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In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems.As in other areas of mathematics, such problems are often made public at professional conferences and meetings.
One can normalize a Cayley table of a quasigroup in the same manner as a reduced Latin square. Then the quasigroup associated to a reduced Latin square has a (two sided) identity element (namely, the first element among the row headers). A quasigroup with a two sided identity is called a loop. Some, but not all, loops are groups.
A quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, (A, +), taking its subtraction operation as quasigroup multiplication yields a pique (A, −) with the group identity (zero) turned into a "pointed ...
Given a loop L, one can define an incidence geometric structure called a 3-net. Conversely, after fixing an origin and an order of the line classes, a 3-net gives rise to a loop. Choosing a different origin or exchanging the line classes may result in nonisomorphic coordinate loops. However, the coordinate loops are always isotopic.
Pages for logged out editors learn more. Contributions; Talk; Problems in loop theory and quasigroup theory
A solution to Kirkman's schoolgirl problem with vertices denoting girls and colours denoting days of the week [1] Kirkman's schoolgirl problem is a problem in combinatorics proposed by Thomas Penyngton Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary (pg.48). The problem states:
Group-like structures Total Associative Identity Divisible Commutative; Partial magma: Unneeded: Unneeded: Unneeded: Unneeded: Unneeded Semigroupoid: Unneeded: Required
The image shown in the section "Loops" suggests strongly that an associative quasigroup is the same as an inverse semigroup, but I don't think this is true; the "inverse" property of an inverse semigroup is weaker than that of a quasigroup. In fact, it seems that a nonempty associative quasigroup is automatically a group; see quasigroup on