What does it mean for a family to be "maximal with respect to"... certain condition? I'm reading this book by Blackadar: "Operator algebras, theory of $C^*$-algebra and von Neumann algebras" (link: http://wolfweb.unr.edu/homepage/bruceb/Cycr.pdf), and I ran into this term: "families that are maximal with respect to..." certain condition, that I couldn't quite figure out, here's a couple of examples where it is used:
On page 246:

... if $\{ p_i \}$ is a family of abelian projections which is maximal
  with respect to the condition that the central carriers $z_{p_i}$ are
  pairwise orthogonal...

And on page 273:

... let $\{ \phi_i: i \in \Omega \}$ be a family of normal linear
  functionals which is maximal with respect to the condition that the
  supports $p_i$ are mutually orthogonal...



*

*I was kind of confused, in general, What do they mean when some family is "maximal with respect to..."?


:)

It means the following:

Let $(S,<)$ be an ordered set. An element $s_m\in S$ is maximal if
  $\nexists s\in S:s_m<s$.

Both your cases, the ordering is given by inclusion of sets.
