \(ub(A)=\{x\in X:\forall a\in A:a\leq x\}\); \(\min (A)=A\cap lb(A)\); \(\max (A)=A\cap ub(A)\); \(\inf (A)=\max (lb(A))\); \(\sup (A)=\min ub(A)\). Further, \(X\) is inf-complete if \(\inf (A)\neq \emptyset \) for all \(A\subset X\); \(X\) is quasi-inf-complete if \(\inf (A)\neq \emptyset \) for all \(A\subset X\) with \(A\neq \emptyset \); \(X\) is pseudo-inf-complete if \(\inf (A)\neq \emptyset \) for all \(A\subset X\) with \( lb(A)\neq \emptyset \); \(X\) is semi-inf-complete if \(\inf (A)\neq \emptyset \) for all \(A\subset X\) with \(A\neq \emptyset \) and \(lb(A)\neq \emptyset \). And dually for sup instead of inf; and complete = inf-complete plus sup-complete. Then a lot of equivalences are established: 3.3 \(X\) is quasi-inf-complete and \(X\neq \emptyset \) \(\Leftrightarrow X\) is semi-inf-complete and \(lb(X)\neq \emptyset \). 3.4 \(X\) is pseudo-inf-complete and \(X\neq \emptyset \Leftrightarrow X\) is semi-inf-complete and \(ub(X)\neq \emptyset \). 3.5 The following are equivalent: 1) \(X\) is inf-complete. 2) \(X\) is quasi-inf-complete and \( ub(X)\neq \emptyset \). 3) \(X\) is pseudo-inf-complete and \(lb(X)\neq \emptyset \). 4) \(X\) is semi-inf-complete and \(lb(X)\neq \emptyset \) and \(ub(X)\neq \emptyset \). Section 4 contains relationships between infimum and supremum completeness: 4.1 \(X\) is inf-complete \(\Leftrightarrow \) \(X\) is sup-complete. 4.3 \(X\) is quasi-inf-complete \( \Leftrightarrow X\) is pseudo-sup-complete. 4.4 \(X\) is quasi-sup-complete \(\Leftrightarrow X \) is pseudo-inf-complete. With the corollary: \(X \) is quasi-complete \(\Leftrightarrow X\) is pseudo-complete. 4.6 \(X\) is semi-inf-complete \(\Leftrightarrow \) \(X\) is semi-sup-complete. 4.7 Corollary: \(X\) is semi-complete \( \Leftrightarrow X\) is semi-inf-complete or semi-sup-complete.