Smarandache isotopy of second Smarandache Bol loops

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1 Scientia Mana Vol. 7 (2011), No. 1, Smarandache isotopy o second Smarandache Bol loops Tèmító. pé. Gbó. láhàn Jaíyéọlá Department o Mathematics, Obaemi Awolowo University, Ile Ie , Nieria jaiyeolatemitope@yahoo.com tjayeola@oauie.edu.n Abstract The pair (G H, ) is called a special loop i (G, ) is a loop with an arbitrary subloop (H, ) called its special subloop. A special loop (G H, ) is called a second Smarandache Bol loop (S 2 ndbl) i and only i it obeys the second Smarandache Bol identity (xs z)s = x(sz s) or all x, z in G and s in H. The popularly known and well studied class o loops called Bol loops all into this class and so S 2 ndbls eneralize Bol loops. The Smarandache isotopy o S 2 ndbls is introduced and studied or the irst time. It is shown that every Smarandache isotope (S isotope) o a special loop is Smarandache isomorphic (S isomorphic) to a S-principal isotope o the special loop. It is established that every special loop that is S- isotopic to a S 2 ndbl is itsel a S 2 ndbl. A special loop is called a Smarandache G-special loop (SGS loop) i and only i every special loop that is S-isotopic to it is S-isomorphic to it. A S 2 ndbl is shown to be a SGS-loop i and only i each element o its special subloop is a S 1 st companion or a S 1 st pseudo-automorphism o the S 2 ndbl. The results in this work eneralize the results on the isotopy o Bol loops as can be ound in the Ph. D. thesis o D. A. Robinson. Keywords Special loop, second Smarandache Bol loop, Smarandache princiapl isotope, Sm -arandache isotopy. 1. Introduction The study o the Smarandache concept in roupoids was initiated by W. B. Vasantha Kandasamy in [24]. In her book [22] and irst paper [23] on Smarandache concept in loops, she deined a Smarandache loop (S loop) as a loop with at least a subloop which orms a subroup under the binary operation o the loop. The present author has contributed to the study o S-quasiroups and S-loops in [5]-[12] by introducin some new concepts immediately ater the works o Muktibodh [15]-[16]. His recent monoraph [14] ives inter-relationships and connections between and amon the various Smarandache concepts and notions that have been developed in the aorementioned papers. But in the quest o developin the concept o Smarandache quasiroups and loops into a theory o its own just as in quasiroups and loop theory (see [1]-[4], [17], [22]), there is the need to introduce identities or types and varieties o Smarandache quasiroups and loops. This led Jaíyéọlá [13] to the introduction o second Smarandache Bol loop (S 2 ndbl) described by the

2 Vol. 7 Smarandache isotopy o second Smarandache Bol loops 83 second Smarandache Bol identity (xs z)s = x(sz s) or all x, z in G and s in H where the pair (G H, ) is called a special loop i (G, ) is a loop with an arbitrary subloop (H, ). For now, a Smarandache loop or Smarandache quasiroup will be called a irst Smarandache loop (S 1 st loop) or irst Smarandache quasiroup (S 1 st quasiroup). Let L be a non-empty set. Deine a binary operation ( ) on L: i x y L or all x, y L, (L, ) is called a roupoid. I the equations; a x = b and y a = b have unique solutions or x and y respectively, then (L, ) is called a quasiroup. For each x L, the elements x ρ = xj ρ, x λ = xj λ L such that xx ρ = e ρ and x λ x = e λ are called the riht, let inverses o x respectively. Furthermore, i there exists a unique element e = e ρ = e λ in L called the identity element such that or all x in L, x e = e x = x, (L, ) is called a loop. We write xy instead o x y, and stipulate that has lower priority than juxtaposition amon actors to be multiplied. For instance, x yz stands or x(yz). A loop is called a riht Bol loop (Bol loop in short) i and only i it obeys the identity (xy z)y = x(yz y). This class o loops was the irst to catch the attention o loop theorists and the irst comprehensive study o this class o loops was carried out by Robinson [19]. The popularly known and well studied class o loops called Bol loops all into the class o S 2 ndbls and so S 2 ndbls eneralize Bol loops. The aim o this work is to introduce and study or the irst time, the Smarandache isotopy o S 2 ndbls. It is shown that every Smarandache isotope (S-isotope) o a special loop is Smarandache isomorphic (S-isomorphic) to a S-principal isotope o the special loop. It is established that every special loop that is S-isotopic to a S 2 ndbl is itsel a S 2 ndbl. A S 2 ndbl is shown to be a Smarandache G-special loop i and only i each element o its special subloop is a S 1 st companion or a S 1 st pseudo-automorphism o the S 2 ndbl. The results in this work eneralize the results on the isotopy o Bol loops as can be ound in the Ph. D. thesis o D. A. Robinson. 2. Preliminaries Deinition 1. Let (G, ) be a quasiroup with an arbitrary non-trivial subquasiroup (H, ). Then, (G H, ) is called a special quasiroup with special subquasiroup (H, ). I (G, ) is a loop with an arbitrary non-trivial subloop (H, ). Then, (G H, ) is called a special loop with special subloop (H, ). I (H, ) is o exponent 2, then (G H, ) is called a special loop o Smarandache exponent 2. A special quasiroup (G H, ) is called a second Smarandache riht Bol quasiroup (S 2 ndriht Bol quasiroup) or simply a second Smarandache Bol quasiroup (S 2 nd-bol quasiroup) and abbreviated S 2 ndrbq or S 2 ndbq i and only i it obeys the second Smarandache Bol identity (S 2 nd-bol identity) i.e S 2 ndbi (xs z)s = x(sz s) or all x, z G and s H. (1) Hence, i (G H, ) is a special loop, and it obeys the S 2 ndbi, it is called a second Smarandache Bol loop(s 2 nd-bol loop) and abbreviated S 2 ndbl.

3 84 Tèmító. pé. Gbó. láhàn Jaíyéọlá No. 1 Remark 1. A Smarandache Bol loop (i.e a loop with at least a non-trivial subloop that is a Bol loop) will now be called a irst Smarandache Bol loop (S 1 st-bol loop). It is easy to see that a S 2 ndbl is a S 1 stbl. But the converse is not enerally true. So S 2 ndbls are particular types o S 1 stbl. Their study can be used to eneralise existin results in the theory o Bol loops by simply orcin H to be equal to G. Deinition 2. Let (G, ) be a quasiroup (loop). It is called a riht inverse property quasiroup (loop) [RIPQ (RIPL)] i and only i it obeys the riht inverse property (RIP) yx x ρ = y or all x, y G. Similarly, it is called a let inverse property quasiroup (loop) [LIPQ (LIPL)] i and only i it obeys the let inverse property (LIP) x λ xy = y or all x, y G. Hence, it is called an inverse property quasiroup (loop) [IPQ (IPL)] i and only i it obeys both the RIP and LIP. (G, ) is called a riht alternative property quasiroup (loop) [RAPQ (RAPL)] i and only i it obeys the riht alternative property (RAP) y xx = yx x or all x, y G. Similarly, it is called a let alternative property quasiroup (loop) [LAPQ (LAPL)] i and only i it obeys the let alternative property (LAP) xx y = x xy or all x, y G. Hence, it is called an alternative property quasiroup (loop) [APQ (APL)] i and only i it obeys both the RAP and LAP. The bijection L x : G G deined as yl x = x y or all x, y G is called a let translation (multiplication) o G while the bijection R x : G G deined as yr x = y x or all x, y G is called a riht translation (multiplication) o G. Let x\y = yl 1 x = yl x and x/y = xr 1 y = xr y, and note that x\y = z x z = y and x/y = z z y = x. The operations \ and / are called the let and riht divisions respectively. We stipulate that / and \ have hiher priority than amon actors to be multiplied. For instance, x y/z and x y\z stand or x(y/z) and x (y\z) respectively. (G, ) is said to be a riht power alternative property loop (RPAPL) i and only i it obeys the riht power alternative property (RPAP) xy n = (((xy)y)y)y y }{{} n-times i.e. R y n = R n y or all x, y G and n Z. The riht nucleus o G denoted by N ρ (G, ) = N ρ (G) = {a G : y xa = yx a x, y G}. Let (G H, ) be a special quasiroup (loop). It is called a second Smarandache riht inverse property quasiroup (loop) [S 2 ndripq (S 2 ndripl)] i and only i it obeys the second Smarandache riht inverse property (S 2 ndrip) ys s ρ = y or all y G and s H. Similarly, it is called a second Smarandache let inverse property quasiroup (loop) [S 2 ndlipq (S 2 ndlipl)] i and only i it obeys the second Smarandache let inverse property (S 2 ndlip) s λ sy = y or all y G and s H. Hence, it is called a second Smarandache inverse property quasiroup (loop) [S 2 ndipq (S 2 ndipl)] i and only i it obeys both the S 2 ndrip and S 2 ndlip. (G H, ) is called a third Smarandache riht inverse property quasiroup (loop) [S 3 rdripq (S 3 rdripl)] i and only i it obeys the third Smarandache riht inverse property (S 3 rdrip) sy y ρ = s or all y G and s H.

4 Vol. 7 Smarandache isotopy o second Smarandache Bol loops 85 (G H, ) is called a second Smarandache riht alternative property quasiroup (loop) [S 2 ndra PQ(S 2 ndrapl)] i and only i it obeys the second Smarandache riht alternative property (S 2 ndrap) y ss = ys s or all y G and s H. Similarly, it is called a second Smarandache let alternative property quasiroup (loop) [S 2 ndlapq (S 2 ndlapl)] i and only i it obeys the second Smarandache let alternative property (S 2 ndlap) ss y = s sy or all y G and s H. Hence, it is called an second Smarandache alternative property quasiroup (loop) [S 2 ndapq (S 2 ndapl)] i and only i it obeys both the S 2 ndrap and S 2 ndlap. (G H, ) is said to be a Smarandache riht power alternative property loop (SRPAPL) i and only i it obeys the Smarandache riht power alternative property (SRPAP) xs n = (((xs)s)s)s s }{{} n-times i.e. R s n = R n s or all x G, s H and n Z. The Smarandache riht nucleus o G H denoted by SN ρ (G H, ) = SN ρ (G H ) = N ρ (G) H. G H is called a Smarandache riht nuclear square special loop i and only i s 2 SN ρ (G H ) or all s H. Remark 2. A Smarandache; RIPQ or LIPQ or IPQ (i.e a loop with at least a non-trivial subquasiroup that is a RIPQ or LIPQ or IPQ) will now be called a irst Smarandache; RIPQ or LIPQ or IPQ (S 1 stripq or S 1 stlipq or S 1 stipq). It is easy to see that a S 2 ndripq or S 2 ndlipq or S 2 ndipq is a S 1 stripq or S 1 stlipq or S 1 stipq respectively. But the converse is not enerally true. Deinition 3. Let (G, ) be a quasiroup (loop). The set SY M(G, ) = SY M(G) o all bijections in G orms a roup called the permutation (symmetric) roup o G. The triple (U, V, W ) such that U, V, W SY M(G, ) is called an autotopism o G i and only i xu yv = (x y)w x, y G. The roup o autotopisms o G is denoted by AUT (G, ) = AUT (G). Let (G H, ) be a special quasiroup (loop). The set SSY M(G H, ) = SSY M(G H ) o all Smarandache bijections (S-bijections) in G H i.e A SY M(G H ) such that A : H H orms a roup called the Smarandache permutation (symmetric) roup [S-permutation roup] o G H. The triple (U, V, W ) such that U, V, W SSY M(G H, ) is called a irst Smarandache autotopism (S 1 st autotopism) o G H i and only i xu yv = (x y)w x, y G H. I their set orms a roup under componentwise multiplication, it is called the irst Smarandache autotopism roup (S 1 st autotopism roup) o G H and is denoted by S 1 staut (G H, ) = S 1 staut (G H ). The triple (U, V, W ) such that U, W SY M(G, ) and V SSY M(G H, ) is called a second riht Smarandache autotopism (S 2 nd riht autotopism) o G H i and only i xu sv = (x s)w x G and s H. I their set orms a roup under componentwise multiplication, it is called the second riht Smarandache autotopism roup (S 2 nd riht autotopism roup) o G H and is denoted by S 2 ndrau T (G H, ) = S 2 ndraut (G H ).

5 86 Tèmító. pé. Gbó. láhàn Jaíyéọlá No. 1 The triple (U, V, W ) such that V, W SY M(G, ) and U SSY M(G H, ) is called a second let Smarandache autotopism (S 2 nd let autotopism) o G H i and only i su yv = (s y)w y G and s H. I their set orms a roup under componentwise multiplication, it is called the second let Smarandache autotopism roup (S 2 nd let autotopism roup) o G H and is denoted by S 2 ndlaut (G H, ) = S 2 ndlaut (G H ). Let (G H, ) be a special quasiroup (loop) with identity element e. A mappin T SSY M(G H ) is called a irst Smarandache semi-automorphism (S 1 st semi-automorphism) i and only i et = e and (xy x)t = (xt yt )xt or all x, y G. A mappin T SSY M(G H ) is called a second Smarandache semi-automorphism (S 2 nd semi-automorphism) i and only i et = e and (sy s)t = (st yt )st or all y G and all s H. A special loop (G H, ) is called a irst Smarandache semi-automorphic inverse property loop (S 1 stsaipl) i and only i J ρ is a S 1 st semi-automorphism. A special loop (G H, ) is called a second Smarandache semi-automorphic inverse property loop (S 2 ndsaipl) i and only i J ρ is a S 2 nd semi-automorphism. Let (G H, ) be a special quasiroup (loop). A mappin A SSY M(G H ) is a 1. First Smarandache pseudo-automorphism (S 1 st pseudo-automorphism) o G H i and only i there exists a c H such that (A, AR c, AR c ) S 1 staut (G H ). c is reered to as the irst Smarandache companion (S 1 st companion) o A. The set o such A s is denoted by S 1 stp AUT (G H, ) = S 1 stp AUT (G H ). 2. Second riht Smarandache pseudo-automorphism (S 2 nd riht pseudo-automorphism) o G H i and only i there exists a c H such that (A, AR c, AR c ) S 2 ndraut (G H ). c is reered to as the second riht Smarandache companion (S 2 nd riht companion) o A. The set o such A s is denoted by S 2 ndrp AUT (G H, ) = S 2 ndrp AUT (G H ). 3. Second let Smarandache pseudo-automorphism (S 2 nd let pseudo-automorphism) o G H i and only i there exists a c H such that (A, AR c, AR c ) S 2 ndlaut (G H ). c is reered to as the second let Smarandache companion (S 2 nd let companion) o A. The set o such A s is denoted by S 2 ndlp AUT (G H, ) = S 2 ndlp AUT (G H ). Let (G H, ) be a special loop. A mappin A SSY M(G H ) is a 1. First Smarandache automorphism (S 1 st automorphism) o G H i and only i A S 1 stp AUT (G H ) such that c = e. Their set is denoted by S 1 staum(g H, ) = S 1 staum(g H ). 2. Second riht Smarandache automorphism (S 2 nd riht automorphism) o G H i and only i A S 2 ndrp AUT (G H ) such that c = e. Their set is denoted by S 2 ndraum(g H, ) = S 2 ndraum(g H ).

6 Vol. 7 Smarandache isotopy o second Smarandache Bol loops Second let Smarandache automorphism (S 2 nd let automorphism) o G H i and only i A S 2 ndlp AUT (G H ) such that c = e. Their set is denoted by S 2 ndlaum(g H, ) = S 2 ndlaum(g H ). A special loop (G H, ) is called a irst Smarandache automorphism inverse property loop (S 1 staipl) i and only i (J ρ, J ρ, J ρ ) AUT (H, ). A special loop (G H, ) is called a second Smarandache riht automorphic inverse property loop (S 2 ndraipl) i and only i J ρ is a S 2 nd riht automorphism. A special loop (G H, ) is called a second Smarandache let automorphic inverse property loop (S 2 ndlaipl) i and only i J ρ is a S 2 nd let automorphism. Deinition 4. Let (G, ) and (L, ) be quasiroups (loops). The triple (U, V, W ) such that U, V, W : G L are bijections is called an isotopism o G onto L i and only i xu yv = (x y)w x, y G. (2) Let (G H, ) and (L M, ) be special roupoids. G H and L M are Smarandache isotopic (S-isotopic) [and we say (L M, ) is a Smarandache isotope o (G H, )] i and only i there exist bijections U, V, W : H M such that the triple (U, V, W ) : (G H, ) (L M, ) is an isotopism. In addition, i U = V = W, then (G H, ) and (L M, ) are said to be Smarandache isomorphic (S-isomorphic) [and we say (L M, ) is a Smarandache isomorph o (G H, ) and thus write (G H, ) (L M, ).]. (G H, ) is called a Smarandache G-special loop (SGS-loop) i and only i every special loop that is S-isotopic to (G H, ) is S-isomorphic to (G H, ). Theorem 1. (Jaíyéọlá [13]) Let the special loop (G H, ) be a S 2 ndbl. Then it is both a S 2 ndripl and a S 2 ndrapl. Theorem 2. (Jaíyéọlá [13]) Let (G H, ) be a special loop. (G H, ) is a S 2 ndbl i and only i (R 1 s, L s R s, R s ) S 1 staut (G H, ). 3. Main results Lemma 1. Let (G H, ) be a special quasiroup and let s, t H. For all x, y G, let x y = xr 1 t yl 1 s. (3) Then, (G H, ) is a special loop and so (G H, ) and (G H, ) are S-isotopic. Proo. It is easy to show that (G H, ) is a quasiroup with a subquasiroup (H, ) since (G H, ) is a special quasiroup. So, (G H, ) is a special quasiroup. It is also easy to see that s t H is the identity element o (G H, ). Thus, (G H, ) is a special loop. With U = R t, V = L s and W = I, the triple (U, V, W ) : (G H, ) (G H, ) is an S-isotopism. Remark 3. (G H, ) will be called a Smarandache principal isotopism (S-principal isotopism) o (G H, ). Theorem 3. I the special quasiroup (G H, ) and special loop (L M, ) are S-isotopic, then (L M, ) is S-isomorphic to a S-principal isotope o (G H, ).

7 88 Tèmító. pé. Gbó. láhàn Jaíyéọlá No. 1 Proo. Let e be the identity element o the special loop (L M, ). Let U, V and W be 1-1 S-mappins o G H onto L M such that xu yv = (x y)w x, y G H. Let t = ev 1 and s = eu 1. Deine x y or all x, y G H by x y = (xw yw )W 1. (4) From (2), with x and y replaced by xw U 1 and yw V 1 respectively, we et (xw yw )W 1 = xw U 1 yw V 1 x, y G H. (5) In (5), with x = ew 1, we et W V 1 = L 1 s Hence, rom (4) and (5), and with y = ew 1, we et W U 1 = R 1 t. x y = xr 1 t yl 1 s and (x y)w = xw yw x, y G H. That is, (G H, ) is a S-principal isotope o (G H, ) and is S-isomorphic to (L M, ). Theorem 4. Let (G H, ) be a S 2 ndripl. Let, H and let (G H, ) be a S-principal isotope o (G H, ). (G H, ) is a S 2 ndripl i and only i α(, ) = (R, L R, R 1 ) S 2 ndraut (G H, ) or all, H. Proo. Let (G H, ) be a special loop that has the S 2 ndrip and let, H. For all x, y G, deine x y = xr 1 yl 1 as in (3). Recall that is the identity in (G H, ), so x x ρ = where xj ρ = x ρ i.e the riht identity element o x in (G H, ). Then, or all x G, x x ρ = xr 1 xj ρl 1 all s H, then sr 1 = and by the S 2 ndrip o (G H, ), since sr 1 )J ρ because (H, ) has the RIP. Thus, = ( ) (sj ρl 1 sr 1 = sj ρl 1 sj ρl 1 = or J ρl sj ρ = sr J λl. (6) (G H, ) has the S 2 ndrip i (x s) sj ρ = s or all s H, x G H i (xr 1 x, or all s H, x G H. Replace x by x and s by s, then (x s)r 1 i (x s)r 1 Usin (6), = (x ) ( s)j ρl 1 (x s)r 1 sl 1 )R 1 sj ρl 1 = ( s)j ρl 1 = x J ρ or all s H, x G H since (G H, ) has the S 2 ndrip. = xr ( s)r (x s)r 1 = xr sl R α(, ) = (R, L R, R 1 ) S 2 ndraut (G H, ) or all, H. Theorem 5. I a special loop (G H, ) is a S 2 ndbl, then any o its S-isotopes is a S 2 ndripl. Proo. By virtue o theorem 3, we need only to concern ourselves with the S-principal isotopes o (G H, ). (G H, ) is a S 2 ndbl i it obeys the S 2 ndbi i (xs z)s = x(sz s) or all x, z G and s H i L xs R s = L s R s L x or all x G and s H i Rs all x G and s H i Rs s Assume that (G H, ) is a S 2 ndbl. Then, by theorem 2, (R 1 s xs = L 1 x Rs s or = L x Rs xs or all x G and s H. (7), L s R s, R s ) S 1 staut (G H, ) (Rs 1, L s R s, R s ) S 2 ndraut (G H, )

8 Vol. 7 Smarandache isotopy o second Smarandache Bol loops 89 (R 1 s, L s R s, R s ) 1 = (R s, R 1 s L 1 s, Rs 1 ) S 2 ndraut (G H, ). By (7), α(x, s) = (R s, L x Rs xs, Rs 1 ) S 2 ndraut (G H, ) or all, H. But (G H, ) has the S 2 ndrip by theorem 1. So, ollowin theorem 4, all special loops that are S-isotopic to (G H, ) are S 2 ndripls. Theorem 6. Suppose that each special loop that is S-isotopic to (G H, ) is a S 2 ndripl, then the identities: 1. ()\ = (x)\x; 2. \(s 1 ) = ()\[(s) 1 ] are satisied or all,, s H and x G. Proo. In particular, (G H, ) has the S 2 ndrip. Then by theorem 3, α(, ) = (R, L R 1 L 1 ) S 2 ndraut (G H, ) or all, H. Let, R 1 Then, Y = L R. (8) x sy = (xs)r 1. (9) Put s = in (9), then x Y = (x)r 1 = x. But, Y = L R = ()\[() 1 ] = ()\. So, x ()\ = x ()\ = (x)\x. Put x = e in (9), then sy L = sr 1 sr = sl R \(s 1 ) = ()\[(s) 1 ]. sy = sr. So, combinin this with (8), Theorem 7. Every special loop that is S-isotopic to a S 2 ndbl is itsel a S 2 ndbl. Proo. Let (G H, ) be a special loop that is S-isotopic to an S 2 ndbl (G H, ). Assume that x y = xα yβ where α, β : H H. Then the S 2 ndbi can be written in terms o ( ) as ollows. (xs z)s = x(sz s) or all x, z G and s H. Replace xα by x, sβ by s and zβ by z, then [(xα sβ)α zβ]α sβ = xα [(sα zβ)α sβ]β. (10) [(x s)α z]α s = x [(sβ 1 α z)α s]β. (11) I x = e, then (sα z)α s = [(sβ 1 α z)α s]β. (12) Substitutin (12) into the RHS o (11) and replacin x, s and z by x, s and z respectively, we have [(x s)α z]α s = x [(sα z)α s]. (13) With s = e, (xα z)α = x (eα z)α. Let (eα z)α = zδ, where δ SSY M(G H ). Then, (xα z)α = x zδ. (14) Applyin (14), then (13) to the expression [(x s) zδ] s, that is [(x s) zδ] s = [(x s)α z]α s = x [(sα z)α s] = x [(s zδ) s].

9 90 Tèmító. pé. Gbó. láhàn Jaíyéọlá No. 1 implies Replace zδ by z, then [(x s) zδ] s = x [(s zδ) s]. [(x s) z] s = x [(s z) s]. Theorem 8. Let (G H, ) be a S 2 ndbl. Each special loop that is S-isotopic to (G H, ) is S-isomorphic to a S-principal isotope (G H, ) where x y = xr yl 1 or all x, y G and some H. Proo. Let e be the identity element o (G H, ). Let (G H, ) be any S-principal isotope o (G H, ) say x y = xrv 1 yl 1 u or all x, y G and some u, v H. Let e be the identity element o (G H, ). That is, e = u v. Now, deine x y by x y = [(xe ) (ye )]e 1 or all x, y G. Then R e is an S-isomorphism o (G H, ) onto (G H, ). Observe that e is also the identity element or (G H, ) and since (G H, ) is a S 2 ndbl, So, usin (15), x y = [(xe ) (ye )]e 1 = [xr e R 1 v implies that (pe )(e 1 q e 1 ) = pq e 1 or all p, q G. (15) yr e L 1 u ]e 1 = xr e Rv 1 R e yr e L 1 u L e 1R e 1 x y = xa yb, A = R e R 1 v R e and B = R e L 1 u L e 1R e 1. (16) Let = ea. then, y = e y = ea yb = yb or all y G. So, B = L 1. In act, eb = ρ = 1. Then, x = x e = xa eb = xa 1 or all x G implies x = (xa 1 ) implies x = xa (S 2 ndrip) implies A = R. Now, (16) becomes x y = xr yl 1. Theorem 9. Let (G H, ) be a S 2 ndbl with the S 2 ndraip or S 2 ndlaip, let H and let x y = xr yl 1 or all x, y G. Then (G H, ) is a S 1 staipl i and only i N λ (H, ). Proo. Since (G H, ) is a S 2 ndbl, J = J λ = J ρ in (H, ). Usin (6) with = 1, (G H, ) is a S 1 staipl i (x y)j ρ = xj ρ yj ρ or all x, y H i sj ρ = sr JL. (17) (xr yl 1 )J ρ = xj ρr yj ρl 1. (18) Let x = ur 1 and y = vl and use (16), then (18) becomes (uv)r JL = ujl R vl R J i α = (JL R, L R J, R JL ) AUT (H, ). Since (G H, ) is a S 1 staipl, so (J, J, J) AUT (H, ). So, α AUT (H, ) β = α(j, J, J)(R 1, L 1 1R 1, R 1) AUT (H, ). Since (G H, ) is a S 2 ndbl, xl R L 1R 1 = [ 1 (x )] 1 = [( 1 x)] 1 = x or all x G. That is, L R L 1R 1 = I in (G H, ). Also, since J AUM(H, ), then R J = JR 1 and L J =

10 Vol. 7 Smarandache isotopy o second Smarandache Bol loops 91 JL 1 in (H, ). So, β = (JL R JR 1 1, L R J 2 L 1R 1, R JL JR 1) = (JL JR 1R 1 1, L R L 1R 1, R L 1R 1) = (L 1, I, R L 1R 1). Hence, (G H, ) is a S 1 staipl i β AUT (H, ). Now, assume that β AUT (H, ). Then, xl 1 y = (xy)r L 1R 1 or all x, y H. For y = e, L 1 = R L 1R 1 in (H, ). so, β = (L 1, I, L 1) AUT (H, ) 1 N λ (H, ) N λ (H, ). On the other hand, i N λ (H, ), then, γ = (L, I, L ) AUT (H, ). But N λ (H, ) L 1 = L 1 = R L 1R 1 in (H, ). Hence, β = γ 1 and β AUT (H, ). Corollary 1. Let (G H, ) be a S 2 ndbl and a S 1 staipl. Then, or any special loop (G H, ) that is S-isotopic to (G H, ), (G H, ) is a S 1 staipl i (G H, ) is a S 1 st-loop and a S 1 st commutative loop. Proo. Suppose every special loop that is S-isotopic to (G H, ) is a S 1 staipl. Then, N λ (H, ) or all H by theorem 9. So, (G H, ) is a S 1 st-loop. Then, y 1 x 1 = (xy) 1 = x 1 y 1 or all x, y H. So, (G H, ) is a S 1 st commutative loop. The proo o the converse is as ollows. I (G H, ) is a S 1 st-loop and a S 1 st commutative loop, then or all x, y H such that x y = xr yl 1, (x y) z = (xr yl 1 )R zl 1 = (x 1 y) 1 z. x (y z) = xr (yr zl 1 )L 1 = x 1 (y 1 z). So, (x y) z = x (y z). Thus, (H, ) is a roup. Furthermore, x y = xr yl 1 = x 1 y = x y = y x = y 1 x = y x. So, (H, ) is commutative and so has the AIP. Thereore, (G H, ) is a S 1 staipl. Lemma 2. Let (G H, ) be a S 2 ndbl. Then, every special loop that is S-isotopic to (G H, ) is S-isomorphic to (G H, ) i and only i (G H, ) obeys the identity (x ) 1 \(y ) = (xy) () or all x, y G H and, H. Proo. Let (G H, ) be an arbitrary S-principal isotope o (G H, ). It is claimed that R (G H, ) (G H, ) i xr yr = (x y)r i (x )R 1 (y )L 1 = (x y)r i (x ) 1 \(y ) = (xy) () or all x, y G H and, H. Theorem 10. Let (G H, ) be a S 2 ndbl, let H, and let x y = xr yl 1 or all x, y G. Then, (G H, ) (G H, ) i and only i there exists a S 1 st pseudo-automorphism o (G H, ) with S 1 st companion. Proo. (G H, ) (G H, ) i and only i there exists T SSY M(G H, ) such that xt yt = (x y)t or all x, y G i xt R yt L 1 = (x y)t or all x, y G i α = (T R, T L 1, T ) S 1 staut (G H ).

11 92 Tèmító. pé. Gbó. láhàn Jaíyéọlá No. 1 Recall that by theorem 2, (G H, ) is a S 2 ndbl i (R 1, L R, R ) S 1 staut (G H, ) or each H. So, with S 1 st companion. α S 1 staut (G H ) β = α(r 1, L R, R ) = (T, T R, T R ) S 1 staut (G H, ) T S 1 stp AUT (G H ) Corollary 2. Let (G H, ) be a S 2 ndbl, let H and let x y = xr yl 1 or all x, y G H. I N ρ (H, ), then, (G H, ) (G H, ). Proo. Followin theorem 10, N ρ (H, ) T S 1 stp AUT (G H ) with S 1 st companion. Corollary 3. Let (G H, ) be a S 2 ndbl. Then, every special loop that is S-isotopic to (G H, ) is S-isomorphic to (G H, ) i and only i each element o H is a S 1 st companion or a S 1 st pseudo-automorphism o (G H, ). Proo. This ollows rom theorem 8 and theorem 10. Corollary 4. Let (G H, ) be a S 2 ndbl. Then, (G H, ) is a SGS-loop i and only i each element o H is a S 1 st companion or a S 1 st pseudo-automorphism o (G H, ). Proo. This is an immediate consequence o corollary 4. Remark 4. Every Bol loop is a S 2 ndbl. Most o the results on isotopy o Bol loops in chapter 3 o [19] can easily be deduced rom the results in this paper by simply orcin H to be equal to G. Reerences [1] R. H. Bruck, A survey o binary systems, Spriner-Verla, Berlin-Göttinen-Heidelber, [2]O. Chein, H. O. Pluelder and J. D. H. Smith, Quasiroups and Loops: Theory and Applications, Heldermann Verla, [3] J. Dene and A. D. Keedwell, Latin squares and their applications, the Enlish University press Lts, [4] E. G. Goodaire, E. Jespers and C. P. Milies, Alternative loop rins, NHMS (184), Elsevier, [5] T. G. Jaíyéọlá, An holomorphic study o the Smarandache concept in loops, Scientia Mana Journal, 2(2006), No. 1, 1-8. [6] T. G. Jaíyéọlá, Parastrophic invariance o Smarandache quasiroups, Scientia Mana Journal, 2(2006), No. 3, [7] T. G. Jaíyéọlá, On the universality o some Smarandache loops o Bol-Mouan type, Scientia Mana Journal, 2(2006), No. 4, [8] T. G. Jaíyéọlá, A Pair O Smarandachely Isotopic Quasiroups And Loops O The Same Variety, International Journal o Mathematical Combinatorics, 1(2008), [9] T. G. Jaíyéọlá, An Holomorphic Study O Smarandache Automorphic and Cross Inverse Property Loops, Proceedins o the 4 th International Conerence on Number Theory and Smarandache Problems, Scientia Mana, 4(2008), No. 1,

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