%0 Journal Article
%T A classification of hull operators in archimedean lattice-ordered groups with unit
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Carrera, Ricardo E.
%A Hager, Anthony W.
%D 2020
%\ 07/01/2020
%V 13
%N 1
%P 83-104
%! A classification of hull operators in archimedean lattice-ordered groups with unit
%K lattice-ordered group
%K Archimedean
%K weak unit
%K bounded monocoreflection
%K essential extension
%K hull operator
%K partially ordered semigroup
%R 10.29252/cgasa.13.1.83
%X 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.
%U https://cgasa.sbu.ac.ir/article_87552_9aa6961ac859a3c87241c8124af70410.pdf