  We initiate the systematic study of loop conditions of arbitrary finite
width. Each loop condition is a finite set of identities of a particular shape,
and satisfaction of these identities in an algebra is characterized by it
forcing a constant tuple into certain invariant relations on powers of the
algebra.
  By showing the equivalence of various loop conditions, we are able to provide
a new and short proof of the recent celebrated result stating the existence of
a weakest non-trivial idempotent strong Mal'cev condition.
  We then consider pseudo-loop conditions, a modification suitable for
oligomorphic algebras, and show the equivalence of various pseudo-loop
conditions within this context. This allows us to provide a new and short proof
of the fact that the satisfaction of non-trivial identities of height 1 in a
closed oligomorphic core implies the satisfaction of a fixed single identity.
