  We provide a general framework for no-arbitrage concepts in topological
vector lattices, which covers many of the well-known no-arbitrage concepts as
particular cases. The main structural condition we impose is that the outcomes
of trading strategies with initial wealth zero and those with positive initial
wealth have the structure of a convex cone. As one consequence of our approach,
the concepts NUPBR, NAA$_1$ and NA$_1$ may fail to be equivalent in our general
setting. Furthermore, we derive abstract versions of the fundamental theorem of
asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and
investigate when the FTAP is warranted in its classical form with a separating
measure. We also consider a financial market with semimartingales which does
not need to have a num\'{e}raire, and derive results which show the links
between the no-arbitrage concepts by only using the theory of topological
vector lattices and well-known results from stochastic analysis in a sequence
of short proofs.
