Quasigroup
In mathematics, a quasigroup is a set Q with a binary operation, here denoted *, with the property that for all a and b in Q there are unique solutions to the equations a * x = b and y * a = b. In this encyclopedia it will also be assumed that a quasigroup is nonempty.Note that quasigroups have the cancellation property: if a * b = a * c, then b = c. This is because x = b is certainly a solution of the equation a * b = a * x, and the solution is required to be unique. Similarly, if a * b = c * b, then a = c.
The multiplication table of a finite quasigroup is called a Latin square: an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
A quasigroup group with an identity element is called a loop. It follows from the definition of a quasigroup that each element of a loop has both a left inverse and a right inverse.
A Moufang loop is a quasigroup Q in which (a * b) * (c * a) = (a * (b * c)) * a, for all a, b and c in Q. As the name suggests, Moufang loops are actually loops, and we will now prove this. Let a be any element of Q, and let e be the element such that a * e = a. Then for any x in Q, (x * a) * x = (x * (a * e)) * x = (x * a) * (e * x), and cancelling gives x = e * x. So e is a left identity element. Now let b be the element such that b * e = e. Then y * b = e * (y * b), as e is a left identity, so (y * b) * e = (e * (y * b)) * e = (e * y) * (b * e) = (e * y) * e = y * e. Cancelling gives y * b = y, so b is a right identity element. Lastly, e = e * b = b, so e is a two-sided identity element.
Every group is a quasigroup, because a * x = b if and only if x = a-1 * b, and y * a = b if and only if y = b * a-1. Moreover, an associative quasigroup must be a Moufang loop, and an associative loop must clearly be a group. This shows that groups are precisely the associative quasigroups. The structure theory of loops is quite analogous to that of groups.
Examples of quasigroups:
- Every group.
- Every Steiner triple system.
- The set of nonzero elements of a finite-dimensional algebra with no zero divisors. (For example, the nonzero octonions.)
- Rn with the operation x * y = (x + y) / 2. More generally, any vector space over a field of characteristic not equal to 2 gives a quasigroup using this operation.
Tretherick is avenged: if Flash drops
thing." During this delicate condition of affairs, Mrs. Tretherick
Hotel, with only the clothes she had on her back. Here she staid
that she bore herself with the strictest propriety.
It was a clear morning in early spring that Mrs. Tretherick,
toward the fringe of dark pines which indicated the extreme limits
preoccupied with the departure of the Wingdown coach at the other
the settlement without discomposing observation. Here she took a
thoroughfare of Fiddletown and passing through a belt of woodland.
The dwellings were few, ambitious, and uninterrupted by shops. And
which usually distinguished him, that his coat was tightly buttoned
arm, swung jauntily, was not entirely at his ease. Mrs.
of her dangerous eyes; and the colonel, with an embarrassed cough
Dutch Flat on a spree. There is no one in the house but a
with a slight inflation of the chest that imperiled the security of
your property."
"I'm sure it's very kind of you, and so disinterested!" simpered
who has soul--someone to sympathize with in a community so hardened
but not until they wrought their perfect and accepted work upon her
up and down the.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|