TY - JOUR
ID - 87552
TI - A classification of hull operators in archimedean lattice-ordered groups with unit
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Carrera, Ricardo E.
AU - Hager, Anthony W.
AD - Department of Mathematics, Nova Southeastern University, 3301 College Ave., Fort Lauderdale, FL, 33314, USA.
AD - Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
Y1 - 2020
PY - 2020
VL - 13
IS - 1
SP - 83
EP - 104
KW - lattice-ordered group
KW - Archimedean
KW - weak unit
KW - bounded monocoreflection
KW - essential extension
KW - hull operator
KW - partially ordered semigroup
DO - 10.29252/cgasa.13.1.83
N2 - 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.
UR - https://cgasa.sbu.ac.ir/article_87552.html
L1 - https://cgasa.sbu.ac.ir/article_87552_9aa6961ac859a3c87241c8124af70410.pdf
ER -