Categories and General Algebraic Structures with Applications
Lattice-ordered group, Archimedean, Weak unit, Bounded monocoreflection, Essential extension, Hull operator, Partially ordered semigroup
The category, or class of algebras, in the title is denoted by W. A hull operator (ho) in W is a reflection in the category consisting of W objects with only essential embeddings as morphisms. The proper class of all of these is hoW. The bounded monocoreflection in W is denoted B. We classify the ho’s by their interaction with B as follows. A “word” is a function w : hoW → WW obtained as a finite composition of B and x a variable ranging in hoW. The set of these,“Word”, is in a natural way a partially ordered semigroup of size 6, order isomorphic to F(2), the free 0 −1 distributive lattice on 2 generators. Then, 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 (≈ F(2)). Of the 6: 1 is of size ≥ 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.
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Carrera, Ricardo Enrique and Hager, Anthony W., "A Classification of Hull Operators in Archimedean Lattice-Ordered Groups With Unit" (2020). Mathematics Faculty Articles. 299.