Algebra Seminar talk
2023-11-10
Johanna Brunar
Identities in finite Taylor algebras
Abstract:
A finite idempotent algebra is said to be Taylor if it satisfies a set of nontrivial identities. Here, an algebra is called idempotent if the identity f(x, ..., x)≈x is witnessed by all of its term operations f. By a strong theorem (Maróti and McKenzie, 2008), a finite idempotent algebra is Taylor if and only if it possesses a weak near-unanimity (WNU) term operation, i.e. a term operation w witnessing the identities w(x, ..., x,y)≈ w(x, ... y, x)≈...≈ w(y, x, ..., x).
The theory of loop conditions provides a method of showing the validity of identities in an algebra via the existence of a constant tuple in an associated relation. From this point of origin, we initiate the systematic study of k-WNU term operations, which generalise the notion of WNU term operations. The relation associated with the identities defining a k-WNU term operation of arity n has the symmetry property of being (n,k)-symmetric. We intend to find sufficient conditions on n that guarantee the existence of a constant tuple in every nonempty (n,k)-symmetric invariant relation defined on a finite idempotent Taylor algebra, and that thus imply the existence of a k-WNU term operation of arity n.