A classification of hull operators in archimedean lattice-ordered groups with unit
Ricardo Enrique Carrera, Anthony W. Hager · Categories and General Algebraic Structures with Application · 2020
The category, or class of algebras, in the title is denoted by $bf W$. A hull operator (ho) in $bf W$ is a reflection in the category consisting of $bf W$ objects with only essential embeddings as morphisms. The proper class of all of these is $bf hoW$. The bounded monocoreflection in $bf W$ is denoted $B$. We classify the ho's by their interaction with $B$ as follows. A ``word'' is a function $w: {bf hoW} longrightarrow {bf W}^{bf W}$ obtained as a finite composition of $B$ and $x$ a variable ranging in $bf hoW$. The set of these,``Word'', is in a natural way a partially ordered semigroup of size $6$, order isomorphic to ${rm F}(2)$, the free $0-1$ distributive lattice on $2$ generators. Then, $bf hoW$ is partitioned into $6$ disjoint pieces, by equations and inequations in words, and each piece is represented by a characteristic order-preserving quotient of Word ($approx {rm F}(2)$). Of the $6$: $1$ is of size $geq 2$, $1$ is at least infinite, $2$ are each proper classes, and of these $4$, all quotients are chains; another $1$ is a proper class with unknown quotients; the remaining $1$ is not known to be nonempty and its quotients would not be chains.