nLab associative quasigroup

Definitions

A (left/right/two-sided) associative quasigroup is a (left/right/two-sided) quasigroup $(G, \cdot,\backslash,/)$ where $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ for all $a$, $b$, and $c$ in $G$.

In every associative quasigroup $G$, $a/a = b\backslash b$ for every $a$ and $b$ in $G$. This is because for every $a$ and $b$ in $G$, $b \cdot a = b \cdot (a/a) \cdot a = b \cdot (b\backslash b) \cdot a$. Left dividing both sides by $b$ and right dividing both sides by $a$ results in $a/a = b\backslash b$. This means in particular that $a/a = a\backslash a$, 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 $G$ with a binary operation $(-)\backslash(-):G \times G \to G$ (a magma) such that:

• For all $a$ and $b$ in $G$, $a\backslash a=b\backslash b$
• For all $a$ in $G$, $(a\backslash (a\backslash a))\backslash (a\backslash a)=a$
• For all $a$, $b$, and $c$ in $G$, $(a\backslash b)\backslash c= b\backslash ((a\backslash (a\backslash a))\backslash c)$.

For any element $a$ in $G$, the element $a\backslash a$ is called a right identity element, and the element $a\backslash (a\backslash a)$ is called the right inverse element of $a$. For all elements $a$ and $b$ in $G$, left multiplication of $a$ and $b$ is defined as $(a\backslash (a\backslash a))\backslash b$.

A right associative quasigroup is a set $G$ with a binary operation $(-)/(-):G \times G \to G$ such that:

• For all $a$ and $b$ in $G$, $a/a=b/b$
• For all $a$ in $G$, $(a/a)/((a/a)/a)=a$
• for all $a$, $b$, and $c$ in $G$, $a/(b/c)=(a/((c/c)/c)/b$

For any element $a$ in a $G$, the element $a/a$ is called a left identity element, and the element $(a/a)/a$ is called the left inverse element of $a$. For all elements $a$ and $b$, right multiplication of $a$ and $b$ is defined as $a/((b/b)/b)$.

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. $a/a = a \backslash a$) for all $a$ in $G$

• left and right inverse elements are equal (i.e. $(a/a)/a = a\backslash (a\backslash a)$) for all $a$ in $G$

• left and right multiplications are equal (i.e. $a/((b/b)/b) = (a\backslash (a\backslash a))\backslash b$) for all $a$ and $b$ in $G$.

This definition first appeared on the heap article and is due to Toby Bartels.

Pseudo-torsors

Every left associative quasigroup $G$ has a pseudo-torsor $t_G:G^3 \to G$ defined as $t_G(x,y,z) = x \cdot (y \backslash z)$. Every right associative quasigroup $H$ has a pseudo-torsor $t_H:H^3 \to H$ defined as $t_H(x,y,z) = (x / y) \cdot z$. 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.

Category of associative quasigroups

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 $AssocQuasiGrp$.

As the category of associative quasigroups is a concrete category, there is a forgetful functor $U:AssocQuasiGrp \to Set$. $U$ has a left adjoint, the free associative quasigroup functor $F:Set \to AssocQuasiGrp$.

The empty associative quasigroup $0$ whose underlying set is the empty set is the initial associative quasigroup, and is strictly initial. The trivial associative quasigroup $1$ whose underlying set is the singleton is the terminal associative quasigroup.

The direct product $G \times H$ of associative quasigroups $G$ and $H$ is the cartesian product of sets $U(G) \times U(H)$ with an associative quasigroup structure defined componentwise by

$$(g_1, h_1) \cdot_{G \times H} (g_2, h_2) = (g_1 \cdot_G g_2, h_1 \cdot_H h_2)$$

$$(g_1, h_1) /_{G \times H} (g_2, h_2) = (g_1 /_G g_2, h_1 /_H h_2)$$

$$(g_1, h_1) \backslash_{G \times H} (g_2, h_2) = (g_1 \backslash_G g_2, h_1 \backslash_H h_2)$$

for all $(g_1, h_1), (g_2, h_2) \in U(G) \times U(H)$, with product projections $p_G: G \times H \to G$ $p_H: G \times H \to H$ where $p_G(g, h) = g$ and $p_H(g,h) = h$ for all $(g,h)\in G \times H$ The direct product of associative quasigroups is thus the cartesian product in $AssocQuasiGrp$. The endofunctor $P(G) = G \times 1$ is the identity functor on $AssocQuasiGrp$ while the endofunctor $P(G) = G \times 0$ is a constant functor that sends every associative quasigroup to $0$.

An associative subquasigroup of an associative quasigroup $G$ is an associative quasigroup $H$ with an associative quasigroup monomorphism

$$H \hookrightarrow G \,.$$

The empty associative is the initial associative subquasigroup of $G$.

Let $G$ and $H$ be associative quasigroups and let $f:G \to H$ be an associative quasigroup homomorphism. Then given an element $h \in H$, there is an associative subquasigroup

$$i: I \hookrightarrow G \,.$$

such that $g \in I$ if and only if $f(g) = h$. $I$ is called the fiber of $f$ over $h$.

Because $AssocQuasiGrp$ has a terminal object, cartesian products, and fibers, it is a finitely complete category.

Every associative quasigroup $G$ and associative subquasigroup $H \hookrightarrow G$ has a set of left ideals $G H$ in $G$ and a set of right ideals $H G$ in $G$.

Examples

• Every group is an associative quasigroup.

• The empty associative quasigroup is an associative quasigroup that is not a group.

Related concepts

• quasigroup

• commutative quasigroup

• loop

• group

• n-ary group

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

References

• The Group With No Elements at the n-category café