Search results
Results from the WOW.Com Content Network
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.
The integers Z (or the rationals Q or the reals R) with subtraction (−) form a quasigroup. These quasigroups are not loops because there is no identity element (0 is a right identity because a − 0 = a, but not a left identity because, in general, 0 − a ≠ a). The nonzero rationals Q × (or the nonzero reals R ×) with division (÷) form ...
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.
Talk: List of problems in loop theory and quasigroup theory. Add languages. Page contents not supported in other languages. ... Download as PDF; Printable version;
Download as PDF; Printable version ... system-- Laguerre form-- Laguerre ... topics-- List of problems in loop theory and quasigroup theory-- List of ...
Download as PDF; Printable version; ... List of problems in loop theory and quasigroup theory; ... Polarization of an algebraic form;
Moufang loops are universal among inverse property loops; that is, a loop Q is a Moufang loop if and only if every loop isotope of Q has the inverse property. It follows that every loop isotope of a Moufang loop is a Moufang loop. One can use inverses to rewrite the left and right Moufang identities in a more useful form:
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