Today, we’ll examine the properties of the multiset intersection that we defined yesterday.

Suppose is a family of multisets over . Then the following relationships hold:

(i) .

(ii) for all .

(iii) .

(iv) .

The proof of part (i) is a straightforward series of equivalencies:

Part (ii) follows directly from the definition of the infimum of a set, since for all and .

Recalling that the cardinal sum is a monotonic nondecreasing operation, we see that

for all , which proves part (iii).

Part (iv) requires only slightly more work. To begin with, consider a family of cardinals. If there exists some such that , then . If, however, no such exists, then

for all .

(This relies on the axiom of choice, which we have been assuming from the outset.) In other words, the product of any family of nonzero cardinals is monotonic nondecreasing.

Let . If there exists some such that , then the multiplicity of in the intersection would be 0, contradicting the fact that is a member of the intersection. Since for all , we have

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