Smarandache isotopy of second Smarandache Bol loops

The pair (GH , ·) is called a special loop if (G, ·) is a loop with an arbitrary subloop (H, ·) called its special subloop. A special loop (GH , ·) is called a second Smarandache Bol loop (S2ndBL) if and only if it obeys the second Smarandache Bol identity (xs · z)s = x(sz · s) for all x, z in G and s in H. The popularly known and well studied class of loops called Bol loops fall into this class and so S2ndBLs generalize Bol loops. The Smarandache isotopy of S2ndBLs is introduced and studied for the first time. It is shown that every Smarandache isotope (S − isotope) of a special loop is Smarandache isomorphic (S − isomorphic) to a S-principal isotope of the special loop. It is established that every special loop that is Sisotopic to a S2ndBL is itself a S2ndBL. A special loop is called a Smarandache G-special loop (SGS − loop) if and only if every special loop that is S-isotopic to it is S-isomorphic to it. A S2ndBL is shown to be a SGS-loop if and only if each element of its special subloop is a S1st companion for a S1st pseudo-automorphism of the S2ndBL. The results in this work generalize the results on the isotopy of Bol loops as can be found in the Ph. D. thesis of D. A. Robinson.