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Residuated Park Theories (1102.1139v1)
Published 6 Feb 2011 in cs.LO and math.LO
Abstract: When $L$ is a complete lattice, the collection $\Mon_L$ of all monotone functions $Lp \to Ln$, $n,p \geq 0$, forms a Lawvere theory. We enrich this Lawvere theory with the binary supremum operation $\vee$, an operation of (left) residuation $\res$ and the parameterized least fixed point operation $\dagger$. We exhibit a system of \emph{equational} axioms which is sound and proves all valid equations of the theories $\Mon_L$ involving only the theory operations, $\vee$ and $\dagger$, i.e., all valid equations not involving residuation. We also present an alternative axiomatization, where $\dagger$ is replaced by a star operation, and provide an application to regular tree languages.