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associative quasigroup

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Algebra

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Definitions

A (left/right/two-sided) associative quasigroup is a (left/right/two-sided) quasigroup (G,,\,/)(G, \cdot,\backslash,/) where a(bc)=(ab)ca \cdot (b \cdot c) = (a \cdot b) \cdot c for all aa, bb, and cc in GG.

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

  • For all aa and bb in GG, a\a=b\ba\backslash a=b\backslash b
  • For all aa in GG, (a\(a\a))\(a\a)=a(a\backslash (a\backslash a))\backslash (a\backslash a)=a
  • For all aa, bb, and cc in GG, (a\b)\c=b\((a\(a\a))\c)(a\backslash b)\backslash c= b\backslash ((a\backslash (a\backslash a))\backslash c).

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

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

  • For all aa and bb in GG, a/a=b/ba/a=b/b
  • For all aa in GG, (a/a)/((a/a)/a)=a(a/a)/((a/a)/a)=a
  • for all aa, bb, and cc in GG, a/(b/c)=(a/((c/c)/c)/ba/(b/c)=(a/((c/c)/c)/b

For any element aa in a GG, the element a/aa/a is called a left identity element, and the element (a/a)/a(a/a)/a is called the left inverse element of aa. For all elements aa and bb, right multiplication of aa and bb is defined as a/((b/b)/b)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\aa/a = a \backslash a) for all aa in GG

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

  • left and right multiplications are equal (i.e. a/((b/b)/b)=(a\(a\a))\ba/((b/b)/b) = (a\backslash (a\backslash a))\backslash b) for all aa and bb in GG.

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

Pseudo-torsors

Every left associative quasigroup GG has a pseudo-torsor t G:G 3Gt_G:G^3 \to G defined as t G(x,y,z)=x(y\z)t_G(x,y,z) = x \cdot (y \backslash z). Every right associative quasigroup HH has a pseudo-torsor t H:H 3Ht_H:H^3 \to H defined as t H(x,y,z)=(x/y)zt_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 AssocQuasiGrpAssocQuasiGrp.

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

The empty associative quasigroup 00 whose underlying set is the empty set is the initial associative quasigroup, and is strictly initial. The trivial associative quasigroup 11 whose underlying set is the singleton is the terminal associative quasigroup.

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

(g 1,h 1) G×H(g 2,h 2)=(g 1 Gg 2,h 1 Hh 2) (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×H(g 2,h 2)=(g 1/ Gg 2,h 1/ Hh 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)\ G×H(g 2,h 2)=(g 1\ Gg 2,h 1\ Hh 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)U(G)×U(H)(g_1, h_1), (g_2, h_2) \in U(G) \times U(H), with product projections p G:G×HGp_G: G \times H \to G p H:G×HHp_H: G \times H \to H where p G(g,h)=gp_G(g, h) = g and p H(g,h)=hp_H(g,h) = h for all (g,h)G×H(g,h)\in G \times H The direct product of associative quasigroups is thus the cartesian product in AssocQuasiGrpAssocQuasiGrp. The endofunctor P(G)=G×1P(G) = G \times 1 is the identity functor on AssocQuasiGrpAssocQuasiGrp while the endofunctor P(G)=G×0P(G) = G \times 0 is a constant functor that sends every associative quasigroup to 00.

An associative subquasigroup of an associative quasigroup GG is an associative quasigroup HH with an associative quasigroup monomorphism

HG. H \hookrightarrow G \,.

The empty associative is the initial associative subquasigroup of GG.

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

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

such that gIg \in I if and only if f(g)=hf(g) = h. II is called the fiber of ff over hh.

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

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

Examples

  • Every group is an associative quasigroup.

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

algebraic structureoidification
truth valuetransitive relation
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
(left,right) cancellative monoid(left,right) cancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
differential ring?differential ringoid?
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
strict monoidal categorystrict 2-category
strict 2-groupstrict 2-groupoid
monoidal poset?2-poset
monoidal groupoid?(2,1)-category
monoidal category2-category/bicategory
2-group2-groupoid/bigroupoid

References

Last revised on May 25, 2021 at 07:44:17. See the history of this page for a list of all contributions to it.