Pseudo-loop conditions

About a decade ago, it was realised that the satisfaction of a given identity (or equation) of the form $$ in an algebra is equivalent to the algebra forcing a loop into any graph on which it acts and which contains a certain finite subgraph associated with the identity. Such identities have since also been called loop conditions, and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a Siggers identity $$ in any arbitrary non-trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form $$ , which we call loop conditions of width $$ . We show that their satisfaction in an algebra is equivalent to any action of the algebra on a certain type of relation forcing a constant tuple into the relation. Proving that for each fixed width $$ there is a weakest loop condition (that is, one entailed by all others), we obtain a new and short proof of the recent celebrated result stating that there exists a concrete loop condition of width 3 which is entailed in any non-trivial idempotent, possibly infinite, algebra. The framework of classical (width 2) loop conditions is insufficient for such proof. We then consider pseudo-loop conditions of finite width, a generalisation suitable for non-idempotent algebras; they are of the form $$ , and of central importance for the structure of algebras associated with $$ -categorical structures. We show that for the latter, satisfaction of a pseudo-loop condition is characterised by pseudo-loops, that is, loops modulo the action of the automorphism group, and that a weakest pseudo-loop condition exists (for $$ -categorical cores). This way we obtain a new and short proof of the theorem that the satisfaction of any non-trivial identities of height 1 in such algebras implies the satisfaction of a fixed single identity.