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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.
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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 ...
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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.
Group-like structures Closure Associative Identity Cancellation Commutative; Partial magma: Unneeded: Unneeded: Unneeded: Unneeded: Unneeded Semigroupoid: Unneeded