Not an isomorphism, but what to call it?
I have two types A and B, and I want to express that the composition of two functions f :: B -> A and g :: A -> B gives me the identity idA = f . g :: A -> A. I don't need g . f :: B -> B to be the identity on B, so I want a weaker statement than isomorphism. I understand that: (1) If I look at it from the perspective of f, then g is the right inverse or section (or split monomorphism). (2) If I look at from g, then f is the left inverse or retraction (or split epimorphism). But I just want two functions that give me an identity on one of the two types and I don't care which function's perspective I'm looking at it from. Is there a word for that? Regards, Sean
On Thu, Jan 19, 2012 at 3:24 PM, Sean Leather
I have two types A and B, and I want to express that the composition of two functions f :: B -> A and g :: A -> B gives me the identity idA = f . g :: A -> A. I don't need g . f :: B -> B to be the identity on B, so I want a weaker statement than isomorphism.
I understand that: (1) If I look at it from the perspective of f, then g is the right inverse or section (or split monomorphism). (2) If I look at from g, then f is the left inverse or retraction (or split epimorphism).
But I just want two functions that give me an identity on one of the two types and I don't care which function's perspective I'm looking at it from. Is there a word for that?
I don't think it makes sense to say you want one label for the situation when looking from either end - the relation you're labeling is non-symmetric.
Regards, Sean
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-----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 On 01/20/2012 07:24 AM, Sean Leather wrote:
I have two types A and B, and I want to express that the composition of two functions f :: B -> A and g :: A -> B gives me the identity idA = f . g :: A -> A. I don't need g . f :: B -> B to be the identity on B, so I want a weaker statement than isomorphism.
I understand that: (1) If I look at it from the perspective of f, then g is the right inverse or section (or split monomorphism). (2) If I look at from g, then f is the left inverse or retraction (or split epimorphism).
But I just want two functions that give me an identity on one of the two types and I don't care which function's perspective I'm looking at it from. Is there a word for that?
Regards, Sean
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It is not clear to me exactly what you are asking, so shot in the dark: injection or surjection? - -- Tony Morris http://tmorris.net/ -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.11 (GNU/Linux) Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org/ iQEcBAEBAgAGBQJPGJNXAAoJEPxHMY3rBz0PHt0IAKP1lVcfDXZm00h4W1WQPDBT h6LB9nLlp0cgAh5CH06FsdQFqdtDVJNMkV7/9q3H/wTFOoscZHCTigr1G+vE/kA8 lh1/Gb3caQByt6rWkgD79998FL5ZCBdHN2HYh1o/RPBwA/BYxA041P92pE0EFTKB 1oylh5ldUfv8rEzvHhQVw0USrJ11uiZfn/T3+UrO2s2xLQZS1oTWNZhsKMccjB95 tYaqEw+20Q+8yBanVnDJFOqD3yPXIRBHkTSJTOFO+Y++oen4gXUzSJJ2lkpXLECE ojMNHD/9Yh43gCm1Jq3Wuz5B6mr+v+RTRuLkxiVMsK7wxW+lfmOgeMyxHyr8pxU= =aPtB -----END PGP SIGNATURE-----
Am 19.01.2012 um 22:24 schrieb Sean Leather:
I have two types A and B, and I want to express that the composition of two functions f :: B -> A and g :: A -> B gives me the identity idA = f . g :: A -> A. I don't need g . f :: B -> B to be the identity on B, so I want a weaker statement than isomorphism.
I understand that: (1) If I look at it from the perspective of f, then g is the right inverse or section (or split monomorphism). (2) If I look at from g, then f is the left inverse or retraction (or split epimorphism).
But I just want two functions that give me an identity on one of the two types and I don't care which function's perspective I'm looking at it from. Is there a word for that?
If (g . f) is a closure operator for some ordering on B, then
A is a retract of B.
http://nlab.mathforge.org/nlab/show/retract
g is the section, f is the rectraction. You seem to have it already.
The definition needn't be biased toward one of the functions.
On Thu, Jan 19, 2012 at 4:24 PM, Sean Leather
I have two types A and B, and I want to express that the composition of two functions f :: B -> A and g :: A -> B gives me the identity idA = f . g :: A -> A. I don't need g . f :: B -> B to be the identity on B, so I want a weaker statement than isomorphism.
I understand that: (1) If I look at it from the perspective of f, then g is the right inverse or section (or split monomorphism). (2) If I look at from g, then f is the left inverse or retraction (or split epimorphism).
But I just want two functions that give me an identity on one of the two types and I don't care which function's perspective I'm looking at it from. Is there a word for that?
Regards, Sean
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On Thu, Jan 19, 2012 at 23:21, Dan Doel wrote:
A is a retract of B.
http://nlab.mathforge.org/nlab/show/retract
g is the section, f is the rectraction. You seem to have it already. The definition needn't be biased toward one of the functions.
Great! That's what I was looking for. Thanks!
participants (5)
-
Antoine Latter -
Dan Doel -
Holger Siegel -
Sean Leather -
Tony Morris