Algebra: exotic identity values?
April 8, 2013 1:48 AM   Subscribe

[mathfilter] please give me some example algebraic structures that are not commutative, with exotic identity values.

I am creating an inventory of example algebraic structures of one operator that feature associativity, commutativity, identity values and all combinations thereof. My table is nearly full but for three entries:

- non-associative, non-commutative with different left and right identities

- non-associative, non-commutative with both left and right identities, where both identities are equal

- associative, non-commutative, with either a left or right identity but not both

(For clarity: a left identity is a value 0l such that 0l . x = x for all x, and a right identity is a value 0r such that x . 0r = x for all x. They may not be both present / the same if the structure is not commutative.)

Do you know of example functions in these three categories?
posted by knz to Science & Nature (5 answers total) 2 users marked this as a favorite
Best answer: What about the quaternions and the octonions? Multiplication in the quaternions isn't commutative (but is associative), and multiplication in the octionions is neither (generally) commutative or associative.
posted by oonh at 3:22 AM on April 8, 2013 [2 favorites]

Response by poster: @oonh: thanks, I can use quaternions/octonions as an example where both left and right identities exist and are equal.

Any idea for structures where the identities are asymmetrical?
posted by knz at 3:29 AM on April 8, 2013

Best answer: Algebraist here. Do you need detailed explanations for each example or just a list?

If it's just a list then you will be able to find many examples of the associative, non-commutative structures with different left/right identities as subsets of matrix rings. You might try a search for pseudo-identity or pseudo-inverse for specific examples. Path algebras (defined over quivers) might give you the "nicest" examples of this category. Monoids and semigroups might be another place to look for "nice" examples.
posted by El_Marto at 3:59 AM on April 8, 2013

Wikipedia has a sort of example of the last one: take a two element set {e,f} with an operation * such that e*e=e*f=e and f*f=f*e=f. Associativity is rather vacuous, though. (I tried quickly to add a third element in to get real associativity and failed, but that's definitely not to say it can't be done.)

I believe you'll run into problems with binary operations, as if you have both a left and a right identity, they have to be the same. (Say e is a left identity and f a right identity. Then ex=x for all x and xf=x for all x. Stick e or f in for x in (the right) one and you have f=ef=e.) You could allow your operation to not be defined for all pairs of elements.
posted by hoyland at 4:24 AM on April 8, 2013

I would suggest that you try asking your question at the mathematics StackExchange site:
posted by number9dream at 2:10 PM on April 8, 2013

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