definition of complete vector lattice

Question

Suppose $M$ is a von-Neumann algebra, $L=M\cap M'$ is the centre of $M$. The last line on page 29, C*-algebras and their automorphism groups, states that the self-adjoint part $L_{sa}$ of $L$ is a complete vector lattice.

Found in wekipeida, a complete lattice is a lattice in which each subset have both a supremum and a infimum.

However, even for $M=\mathbb{C}$, $L_{sa}=\mathbb{R}$ is not complete according to the above definition, as $\mathbb{R}$ itself has neither supremum or infimum.

Answer 1

Well, $(\Bbb R, \min, \max)$ is indeed not a complete lattice, but it is almost complete: $\bar{\Bbb R}:=\Bbb R\cup\{\pm\infty\} $ is complete, or put in other words,

Every bounded above subset has maximum.

which implies that every bounded below subset has minimum.

And actually this is the definition of (Dedekind) completeness for vector lattices, cf. wikipedia.