Intrinsic entropy for generalized quasimetric semilattices

Abstract

We introduce the notion of intrinsic semilattice entropy $\tilde{h}$ in the category ${\mathcal{L}}_{qm}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $\mathfrak{X}$ and functors $F:\mathfrak{X}\to {\mathcal{L}}_{qm}$ we find specific known entropies ${\tilde{h}}_{\mathfrak{X}}$ on $\mathfrak{X}$ as intrinsic functorial entropies, that is, as ${\tilde{h}}_{\mathfrak{X}}=\tilde{h}\circ F$. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for locally compact totally disconnected groups and the algebraic entropy for locally compact compactly covered abelian groups.