Closure operators and their middle-interchange law

摘要

The paper presents a new definition of closure operator which encompasses the standard Dikranjan-Giuli notion, as well as the Bourn-Gran notion of normal closure operator. As is well known, any two closure operators C, D in a category may be composed in, within order, two different ways. For a subobject M→X one may consider D_x(C_xM) or D_(cx)(M)(M) as the value at M of a new closure operator D · C or D * C, respectively. The two binary operations are linked by a lax middle-interchange law. This paper explores situations in which the law holds strictly.