Add the category of SemiRings with an example : NonNegativeIntegers()

[sagetrac-nborie]

All is in the title, we want :

sage: Semirings()
sage: Category of semirings
sage: NN = NonNegativeIntegerSemiring()
sage: NN.category()
Join of Category of semirings and Category of infinite enumerated sets

CC: @sagetrac-sage-combinat @nthiery

Component: categories

Keywords: semiring

Author: Nicolas Borie

Reviewer: Nicolas M. Thiéry

Merged: sage-4.4.4.alpha1

Issue created by migration from https://trac.sagemath.org/ticket/9056

[sagetrac-nborie]

Description changed:

--- 
+++ 
@@ -3,7 +3,7 @@
 ```
 sage: SemiRings()
 sage: Category of semi rings
-sage: NN = NonNegativeIntegers()
+sage: NN = NonNegativeIntegersSemiRing()
 sage: NN.category()
-Category of semi rings
+Join of Category of semirings and Category of infinite enumerated sets
 ```

[sagetrac-nborie]

Description changed:

--- 
+++ 
@@ -1,9 +1,9 @@
 All is in the title, we want :
 
 ```
-sage: SemiRings()
-sage: Category of semi rings
-sage: NN = NonNegativeIntegersSemiRing()
+sage: Semirings()
+sage: Category of semirings
+sage: NN = NonNegativeIntegersSemiring()
 sage: NN.category()
 Join of Category of semirings and Category of infinite enumerated sets
 ```

[sagetrac-nborie]

Attachment: trac_9056_semirings_category-nb.2.patch.gz

[nthiery]

Author: Nicolas Borie

[nthiery]

Reviewer: Nicolas M. Thiéry

[nthiery]

comment:4

I reviewed this patch while Nicolas was writting it. I'll run the tests shortly, and report.

Florent: please have a look to see if you agree with the concept, and if yes set a positive review.

[nthiery]

comment:5

Hi Nicolas B.,

There are a couple failures in the category tests (due to the new test and new category); please fix them. Note that I made a small change to your patch, so you should make sure to first pull.

You might get a conflict with the primer improvement; in that case, move your patch just after it.