System [DAMMA] is a left inverse of [summation] if for any S-acceptable u there exists y such that (y,u) is S-acceptable and solves [summation] and after substituting y = y to [GAMMA] and setting [u.sub.k](l) = [u.sub.k](l), l = 0,..., [[sigma].sub.k] - 1, we get solution u of [GAMMA], satisfying [u.sub.k](l) = [u.sub.k](l), l [greater than or equal to] [[sigma].sub.k].

System [summation] is left invertible if there exists a left inverse of [summation] in the sense of Definition 9.

So [mu] is a

left inverse for [theta]; in order to show that [mu] is a right inverse we compute as follows:

Most of the AIC schemes estimate right inverse [[??].sub.R]([q.sup.-1]) and then it is used as

left inverse [[??].sub.L]([q.sup.-1]) by considering left and right inverse are equal.

The inverse (a

left inverse, a right inverse) operator is given by (2.9).

Similarly, it is called a

left inverse property quasigroup (loop) [LIPQ (LIPL)] if and only if it obeys the

left inverse property (LIP) [x.sup.[lambda]] * xy = y for all x,y [member of] G.

A quasigroup (Q, -)has the

left inverse property,the right inverse property or the cross inverse property, if for any x [member of] Q there exists an element x-1 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], respectively.

where the matrix [G.sup.-L.sub.*] is

left inverse to [G.sub.*] .

A loop is called a Smarandache

left inverse property loop (SLIPL) if it has at least a non-trivial subloop with the LIP.

For each x [member of] L, the elements [x.sup.P] = x[J.sub.P], [x.sup.[lambda] = x[J.sub.[lambda][member of] L such that x[x.sup.P] = [e.sup.P] and [x.sup.[lambda]]x = [e.sup.[lambda]] are called the right,

left inverses of x respectively.