  The theory of limits of discrete combinatorial objects has been thriving for
the last decade or so. The syntactic, algebraic approach to the subject is
popularly known as "flag algebras", while the semantic, geometric one is often
associated with the name ``graph limits''. The language of graph limits is
generally more intuitive and expressible, but a price that one has to pay for
it is that it is better suited for the case of ordinary graphs than for more
general combinatorial objects. Accordingly, there have been several attempts in
the literature, of varying degree of generality, to define limit objects for
more complicated combinatorial structures.
  This paper is another attempt at a workable general theory of dense limit
objects. Unlike previous efforts in this direction (with notable exception of
[Ashwini Aroskar and James Cummings. Limits, regularity and removal for finite
structures. Technical Report arXiv:1412.2014 [math.LO], arXiv e-print, 2014.]),
we base our account on the same concepts from the first-order logic and the
model theory as in the theory of flag algebras.
  We show how our definition naturally encompasses a host of previously
considered cases (graphons, hypergraphons, digraphons, permutons, posetons,
colored graphs, etc.), and we extend the fundamental properties of existence
and uniqueness to this more general case. We also give an intuitive general
proof of the continuous version of the Induced Removal Lemma based on the
completeness theorem for propositional calculus. We capitalize on the notion of
an open interpretation that often allows to transfer methods and results from
one situation to another. Again, we show that some previous arguments can be
quite naturally framed using this language.
