On a representation of t-norms on bounded meet semilattices

We investigate how and when a t-norm on a bounded meet semilattice can be represented by a t-norm on some sub-semilattice and a collection of interior operators. We present such a representation using a novel notion of consistent triplet consisting of a t-norm, a commutative semigroup of interior operators, and a meet semilattice of symmetric lower sets. Furthermore, we study a related problem of extension of a t-norm from a sub-semilattice to the whole semilattice given a consistent triplet. This gives us a new construction method for t-norms on bounded semilattices that covers several known methods. Our results are illustrated by several examples.