Title of article :
A classification of hull operators in archimedean lattice-ordered groups with unit
Carrera, Ricardo E. Department of Mathematics - Nova Southeastern University, 3301 College Ave., Fort Lauderdale, FL, 33314, USA. , Hager, Anthony W. Department of Mathematics and CS - Wesleyan University, Middletown
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.
Lattice-ordered group , Archimedean , weak unit , bounded monocoreflection , essential extension , hull operator , partially ordered semigroup
Journal title :
Categories and General Algebraic Structures with Applications