Some Modifications of Hull Operators in Archimedean Lattice-Ordered Groups with Weak Unit

• Published in: Applied categorical structures 2023-04, Vol.31 (2), Article 12
• Main Authors: Carrera, Ricardo E., Hager, Anthony W.
• Format: Article
• Language: English
• Subjects: Convex and Discrete Geometry; Geometry; Mathematical analysis; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Operators (mathematics); Theory of Computation
• ISSN: 0927-2852; 1572-9095
• DOI: 10.1007/s10485-023-09710-7

W denotes the category, or class of algebras, in the title. A hull operator (ho) in W is a function ho W ⟶ h W which can be called an essential closure operator. The family of these, denoted ho W , is a proper class and a complete lattice in the ordering as functions “pointwise", with the bottom Id W and top Conrad’s essential completion e . Other much studied hull operators are the divisible hull, maximum essential reflection, projectable hull, and Dedekind completion. This paper is the authors’ latest efforts to understand/create structure in ho W through the nature of the interaction that an h might have with B , the bounded monocoreflection in W (e.g., Bh=hB). We define and investigate three functions ho W ⟶ ho W which stand in the relation Id W ≤ α ¯ ( h ) ≤ λ ¯ ( h ) ≤ c ¯ ( h ) ≤ h . General properties that an h might have, and particular choices of h , show various assignments of < and = in this chain.