Talk:Monoid
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could you give some examples related to computer ?
Uniqueness of identity element
Under what situation is the identity element guaranteed to be unique? My off-the-cuff guess is that a cancellative monoid has a unique identity. Is there more to it than that? (Thanks!)
The identity is unique in every monoid. If e and f are two identities, then e = ef = f, so they are equal. -- Fropuff 15:13, 2004 Jul 29 (UTC)
Fixed Identity under Mappings
Can someone show me an example of when a homomorphism would not preserve the identity (assuming that wasn't given as an axiom)?
In my graduate abstract book, it doesn't state that a homomorphism must preserve identity (only an isomorphism) and leaves it as an exercise to construct an example to illustrate. However, in a different class, I was asked to show that a ring isomorphism preserves multiplicative identities (when present, obviously). As far as I can tell, I didn't make use of the fact that my mapping was bijective (1-1 and onto). Since my rings are additive groups and multiplicative monoids, then there should be no difference, unless the distributive law somehow differentiates the two types of monoids. -- Ub3rm4th 13:05, 2005 Jan 25 (UTC-6)
- That is very strange. The whole point of a homomorphism is that it preserves the essential properties of the algebraic structure. If a monoid homomorphism doesn't preserve the identity, then it's just a semigroup homomorphism. --MarkSweep 19:24, 25 Jan 2005 (UTC)
Yep. Fairly standard terminology (although I'm sure you'll find somebody who disagrees!) is that a monoid homomorphism is a map between monoids which preserves the multiplication and the identity. Since monoids are also semigroups, there is also a notion of a "semigroup homomorphism of monoids", which as it turns out will preserve multiplication but not necessarily the identity. You can easily show that a homomorphic image of a monoid identity will act as a local identity for the image of the map, which is another way of saying that a surjective semigroup homomorphism always preserves an identity if present; hence the result for rings. Cambyses 05:37, 26 Jan 2005 (UTC)
- Thank you very much. From my past experiences with wikipedia, I didn't expect answers so soon, so this is the first I've checked. For some reason, I like the answer that it would have to be a "semigroup homomorphism of monoids". My interpretation of homomorphisms was exactly that they preserve the structure of the system, thus I was deeply disturbed. I don't think my book (Hungeford) differentiates between a monoid homomorphism and a semigroup homomorphism of monoids. Much gratitude! -- Ub3rm4th 10:38, 07 Feb 2005 (UTC-6)