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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.
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.
Quasigroup: A magma where division is always possible. Loop: A quasigroup with an identity element. Semigroup: A magma where the operation is associative. Monoid: A semigroup with an identity element. Group: A magma with inverse, associativity, and an identity element. Note that each of divisibility and invertibility imply the cancellation ...
In its most general form a loop group is a group of continuous mappings from a manifold M to a topological group G.. More specifically, [1] let M = S 1, the circle in the complex plane, and let LG denote the space of continuous maps S 1 → G, i.e.
Pages for logged out editors learn more. Contributions; Talk; Problems in loop theory and quasigroup theory
Perceptual control theory (PCT) is a model of behavior based on the properties of negative feedback control loops. A control loop maintains a sensed variable at or near a reference value by means of the effects of its outputs upon that variable, as mediated by physical properties of the environment.