Pseudo-loop conditions

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.