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### Refinement

A refinement of a cover is a cover such that every element is a subset of an element .

### Proper cover

Proper covers are defined as covers of a set which do not contain the entire set itself as a subset (Macula 1994). Of the five covers of , namely , , , , and , only does not contain the subset and so is the unique proper cover of two elements. In general, the number of proper covers for a set of elements is(1)(2)the first few of which are 0, 1, 45, 15913, 1073579193, ... (OEIS A007537).

### Minimal cover

A minimal cover is a cover for which removal of any single member destroys the covering property. For example, of the five covers of , namely , , , , and , only and are minimal covers.Similarly, the minimal covers of are given by , , , , , , , and . The numbers of minimal covers of members for , 2, ..., are 1, 2, 8, 49, 462, 6424, 129425, ... (OEIS A046165).Let be the number of minimal covers of with members. Thenwhere is a binomial coefficient, is a Stirling number of the second kind, andSpecial cases include and . The table below gives the a triangle of (OEIS A035348).SloaneA000392A003468A016111A046166A046167A057668112113161412522151903056516130134102540171171966336217735017066420181302530538220229511298346100814988

### Cover

A family of nonempty subsets of whose union contains the given set (and which contains no duplicated subsets) is called a cover (or covering) of . For example, there is only a single cover of , namely . However, there are five covers of , namely , , , , and .A minimal cover is a cover for which removal of one member destroys the covering property. For example, of the five covers of , only and are minimal covers. There are various other types of specialized covers, including proper covers, antichain covers, -covers, and -covers (Macula 1994).The number of possible covers for a set of elements arethe first few of which are 1, 5, 109, 32297, 2147321017, 9223372023970362989, ...(OEIS A003465).

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