A classification of hull operators in archimedean lattice-ordered groups with unit | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 6، دوره 13، شماره 1، مهر 2020، صفحه 83-104 اصل مقاله (734.86 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.29252/cgasa.13.1.83 | ||
| نویسندگان | ||
| Ricardo E. Carrera* 1؛ Anthony W. Hager2 | ||
| 1Department of Mathematics, Nova Southeastern University, 3301 College Ave., Fort Lauderdale, FL, 33314, USA. | ||
| 2Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459. | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Lattice-ordered group؛ Archimedean؛ weak unit؛ bounded monocoreflection؛ essential extension؛ hull operator؛ partially ordered semigroup | ||
| مراجع | ||
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