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for differentiation of a function. The context in such a course of study involves emphasis on functions rather than their arguments, giving the notation f(x), putting the function f in front of the argument x. Then composition with another function g is written g(f(x)), beginning a backwards
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The unicode notation is left untranslated on my computer, even though I have
Unicode Arial which works well most of the time. I suggest that someone with knowledge of this add an explanation about where to get the font that would render this symbol. Better yet, why not just refer to it as
617:, use row vectors instead. The input of a row vector to the left of a matrix results in a row vector. Two composed linear transformations can be multiplied as matrices, or can be applied one after another to give the same result. Backwardness in linear algebra
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be accomplished with the standard matrix multiplication. Perpetuating backward notation for composition of relations is unnecessary, serves no purpose, and contributes to confusion and mistakes. It is suggested that such notation should not be used. —
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Could you elaborate your issues with linear algebra? If the vectors are columns and application of linear maps as well as matrix multiplication happens from the left, the composition of functions results in the product of matrices in the same order.
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Note that Rel is the
Kleisli category of the powerset monad. I'm not sure how standard the lollypop is for Kleisli categories, though, so I'm not arguing for its use. For relations, Paul Taylor, in his book, uses a symbol like
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of relations, I see nothing wrong by itself with using the arrow notation, as long as it is clear that the entity being typed is a relation. An advantage is that this gives a natural way to fix the embedding of a function
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A further variation encountered in computer science is the Z notation: is used to denote the traditional (right) composition, but ⨾ (a fat semicolon with
Unicode code point U+2A3E) denotes left composition.
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And I don't think that the sentence 'composition of morphisms in category theory is coined on composition of relations' is true. Do you have a reference for this?
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It says 'Composition of relations can be seen as a special case of the composition of morphisms in the category of binary relations.' But composition of relations
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notation. A similar backwardness shows up in the study of linear algebra where column vectors are sometimes transformed by a matrix. Thoughtful authors, cited at
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In lectures
Johnstone uses a regular arrow, but in green chalk. He even sent round an email telling everyone to bring a different-coloured pen for the purpose.
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here, and include the explanation that Hans provided. I am not very keen on the tilde because it is so different from the usual arrow notation for morphisms.
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In
English we read from the left margin of a page to its right margin. This ordering has implications for mathematical notation. After page 18,
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except the two arrows are one. He says (Notation 1.3.4) that he invented the symbol for this purpose. Peter
Johnstone, in the Elephant, uses
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was put in the § Definition. A subsection "Notational variations" was inserted to acknowledge differences in use. Comments expected. —
455:. Others use a right arrow with a vertical line through it. Since there is no universal standard, it seems reasonable to stick with
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in the world of relations, where a choice must be made between two dual isomorphic views. Alternatively, I have seen the notation
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of the monad? The notation might be confusing for people who know the other meaning. Since typed relations are arrows in the
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Knowledge. If you would like to participate, please visit the project page, where you can join
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for illustrative purposes, and I would be much more happy with ~. (I didn't think of this.)
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It is odd to call the final section 'Further
Reading': there are no references there!
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for binary relations, but I believe it is standard to reserve this for functions from
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No objections, so I deleted this section. We can discuss here if that is a problem.
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uses juxtaposition for composition of relations. The other major textbook,
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composition of morphisms in the category of relations, surely.
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Solution: (1,u), (1,t), (2,s), (2,t), (3,s), (3,t), (3,u). —
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I propose to delete this final section. Any objections?
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special subject that may not belong in this article.
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267:{\displaystyle f\colon A\multimap B}
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