Operations on Closure Operators
Despite the powerful continuity condition, the notion of closure operator is very general. It is therefore important to provide tools for improving a given operator. Fortunately, there is a natural lattice structure for closure operators that allows us to distinguish between properties stable under meet (idempotency, hereditariness, productivity), and those stable under join (weak hereditariness, minimality, additivity) . Hence it is clear that each closure operator has an idempotent hull and a weakly hereditary core, and analogously for the other properties. The passage to the hull w.r.t. to a meet-stable property will normally not destroy already existing join-stable properties, and the passage to the core w.r.t. to a join-stable property will normally preserve meet-stable properties.
KeywordsClosure Operator Finite Product Minimal Core Additive Core Dosure Operator
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- The constructions for both the idempotent hull and the weakly hereditary core of a closure operator via infinite (co-)iterations can be found in Dikranjan and Giuli [1987a] who, however, do not formally introduce the binary (co-)composition of closure operators. Additive cores appear in the context of topological categories in Dikranjan . The categorical constructions for the additive core, the fully additive core and the hereditary hull of a closure operator do not seem to have been published previously, and the same is true for the general construction of the indiscrete operator and for the sufficient criteria for (finite) productivity given in 4.11.Google Scholar