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A (left/right/two-sided) associative quasigroup is a (left/right/two-sided) quasigroup where for all , , and in .
In every associative quasigroup , for every and in . This is because for every and in , . Left dividing both sides by and right dividing both sides by results in . This means in particular that , which means that every associative quasigroup is a possibly empty loop. Thus, an associative quasigroup is a associative possibly empty loop, or a possibly empty group. In particular, there is an additional definition of an associative quasigroup in terms of the left and right divisions alone, without any semigroup operation at all:
A left associative quasigroup is a set with a binary operation (a magma) such that:
For any element in , the element is called a right identity element, and the element is called the right inverse element of . For all elements and in , left multiplication of and is defined as .
A right associative quasigroup is a set with a binary operation such that:
For any element in a , the element is called a left identity element, and the element is called the left inverse element of . For all elements and , right multiplication of and is defined as .
An associative quasigroup is a possibly empty left and right group as defined above such that the following are true:
left and right identity elements are equal (i.e. ) for all in
left and right inverse elements are equal (i.e. ) for all in
left and right multiplications are equal (i.e. ) for all and in .
This definition first appeared on the heap article and is due to Toby Bartels.
Every left associative quasigroup has a pseudo-torsor defined as . Every right associative quasigroup has a pseudo-torsor defined as . This means every associative quasigroup has two pseudo-torsors. If the (left or right) associative quasigroup is inhabited, then those pseudo-torsors are actually torsors or heaps.
An associative quasigroup homomorphism is a semigroup homomorphism between associative quasigroups that preserves left and right quotients. Associative quasigroup homomorphisms are the morphisms in the category of associative quasigroups .
As the category of associative quasigroups is a concrete category, there is a forgetful functor . has a left adjoint, the free associative quasigroup functor .
The empty associative quasigroup whose underlying set is the empty set is the initial associative quasigroup, and is strictly initial. The trivial associative quasigroup whose underlying set is the singleton is the terminal associative quasigroup.
The direct product of associative quasigroups and is the cartesian product of sets with an associative quasigroup structure defined componentwise by
for all , with product projections where and for all The direct product of associative quasigroups is thus the cartesian product in . The endofunctor is the identity functor on while the endofunctor is a constant functor that sends every associative quasigroup to .
An associative subquasigroup of an associative quasigroup is an associative quasigroup with an associative quasigroup monomorphism
The empty associative is the initial associative subquasigroup of .
Let and be associative quasigroups and let be an associative quasigroup homomorphism. Then given an element , there is an associative subquasigroup
such that if and only if . is called the fiber of over .
Because has a terminal object, cartesian products, and fibers, it is a finitely complete category.
Every associative quasigroup and associative subquasigroup has a set of left ideals in and a set of right ideals in .
Every group is an associative quasigroup.
The empty associative quasigroup is an associative quasigroup that is not a group.
Last revised on August 21, 2024 at 02:30:44. See the history of this page for a list of all contributions to it.