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1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeJan 11th 2016
   • PermaLink
   Author: porton
   Format: MarkdownItex[Product order](http://ncatlab.org/nlab/show/product+order) is missing (there are no such page at ncatlab.org). First, I propose to create this page. Second, it looks like that operations on product order can be described through operations on the multipliers. For example, as easy to prove, lattice-theoretic subtraction on product (indexed by an index set $N$) order is described by the formula: $a\setminus b = \lambda i\in N: a_i\setminus b_i$ (whenever every $a_i\setminus b_i$ is defined). So, it looks like that for certain (determined by the order?) partial functions $F$ we have $F(x_0,\dots x_n) = \lambda i\in N:F(x_{0,i},\dots x_{n,i})$. Now we should formulate the exact theorem. I think, it is worth your attention.

   Product order is missing (there are no such page at ncatlab.org).

   First, I propose to create this page.

   Second, it looks like that operations on product order can be described through operations on the multipliers.

   For example, as easy to prove, lattice-theoretic subtraction on product (indexed by an index set $N$) order is described by the formula:

   $a\setminus b = \lambda i\in N: a_i\setminus b_i$ (whenever every $a_i\setminus b_i$ is defined).

   So, it looks like that for certain (determined by the order?) partial functions $F$ we have

   $F(x_0,\dots x_n) = \lambda i\in N:F(x_{0,i},\dots x_{n,i})$.

   Now we should formulate the exact theorem. I think, it is worth your attention.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 11th 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIf by "product order", a term I have not heard, you mean a cartesian product in the category of "orders" (preorders? partial orders? what exactly?) then my vote is no, we don't require such a page. A basic lesson of category theory is that an omnibus concept like "product" may be formulated once and for all to apply to any category one might dream up, and you don't have to specify it individually for each of the zillions of categories out there -- it's totally unnecessary. We have also a page [[co-Heyting algebra]] where the difference operator $\setminus$ is specified by a universal property.

   If by “product order”, a term I have not heard, you mean a cartesian product in the category of “orders” (preorders? partial orders? what exactly?) then my vote is no, we don’t require such a page. A basic lesson of category theory is that an omnibus concept like “product” may be formulated once and for all to apply to any category one might dream up, and you don’t have to specify it individually for each of the zillions of categories out there – it’s totally unnecessary.

   We have also a page co-Heyting algebra where the difference operator $\setminus$ is specified by a universal property.

3. • CommentRowNumber3.
   • CommentAuthorporton
   • CommentTimeJan 11th 2016
   • PermaLink
   Author: porton
   Format: MarkdownItexLet we have a partial order on each set $A_i$ for every $i\in N$ for some index set $N$. By definition the product order (partial order) on the set $\prod_{i\in N} A$ (sometimes called "function space" of posets) is the order defined by the formula $a\le b \Leftrightarrow \forall i\in N:a_i\le b_i$. I expect that this particular order is a product in the category of partial orders, but I am not 100% sure. Even if it is a categorical product, product order has an importance by itself, by the reason that it is defined **not** up to isomorphism but exactly, a feature which cannot be done with categorical product only. Because it is explicitly defined without mentioning categories, but has nice category theory properties, it is definitely worth to create "product order" page. Maybe it is a specific case, but a very important specific case. Also note that $\setminus$ is only one of many specific examples for my vaguely formulated conjecture.

   Let we have a partial order on each set $A_i$ for every $i\in N$ for some index set $N$.

   By definition the product order (partial order) on the set $\prod_{i\in N} A$ (sometimes called “function space” of posets) is the order defined by the formula $a\le b \Leftrightarrow \forall i\in N:a_i\le b_i$.

   I expect that this particular order is a product in the category of partial orders, but I am not 100% sure.

   Even if it is a categorical product, product order has an importance by itself, by the reason that it is defined not up to isomorphism but exactly, a feature which cannot be done with categorical product only.

   Because it is explicitly defined without mentioning categories, but has nice category theory properties, it is definitely worth to create “product order” page. Maybe it is a specific case, but a very important specific case.

   Also note that $\setminus$ is only one of many specific examples for my vaguely formulated conjecture.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 11th 2016
   • (edited Jan 11th 2016)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThat is the product in the category of partial orders, and such is unique up to unique isomorphism, as is well-known. Of course one can define (in a wide variety of foundational set-ups) an exact specification, but another basic lesson of category theory is that that such exact specifications are unnecessary. If other well-established nLab authors feel such a page is really needed, then I guess I won't stand in the way. But where would one stop with such an undertaking. Do we need "product abelian group", "product Banach space", "product modular lattice", etc. etc. (Plus, the phrase "product ___" is not good English, as these examples should indicate.)

   That is the product in the category of partial orders, and such is unique up to unique isomorphism, as is well-known. Of course one can define (in a wide variety of foundational set-ups) an exact specification, but another basic lesson of category theory is that that such exact specifications are unnecessary.

   If other well-established nLab authors feel such a page is really needed, then I guess I won’t stand in the way. But where would one stop with such an undertaking. Do we need “product abelian group”, “product Banach space”, “product modular lattice”, etc. etc. (Plus, the phrase “product ___” is not good English, as these examples should indicate.)

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 11th 2016
   • (edited Jan 11th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe common practice for recording "product-X", in the above sense, is to have on the page for "X" a section "Properties" and to record there the statement "The Cartesian product in the category of Xs is given by....".

   The common practice for recording “product-X”, in the above sense, is to have on the page for “X” a section “Properties” and to record there the statement “The Cartesian product in the category of Xs is given by….”.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 11th 2016
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexWe don't have pages for each of the natural numbers, like Wikipedia... What are we going to do!?! `</sarcasm>`

   We don’t have pages for each of the natural numbers, like Wikipedia… What are we going to do!?! </sarcasm>

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 11th 2016
   • (edited Jan 11th 2016)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRe #5: that would be more reasonable and more in line with usual practice here, I agree, but again I think the explicit description of the product is a bit on the trivial side and I certainly don't feel a strong need to give it. For any essentially algebraic theory, for example the theory of partial orders, the product of models is just the obvious thing.

   Re #5: that would be more reasonable and more in line with usual practice here, I agree, but again I think the explicit description of the product is a bit on the trivial side and I certainly don’t feel a strong need to give it. For any essentially algebraic theory, for example the theory of partial orders, the product of models is just the obvious thing.

8. • CommentRowNumber8.
   • CommentAuthorporton
   • CommentTimeJan 12th 2016
   • (edited Jan 13th 2016)
   • PermaLink
   Author: porton
   Format: MarkdownItexSee also [blog post](https://portonmath.wordpress.com/2016/01/12/a-conjecture-about-product-order-and-logic/) where I formulated the idea about $F(x_0,\dots x_n) = \lambda i\in N:F(x_{0,i},\dots x_{n,i})$. The idea was with an error, but below in a comment to the blog post I propose a way to correct this reasoning.

   See also blog post where I formulated the idea about $F(x_0,\dots x_n) = \lambda i\in N:F(x_{0,i},\dots x_{n,i})$.

   The idea was with an error, but below in a comment to the blog post I propose a way to correct this reasoning.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 12th 2016
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexSo why did you link to a) a deleted MO question and b) a posting full of erroneous material? If you discovered this after, the polite thing to do is delete your links and your comment. People do that here if you'd been paying attention. Some might consider unnecessarily linking to one's own page from here as being PageRank farming to some extent.

   So why did you link to a) a deleted MO question and b) a posting full of erroneous material? If you discovered this after, the polite thing to do is delete your links and your comment. People do that here if you’d been paying attention. Some might consider unnecessarily linking to one’s own page from here as being PageRank farming to some extent.

10. • CommentRowNumber10.
    • CommentAuthorporton
    • CommentTimeJan 13th 2016
    • PermaLink
    Author: porton
    Format: MarkdownItex@DavidRoberts Yes, I posted this before I found an error. I don't know how to delete a comment. The blog post contains not only the erroneous idea but also an idea how to correct it. I will edit the comment.

    @DavidRoberts Yes, I posted this before I found an error.

    I don’t know how to delete a comment.

    The blog post contains not only the erroneous idea but also an idea how to correct it.

    I will edit the comment.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 13th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexFor reference, you edit the comment, delete all the text and write something like 'never mind' or 'deleted'. If you find yourself doing this a lot, it's time to reassess your approach to being here. It's tiring to keep reading things that don't turn out or are straightforward observations from well-known results in category theory, etc etc. It's signal-to-noise ratio, and I'm afraid apart from concerted effort in the past on behalf of a few here, no one is getting much signal from what you are posting here about. I'm afraid it's not winning you friends.

    For reference, you edit the comment, delete all the text and write something like ’never mind’ or ’deleted’. If you find yourself doing this a lot, it’s time to reassess your approach to being here. It’s tiring to keep reading things that don’t turn out or are straightforward observations from well-known results in category theory, etc etc. It’s signal-to-noise ratio, and I’m afraid apart from concerted effort in the past on behalf of a few here, no one is getting much signal from what you are posting here about. I’m afraid it’s not winning you friends.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 13th 2016
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexAs best as I can make out, the question is about the structure of products in certain types of categories, and particularly in categories of structures given by sets and operations (possibly partially defined) satisfying specified equations, where one wants to know whether the operations are indeed defined componentwise, as in the example of comment #1. And as I suggested, the answer is yes if the structures are models of an [[essentially algebraic theory]], i.e., form a [[locally finitely presentable category]]. Or even if they form a [[locally presentable category]], a more general concept. Not just products, but arbitrary limits, are constructed "pointwise", and their operations (totally or partially defined) are correspondingly prescribed in pointwise fashion. This has been known for I don't know how long. Longer than some of us here have been alive. So to mirror comment #1, I think it would be "worth VP's attention" to look this stuff up and learn something about it; their is a wealth of material already in the nLab and in references recorded there. (I don't think it's worth *my* time to go into this any further, since it's well known, and since my past attempts to explain some basic points of category theory carefully and at length have been met with responses like "no particular results of your proofs are of any specific interest to me", which taught me a lesson I won't soon forget.)

    As best as I can make out, the question is about the structure of products in certain types of categories, and particularly in categories of structures given by sets and operations (possibly partially defined) satisfying specified equations, where one wants to know whether the operations are indeed defined componentwise, as in the example of comment #1.

    And as I suggested, the answer is yes if the structures are models of an essentially algebraic theory, i.e., form a locally finitely presentable category. Or even if they form a locally presentable category, a more general concept. Not just products, but arbitrary limits, are constructed “pointwise”, and their operations (totally or partially defined) are correspondingly prescribed in pointwise fashion. This has been known for I don’t know how long. Longer than some of us here have been alive.

    So to mirror comment #1, I think it would be “worth VP’s attention” to look this stuff up and learn something about it; their is a wealth of material already in the nLab and in references recorded there. (I don’t think it’s worth my time to go into this any further, since it’s well known, and since my past attempts to explain some basic points of category theory carefully and at length have been met with responses like “no particular results of your proofs are of any specific interest to me”, which taught me a lesson I won’t soon forget.)

13. • CommentRowNumber13.
    • CommentAuthorporton
    • CommentTimeJan 13th 2016
    • PermaLink
    Author: porton
    Format: MarkdownItex@Todd_Trimble Dear Todd, You have certainly answered my question about partial functions defined by equations. (However I do not see how to prove it.) However, the answer is more complex that I expected. Even despite my current research does not use any of this, I am interested to learn it further. I've read (or, well, skimmed) some category theory books, including McLane and Awodey. However, it seems for me that what you suggest me now is far outside of my knowledge. What can you suggest to read to increase my CT knowledge? (Well, first I need to re-read MacLane as my brain does not contain the entire book.)

    @Todd_Trimble Dear Todd,

    You have certainly answered my question about partial functions defined by equations. (However I do not see how to prove it.) However, the answer is more complex that I expected.

    Even despite my current research does not use any of this, I am interested to learn it further.

    I’ve read (or, well, skimmed) some category theory books, including McLane and Awodey. However, it seems for me that what you suggest me now is far outside of my knowledge. What can you suggest to read to increase my CT knowledge? (Well, first I need to re-read MacLane as my brain does not contain the entire book.)

14. • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 13th 2016
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexIt is neither complex nor difficult, although that doesn't mean it doesn't take time to digest. In any case, I'm convinced by now I'm the wrong person to explain this to you, and so I won't. (Plus, there is no royal road to acquiring the insights of category theory.) Off hand I might suggest not shooting for maximal generality at first, but learning about the important special case of general algebraic theories and the connection with monads, say by reading Algebraic Theories by Manes, and seeing that limits of algebras of monads are created at the underlying set level (this is also in Mac Lane); the case of "essentially algebraic theories" is not too much of a conceptual leap from there. Possibly read the article [[algebraic theories]] in the nLab, but I'm not sure how easy or polished it is. I think the book by Adamek and Rosicky (Locally Presentable and Accessible Categories) is pretty good for the leap, although I can't promise it would be super-accessible to relative beginners.

    It is neither complex nor difficult, although that doesn’t mean it doesn’t take time to digest.

    In any case, I’m convinced by now I’m the wrong person to explain this to you, and so I won’t. (Plus, there is no royal road to acquiring the insights of category theory.) Off hand I might suggest not shooting for maximal generality at first, but learning about the important special case of general algebraic theories and the connection with monads, say by reading Algebraic Theories by Manes, and seeing that limits of algebras of monads are created at the underlying set level (this is also in Mac Lane); the case of “essentially algebraic theories” is not too much of a conceptual leap from there. Possibly read the article algebraic theories in the nLab, but I’m not sure how easy or polished it is. I think the book by Adamek and Rosicky (Locally Presentable and Accessible Categories) is pretty good for the leap, although I can’t promise it would be super-accessible to relative beginners.

15. • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 13th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex> I can’t promise it would be super-accessible to relative beginners. ;-)

    I can’t promise it would be super-accessible to relative beginners.

    ;-)