  We introduce an algebraic concept of the frame for abstract conditional
independence (CI) models, together with basic operations with respect to which
such a frame should be closed: copying and marginalization. Three standard
examples of such frames are (discrete) probabilistic CI structures,
semi-graphoids and structural semi-graphoids. We concentrate on those frames
which are closed under the operation of set-theoretical intersection because,
for these, the respective families of CI models are lattices. This allows one
to apply the results from lattice theory and formal concept analysis to
describe such families in terms of implications among CI statements.
  The central concept of this paper is that of self-adhesivity defined in
algebraic terms, which is a combinatorial reflection of the self-adhesivity
concept studied earlier in context of polymatroids and information theory. The
generalization also leads to a self-adhesivity operator defined on the
hyper-level of CI frames. We answer some of the questions related to this
approach and raise other open questions.
  The core of the paper is in computations. The combinatorial approach to
computation might overcome some memory and space limitation of software
packages based on polyhedral geometry, in particular, if SAT solvers are
utilized. We characterize some basic CI families over 4 variables in terms of
canonical implications among CI statements. We apply our method in
information-theoretical context to the task of entropic region demarcation over
5 variables.
