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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.
Talk: List of problems in loop theory and quasigroup theory. Add languages. Page contents not supported in other languages. ... Download as PDF; Printable version;
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 ...
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.
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.
In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension E of a simple group S.In other words, there is a short exact sequence
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
In mathematics, a CH-quasigroup, introduced by Manin (1986, definition 1.3), is a symmetric quasigroup in which any three elements generate an abelian quasigroup. "CH" stands for cubic hypersurface .