A classification of hull operators in archimedean lattice-ordered groups with unit

Document Type : Research Paper

Authors

1 Department of Mathematics, Nova Southeastern University, 3301 College Ave., Fort Lauderdale, FL, 33314, USA.

2 Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.

10.29252/cgasa.13.1.83

Abstract

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.

Keywords

References

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Categories and General Algebraic Structures with Applications
Volume 13, Issue 1
July 2020
Pages 83-104

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