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The varieties of loops of Bol-Moufang type. (English) Zbl 1102.20054

The authors introduce a system of loop identities of Bol-Moufang type, i.e., identities in which two of their three variables occur once on each side, the third variable occurs twice on each side, and the order in which the variables appear on both sides is the same, but both sides differ in bracketing. The aim of the paper is to characterise the varieties of loops defined by one identity of Bol-Moufang type. There are exactly 14 varieties of such loops, namely:
\(\bullet\) groups, defined by \(x(yz)=(xy)z\),
\(\bullet\) extra loops, defined by \(x(y(zx))=((xy)z)x\),
\(\bullet\) Moufang loops, defined by \((xy)(zx)=(x(yz))x\),
\(\bullet\) left Bol loops, defined by \(x(y(xz))=(x(yx))z\),
\(\bullet\) right Bol loops, defined by \(x((yz)y)=((xy)z)y\),
\(\bullet\) C-loops, defined by \(x(y (yz))=((xy)y)z\),
\(\bullet\) LC-loops, defined by \((xx)(yz)=(x(xy))z\),
\(\bullet\) RC-loops, defined by \(x((yz)z)=(xy)(zz)\),
\(\bullet\) left alternative loops, defined by \(x(xy)=(xx)y\),
\(\bullet\) right alternative loops, defined by \(x(yy)=(xy)y\),
\(\bullet\) flexible loops, defined by \(x(yx)=(xy)x\),
\(\bullet\) left nuclear square loops, defined by \((xx)(yz)=((xx)y)z\),
\(\bullet\) middle nuclear square loops, defined by \(x((yy)z)=(x(yy))z\),
\(\bullet\) right nuclear square loops, defined by \(x(y(zz)=(xy)(zz)\).
Moreover, there is the variety of \(\bullet\) 3-power associative loops, defined by \(x(xx)=(xx)x\), which is introduced only for technical reasons.
The following inclusions hold among these varieties of loops of Bol-Moufang type and 3-power associative loops:
\(\bullet\) The variety of groups is contained in any variety of loops listed above.
\(\bullet\) Extra loops are both Moufang loops and C-loops.
\(\bullet\) Moufang loops are left Bol loops, right Bol loops and flexible loops.
\(\bullet\) C-loops are both LC- and RC-loops.
\(\bullet\) Left [right] Bol loops are left [right] alternative.
\(\bullet\) RC-loops [LC-loops] are right [left] alternative loops, middle nuclear square loops, and right [left] nuclear square loops.
\(\bullet\) The variety of 3-power associative loops contains the varieties of flexible loops, left alternative loops and right alternative loops.
The authors also provide all necessary counterexamples. Thus the programme of Fenyves of the classification of loops of Bol-Moufang type is completed.

MSC:

20N05 Loops, quasigroups
08B15 Lattices of varieties
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Online Encyclopedia of Integer Sequences:

Number of nilpotent loops of order 2*prime(n) up to isotopism.