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Matrices are morphisms of a category having natural numbers as objects.

We can consider a lattice of matrices of a given order $n\times m$ (for example component-wise).

My question is about English language usage:

Can we say: the lattice of matrices meaning a lattice for arbitrary fixed $n$, $m$?

Or should we say: a lattice of matrices?

(Matrices was an arbitrary example of a category, the same consideration is for many different more interesting categories.)

Zhen Lin
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porton
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    You would use "a" if there is more than one, or the possibility of there being more than one, lattice. And use "the" if there is just one lattice, or if you are specifying one lattice in particular. – anon Aug 24 '13 at 14:19
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    Should be migrated to english.se, which is not on the above list. – Charles Aug 25 '13 at 03:28

1 Answers1

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Context dictates which article "a" or "the" is appropriate. Use of "the" implies that a uniquely defined object is being referenced (unique up to isomorphism, perhaps, as this is mathematical discussion).

In beginning you would define "a lattice" that you intend to discuss. Once the intention is clear "from context", referring to it as "the lattice" reinforces the reader's understanding that you are not changing the subject.

Where the specific lattice depends on a choice of parameters, say $m,n$ in your matrix example, continued reference to a particular such choice is not necessary unless or until you have occasion to discuss modifying that choice, e.g. to $m',n'$ where (just as illustration) some common factor had been removed from the original $m,n$.

hardmath
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