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Construct a finite nilpotent loop with no finite basis for its laws. Proposed: by M. R. Vaughan-Lee in the Kourovka Notebook of Unsolved Problems in Group Theory; Comment: There is a finite loop with no finite basis for its laws (Vaughan-Lee, 1979) but it is not nilpotent.
Problems in loop theory and quasigroup theory. Add languages. ... Download QR code; Print/export Download as PDF; Printable version;
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.
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Group-like structures Closure Associative Identity Cancellation Commutative; Partial magma: Unneeded: Unneeded: Unneeded: Unneeded: Unneeded Semigroupoid: Unneeded
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.
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:
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