Nilpotent semi-group

A semi-group with zero for which there is an such that ; this is equivalent to the identity

in . The smallest with this property for a given semi-group is called the step (sometimes class) of nilpotency. If , then is called a semi-group with zero multiplication. The following conditions on a semi-group are equivalent: 1) is nilpotent; 2) has a finite annihilator series (that is, an ascending annihilator series of finite length, see Nil semi-group); or 3) there is a such that every sub-semi-group of can be imbedded as an ideal series of length .

A wider concept is that of a nilpotent semi-group in the sense of Mal'tsev [2]. This is the name for a semi-group satisfying for some the identity

where the words and are defined inductively as follows: , , , , where , and are variables. A group is a nilpotent semi-group in the sense of Mal'tsev if and only if it is nilpotent in the usual group-theoretical sense (see Nilpotent group), and the identity is equivalent to the fact that its class of nilpotency is . Every cancellation semi-group satisfying the identity can be imbedded in a group satisfying the same identity.

References

[1]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[2]	A.I. Mal'tsev, "Nilpotent semi-groups" Uchen. Zap. Ivanov. Gos. Ped. Inst. , 4 (1953) pp. 107–111 (In Russian)
[3]	L.N. Shevrin, "On the general theory of semi-groups" Mat. Sb. , 53 : 3 (1961) pp. 367–386 (In Russian)
[4]	L.N. Shevrin, "Semi-groups all sub-semi-groups of which are accessible" Mat. Sb. , 61 : 2 (1963) pp. 253–256 (In Russian)