ON THE UNIVERSALITY OF SOME SMARANDACHE LOOPS OF BOL-MOUFANG TYPE

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1 arxiv:math/ v2 [math.gm] 8 Sep 2007 ON THE UNIVERSALITY OF SOME SMARANDACHE LOOPS OF BOL-MOUFANG TYPE Tèmítópé Gbóláhàn Jaíyéọlá Department o Mathematics, Obaemi Awolowo University, Ile Ie, Nieria. jaiyeolatemitope@yahoo.com, tjayeola@oauie.edu.n Abstract A Smarandache quasiroup(loop) is shown to be universal i all its, -principal isotopes are Smarandache, -principal isotopes. Also, weak Smarandache loops o Bol-Mouan type such as Smarandache: let(riht) Bol, Mouan and extra loops are shown to be universal i all their,-principal isotopes are Smarandache,principal isotopes. Conversely, it is shown that i these weak Smarandache loops o Bol-Mouan type are universal, then some autotopisms are true in the weak Smarandache sub-loops o the weak Smarandache loops o Bol-Mouan type relative to some Smarandache Futhermore, a Smarandache let(riht) inverse property loop in which all its, -principal isotopes are Smarandache, -principal isotopes is shown to be universal i and only i it is a Smarandache let(riht) Bol loop in which all its, -principal isotopes are Smarandache, -principal isotopes. Also, it is established that a Smarandache inverse property loop in which all its, -principal isotopes are Smarandache, -principal isotopes is universal i and only i it is a Smarandache Mouan loop in which all its, -principal isotopes are Smarandache, -principal isotopes. Hence, some o the autotopisms earlier mentioned are ound to be true in the Smarandache sub-loops o universal Smarandache: let(riht) inverse property loops and inverse property loops. 1 Introduction W. B. Vasantha Kandasamy initiated the study o Smarandache loops (S-loop) in In her book [27], she deined a Smarandache loop (S-loop) as a loop with at least a subloop 2000 Mathematics Subject Classiication. Primary 20NO5 ; Secondary 08A05. Keywords and Phrases : Smarandache quasiroups, Smarandache loops, universality,, -principal isotopes On Doctorate Proramme at the University o Ariculture Abeokuta, Nieria. All correspondence to be addressed to this author 1

2 which orms a subroup under the binary operation o the loop called a Smarandache subloop (S-subloop). In [11], the present author deined a Smarandache quasiroup (S-quasiroup) to be a quasiroup with at least a non-trivial associative subquasiroup called a Smarandache subquasiroup (S-subquasiroup). Examples o Smarandache quasiroups are iven in Muktibodh [21]. For more on quasiroups, loops and their properties, readers should check [24], [2],[4], [5], [8] and [27]. In her (W.B. Vasantha Kandasamy) irst paper [28], she introduced Smarandache : let(riht) alternative loops, Bol loops, Mouan loops, and Bruck loops. But in [10], the present author introduced Smarandache : inverse property loops (IPL), weak inverse property loops (WIPL), G-loops, conjuacy closed loops (CCloop), central loops, extra loops, A-loops, K-loops, Bruck loops, Kikkawa loops, Burn loops and homoeneous loops. The isotopic invariance o types and varieties o quasiroups and loops described by one or more equivalent identities, especially those that all in the class o Bol-Mouan type loops as irst named by Fenyves [7] and [6] in the 1960s and later on in this 21 st century by Phillips and Vojtěchovský [25], [26] and [18] have been o interest to researchers in loop theory in the recent past. For example, loops such as Bol loops, Mouan loops, central loops and extra loops are the most popular loops o Bol-Mouan type whose isotopic invariance have been considered. Their identities relative to quasiroups and loops have also been investiated by Kunen [20] and [19]. A loop is said to be universal relative to a property P i it is isotopic invariant relative to P, hence such a loop is called a universal P loop. This lanuae is well used in [22]. The universality o most loops o Bol-Mouan types have been studied as summarised in [24]. Let(Riht) Bol loops, Mouan loops, and extra loops have all been ound to be isotopic invariant. But some types o central loops were shown to be universal in Jaíyéọlá [13] and [12] under some conditions. Some other types o loops such as A-loops, weak inverse property loops and cross inverse property loops (CIPL) have been ound be universal under some neccessary and suicient conditions in [3], [23] and [1] respectively. Recently, Michael Kinyon et. al. [16], [14], [15] solved the Belousov problem concernin the universality o F-quasiroups which has been open since 1967 by showin that all the isotopes o F-quasiroups are Mouan loops. In this work, the universality o the Smarandache concept in loops is investiated. That is, will all isotopes o an S-loop be an S-loop? The answer to this could be yes since every isotope o a roup is a roup (roups are G-loops). Also, the universality o weak Smarandache loops, such as Smarandache Bol loops (SBL), Smarandache Mouan loops (SML) and Smarandache extra loops (SEL) will also be investiated despite the act that it could be expected to be true since Bol loops, Mouan loops and extra loops are universal. The universality o a Smarandache inverse property loop (SIPL) will also be considered. 2 Preliminaries Deinition 2.1 A loop is called a Smarandache let inverse property loop (SLIPL) i it has at least a non-trivial subloop with the LIP. A loop is called a Smarandache riht inverse property loop (SRIPL) i it has at least a non-trivial subloop with the RIP. 2

3 A loop is called a Smarandache inverse property loop (SIPL) i it has at least a non-trivial subloop with the IP. A loop is called a Smarandache riht Bol-loop (SRBL) i it has at least a non-trivial subloop that is a riht Bol(RB)-loop. A loop is called a Smarandache let Bol-loop (SLBL) i it has at least a non-trivial subloop that is a let Bol(LB)-loop. A loop is called a Smarandache central-loop (SCL) i it has at least a non-trivial subloop that is a central-loop. A loop is called a Smarandache extra-loop (SEL) i it has at least a non-trivial subloop that is a extra-loop. A loop is called a Smarandache A-loop (SAL) i it has at least a non-trivial subloop that is a A-loop. A loop is called a Smarandache Mouan-loop (SML) i it has at least a non-trivial subloop that is a Mouan-loop. Deinition 2.2 Let (G, ) and (H, ) be two distinct quasiroups. The triple (A, B, C) such that A, B, C : (G, ) (H, ) are bijections is said to be an isotopism i and only i xa yb = (x y)c x, y G. H is called an isotope o G and they are said to be isotopic. I C = I, then the triple is called a principal isotopism and (H, ) = (G, ) is called a principal isotope o (G, ). I in addition, A = R, B = L, then the triple is called an, -principal isotopism, thus (G, ) is reered to as the, -principal isotope o (G, ). A subloop(subquasiroup) (S, ) o a loop(quasiroup) (G, ) is called a Smarandache, -principal isotope o the subloop(subquasiroup) (S, ) o a loop(quasiroup) (G, ) i or some, S, xr yl = (x y) x, y S. On the other hand (G, ) is called a Smarandache, -principal isotope o (G, ) i or some, S, xr yl = (x y) x, y G (S, ) is a S-subquasiroup(S-subloop) o (G, ). In these cases, and are called Smarandache elements(s-elements). Theorem 2.1 ([2]) Let (G, ) and (H, ) be two distinct isotopic loops(quasiroups). There exists an, -principal isotope (G, ) o (G, ) such that (H, ) = (G, ). Corollary 2.1 Let P be an isotopic invariant property in loops(quasiroups). I (G, ) is a loop(quasiroup) with the property P, then (G, ) is a universal loop(quasiroup) relative to the property P i and only i every, -principal isotope (G, ) o (G, ) has the property P. 3

4 Proo I (G, ) is a universal loop relative to the property P then every distinct loop isotope (H, ) o (G, ) has the property P. By Theorem 2.1, there exists an, -principal isotope (G, ) o (G, ) such that (H, ) = (G, ). Hence, since P is an isomorphic invariant property, every (G, ) has it. Conversely, i every, -principal isotope (G, ) o (G, ) has the property P and since by Theorem 2.1 or each distinct isotope (H, ) there exists an, -principal isotope (G, ) o (G, ) such that (H, ) = (G, ), then all (H, ) has the property, (G, ) is a universal loop relative to the property P. Lemma 2.1 Let (G, ) be a loop(quasiroup) with a subloop(subquasiroup) (S, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then (S, ) is a subloop(subquasiroup) o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Proo I (S, ) is a Smarandache, -principal isotope o (S, ), then or some, S, xr yl = (x y) x, y S S x, y S since, S. So, (S, ) is a subroupoid o (G, ). (S, ) is a subquasiroup ollows rom the act that (S, ) is a subquasiroup. is a two sided identity element in (S, ). (S, ) is a subloop o (G, ). 3 Main Results Universality o Smarandache Loops Theorem 3.1 A Smarandache quasiroup is universal i all its, -principal isotopes are Smarandache, -principal isotopes. Proo Let (G, ) be a Smarandache quasiroup with a S-subquasiroup (S, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subquasiroup o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, It shall now be shown that (x y) z = x (y z) x, y, z S. But in the quasiroup (G, ), xy will have preerence over x y x, y G. (x y) z = (xr 1 ) z = (x 1 1 y) z = (x 1 1 y)r 4

5 x (y z) = x (yr 1 = (x 1 1 y) 1 1 z = x 1 1 y 1 1 z. zl 1 ) = x (y 1 1 z) = xr 1 (y 1 1 z)l 1 = x 1 1 (y 1 1 z) = x 1 1 y 1 1 z. (S, ) is an S-subquasiroup o (G, ) hence, (G, ) is a S-quasiroup. By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) which will now be an S-subquasiroup in (H, ). So, (H, ) is an S-quasiroup. This conclusion can also be drawn straiht rom Corollary 2.1. Theorem 3.2 A Smarandache loop is universal i all its, -principal isotopes are Smarandache, -principal isotopes. Conversely, i a Smarandache loop is universal then (I, L R 1 R ρl 1, R 1 R ρ) is an autotopism o an S-subloop o the S-loop such that and are S- Proo Every loop is a quasiroup. Hence, the irst claim ollows rom Theorem 3.1. The proo o the converse is as ollows. I a Smarandache loop (G, ) is universal then every isotope (H, ) is an S-loop i.e there exists an S-subloop (S, ) in (H, ). Let (G, ) be the, -principal isotope o (G, ), then by Corollary 2.1, (G, ) is an S-loop with say an S-subloop (S, ). So, (x y) z = x (y z) x, y, z S Replacin xr 1 (xr 1 by x, yl 1 )R 1 zl 1 = xr 1 (yr )L 1. by y and takin z = e in (S, ) we have; (x y )R 1 R ρ = x y L R 1 R ρl 1 (I, L R 1 R ρl 1, R 1 R ρ) is an autotopism o an S-subloop (S, ) o the S-loop (G, ) such that and are S- Universality o Smarandache Bol, Mouan and Extra Loops Theorem 3.3 A Smarandache riht(let)bol loop is universal i all its, -principal isotopes are Smarandache, -principal isotopes. Conversely, i a Smarandache riht(let)bol loop is universal then ( ) T 1 = (R R 1, L ρ λr 1 R ρl 1, R 1 R ρ) T 2 = (R ρl 1 L λr 1, L L 1, L 1 λ L λ) is an autotopism o an SRB(SLB)-subloop o the SRBL(SLBL) such that and are S- 5

6 Proo Let (G, ) be a SRBL(SLBL) with a S-RB(LB)-subloop (S, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, It is already known rom [24] that RB(LB) loops are universal, hence (S, ) is a RB(LB) loop thus an S-RB(LB)-subloop o (G, ). By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) which will now be an S-RB(LB)-subloop in (H, ). So, (H, ) is an SRBL(SLBL). This conclusion can also be drawn straiht rom Corollary 2.1. The proo o the converse is as ollows. I a SRBL(SLBL) (G, ) is universal then every isotope (H, ) is an SRBL(SLBL) i.e there exists an S-RB(LB)-subloop (S, ) in (H, ). Let (G, ) be the, -principal isotope o (G, ), then by Corollary 2.1, (G, ) is an SRBL(SLBL) with say an SRB(SLB)-subloop (S, ). So or an SRB-subloop (S, ), [(y x) z] x = y [(x z) x] x, y, z S [(yr 1 xl 1 )R 1 zl 1 ]R 1 xl 1 = yr 1 [(xr )R 1 xl 1 ]L 1. Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have (y R ρr 1 z )R 1 R ρ = y z L λr 1 R ρl 1. Aain, replace y R ρr 1 by y so that (y z )R 1 R ρ = y R R 1 ρ z L λr 1 R ρl 1 (R R 1, L ρ λr 1 R ρl 1, R 1 R ρ) is an autotopism o an SRB-subloop (S, ) o the S-loop (G, ) such that and are S- On the other hand, or a SLB-subloop (S, ), [x (y x)] z = x [y (x z)] x, y, z S [xr 1 (yr 1 xl 1 )L 1 ]R 1 zl 1 = xr 1 [yr 1 (xr )L 1 ]L 1. 6

7 Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have y R ρl 1 L λr 1 z = (y z L λl 1 )L 1 L λ. Aain, replace z L λl 1 by z so that y R ρl 1 L λr 1 z L L 1 = (y z )L 1 λ L (R λ ρl 1 L λr 1, L L 1, L 1 λ L λ) is an autotopism o an SLB-subloop (S, ) o the S-loop (G, ) such that and are S- Theorem 3.4 A Smarandache Mouan loop is universal i all its, -principal isotopes are Smarandache, -principal isotopes. Conversely, i a Smarandache Mouan loop is universal then (R L 1 L λr 1, L R 1 R ρl 1, L 1 L λr 1 R ρ), (R L 1 L λr 1, L R 1 R ρl 1, R 1 R ρl 1 L λ), (R L 1 L λr 1 R ρr 1, L L 1 λ, L 1 L λ), (R R 1 ρ, L R 1 R ρl 1 L λl 1, R 1 R ρ), (R L 1 L λr 1, L λr 1 R ρl 1 λ, R 1 R ρl 1 L λ), (R ρl 1 L λr 1 ρ, L R 1 R ρl 1, L 1 L λr 1 R ρ) are autotopisms o an SM-subloop o the SML such that and are S- Proo Let (G, ) be a SML with a SM-subloop (S, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, It is already known rom [24] that Mouan loops are universal, hence (S, ) is a Mouan loop thus an SM-subloop o (G, ). By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) which will now be an SM-subloop in (H, ). So, (H, ) is an SML. This conclusion can also be drawn straiht rom Corollary 2.1. The proo o the converse is as ollows. I a SML (G, ) is universal then every isotope (H, ) is an SML i.e there exists an SM-subloop (S, ) in (H, ). Let (G, ) be the, - principal isotope o (G, ), then by Corollary 2.1, (G, ) is an SML with say an SM-subloop (S, ). For an SM-subloop (S, ), (x y) (z x) = [x (y z)] x x, y, z S (xr 1 )R 1 (zr 1 xl 1 )L 1 = [xr 1 (yr )L 1 ]R 1 xl 1. 7

8 Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have y R L 1 L λr 1 z L R 1 R ρl 1 = (y z )L 1 L λr 1 R ρ (R L 1 L λr 1, L R 1 R ρl 1, L 1 L λr 1 R ρ) is an autotopism o an SM-subloop (S, ) o the S-loop (G, ) such that and are S- Aain, or an SM-subloop (S, ), (x y) (z x) = x [(y z) x] x, y, z S (xr 1 )R 1 (zr 1 xl 1 )L 1 = xr 1 [(yr )R 1 xl 1 ]L 1. Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have y R L 1 L λr 1 z L R 1 R ρl 1 = (y z )R 1 R ρl 1 L λ (R L 1 L λr 1, L R 1 R ρl 1, R 1 R ρl 1 L λ) is an autotopism o an SM-subloop (S, ) o the S-loop (G, ) such that and are S- Also, i (S, ) is an SM-subloop then, [(x y) x] z = x [y (x z)] x, y, z S [(xr 1 )R 1 xl 1 ]R 1 zl 1 = xr 1 [yr 1 (xr )L 1 ]L 1. Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have y R L 1 L λr 1 R ρr 1 z = (y z L λl 1 )L 1 L λ. Aain, replace z L λl 1 by z so that y R L 1 L λr 1 R ρr 1 z L L 1 λ = (y z )L 1 L λ (R L 1 L λr 1 R ρr 1, L L 1 λ, L 1 L λ) is an autotopism o an SM-subloop (S, ) o the S-loop (G, ) such that and are S- 8

9 Furthermore, i (S, ) is an SM-subloop then, [(y x) z] x = y [x (z x)] x, y, z S [(yr 1 xl 1 )R 1 zl 1 ]R 1 xl 1 = yr 1 [xr 1 (zr 1 xl 1 )L 1 ]L 1. Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have (y R ρr 1 z )R 1 R ρ = y z L R 1 R ρl 1 L λl 1. Aain, replace y R ρr 1 by y so that (y z )R 1 R ρ = y R R 1 ρ z L R 1 R ρl 1 L λl 1 (R R 1, L R 1 ρ R ρl 1 L λl 1, R 1 R ρ) is an autotopism o an SM-subloop (S, ) o the S-loop (G, ) such that and are S- Lastly, (S, ) is an SM-subloop i and only i (S, ) is an SRB-subloop and an SLBsubloop. So by Theorem 3.3, T 1 and T 2 are autotopisms in (S, ), hence T 1 T 2 and T 2 T 1 are autotopisms in (S, ). Theorem 3.5 A Smarandache extra loop is universal i all its, -principal isotopes are Smarandache, -principal isotopes. Conversely, i a Smarandache extra loop is universal then (R L 1 L λr 1, L R 1 R L 1 ρ, L 1 L λr 1 R ), ρ (R R 1 ρ R L 1 L λr 1, L λl 1, L 1 L λ), (R ρr 1, L L 1 λ L R 1 R ρl 1, R 1 R ρ) (R L 1 L λr 1, L R 1 R ρl 1, L 1 L λr 1 R ρ), (R L 1 L λr 1, L R 1 R ρl 1, R 1 R ρl 1 L λ), (R L 1 L λr 1 R ρr 1, L L 1 λ, L 1 L λ), (R R 1 ρ, L R 1 R ρl 1 L λl 1, R 1 R ρ), (R L 1 L λr 1, L λr 1 R ρl 1 λ, R 1 R ρl 1 L λ), (R ρl 1 L λr 1 ρ, L R 1 R ρl 1, L 1 L λr 1 R ρ), are autotopisms o an SE-subloop o the SEL such that and are S- Proo Let (G, ) be a SEL with a SE-subloop (S, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, - principal isotope o (S, ). Let us choose all (S, ) in this manner. So, In [9] and [17] respectively, it was shown and stated that a loop is an extra loop i and only i it is a Mouan loop and a CC-loop. But since CC-loops are G-loops(they are isomorphic 9

10 to all loop isotopes) then extra loops are universal, hence (S, ) is an extra loop thus an SE-subloop o (G, ). By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) which will now be an SE-subloop in (H, ). So, (H, ) is an SEL. This conclusion can also be drawn straiht rom Corollary 2.1. The proo o the converse is as ollows. I a SEL (G, ) is universal then every isotope (H, ) is an SEL i.e there exists an SE-subloop (S, ) in (H, ). Let (G, ) be the, - principal isotope o (G, ), then by Corollary 2.1, (G, ) is an SEL with say an SE-subloop (S, ). For an SE-subloop (S, ), [(x y) z] x = x [y (z x)] x, y, z S [(xr 1 )R 1 zl 1 ]R 1 xl 1 = xr 1 [yr 1 (zr 1 xl 1 )L 1 ]L 1. Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have (y R L 1 L λr 1 z )R 1 R ρ = (y z L R 1 R ρl 1 )L 1 L λ. Aain, replace z L R 1 R ρl 1 by z so that y R L 1 L λr 1 z L R 1 R L 1 ρ = (y z )L 1 L λr 1 R ρ (R L 1 L λr 1, L R 1 ρ R L 1, L 1 L λr 1 ρ R ) is an autotopism o an SE-subloop (S, ) o the S-loop (G, ) such that and are S- Aain, or an SE-subloop (S, ), (x y) (x z) = x [(y x) z] x, y, z S (xr 1 )R 1 (xr )L 1 = xr 1 [(yr 1 xl 1 )R 1 zl 1 ]L 1. Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have y R L 1 L λr 1 z L λl 1 = (y R ρr 1 z )L 1 L λ. Aain, replace y R ρr 1 by y so that y R R 1 R L 1 ρ L λr 1 z L λl 1 = (y z )L 1 L (R R 1 λ R L 1 ρ L λr 1, L λl 1, L 1 L λ) 10

11 is an autotopism o an SE-subloop (S, ) o the S-loop (G, ) such that and are S- Also, i (S, ) is an SE-subloop then, (y x) (z x) = [y (x z)] x x, y, z S (yr 1 xl 1 )R 1 (zr 1 xl 1 )L 1 = [(yr 1 (xr )L 1 ]R 1 xl 1. Replacin yr 1 by y, zl 1 by z and takin x = e in (S, ) we have y R ρr 1 z L R 1 R ρl 1 = (y z L λl 1 )R 1 R ρ. Aain, replace z L λl 1 by z so that y R ρr 1 z L L 1 L λ R 1 R ρl 1 = (y z )R 1 R ρ (R ρr 1, L L 1 L λ R 1 R ρl 1, R 1 R ρ) is an autotopism o an SE-subloop (S, ) o the S-loop (G, ) such that and are S- Lastly, (S, ) is an SE-subloop i and only i (S, ) is an SM-subloop and an SCC-subloop. So by Theorem 3.4, the six remainin triples are autotopisms in (S, ). Universality o Smarandache Inverse Property Loops Theorem 3.6 A Smarandache let(riht) inverse property loop in which all its, -principal isotopes are Smarandache, -principal isotopes is universal i and only i it is a Smarandache let(riht) Bol loop in which all its, -principal isotopes are Smarandache, - principal isotopes. Proo Let (G, ) be a SLIPL with a SLIP-subloop (S, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, (G, ) is a universal SLIPL i and only i every isotope (H, ) is a SLIPL. (H, ) is a SLIPL i and only i it has at least a SLIP-subloop (S, ). By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) to be (S, ) which is already a SLIP-subloop in (H, ). So, (S, ) is also a SLIP-subloop in (G, ). As shown in [24], (S, ) and its, -isotope(smarandache, - isotope) (S, ) are SLIP-subloops i and only i (S, ) is a let Bol subloop(i.e a SLB-subloop). So, (G, ) is SLBL. 11

12 Conversely, i (G, ) is SLBL, then there exists a SLB-subloop (S, ) in (G, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) to be (S, ) which is a SLB-subloop in (H, ) usin the same reasonin in Theorem 3.3. So, (S, ) is a SLB-subloop in (G, ). Let Bol loops have the let inverse property(lip), hence, (S, ) and (S, ) are SLIP-subloops in (G, ) and (G, ) respectively. Thence, (G, ) and (G, ) are SLBLs. Thereore, (G, ) is a universal SLIPL. The proo or a Smarandache riht inverse property loop is similar and is as ollows. Let (G, ) be a SRIPL with a SRIP-subloop (S, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, (G, ) is a universal SRIPL i and only i every isotope (H, ) is a SRIPL. (H, ) is a SRIPL i and only i it has at least a SRIP-subloop (S, ). By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) to be (S, ) which is already a SRIP-subloop in (H, ). So, (S, ) is also a SRIP-subloop in (G, ). As shown in [24], (S, ) and its, -isotope(smarandache, -isotope) (S, ) are SRIP-subloops i and only i (S, ) is a riht Bol subloop(i.e a SRBsubloop). So, (G, ) is SRBL. Conversely, i (G, ) is SRBL, then there exists a SRB-subloop (S, ) in (G, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) to be (S, ) which is a SRBsubloop in (H, ) usin the same reasonin in Theorem 3.3. So, (S, ) is a SRB-subloop in (G, ). Riht Bol loops have the riht inverse property(rip), hence, (S, ) and (S, ) are SRIP-subloops in (G, ) and (G, ) respectively. Thence, (G, ) and (G, ) are SRBLs. Thereore, (G, ) is a universal SRIPL. Theorem 3.7 A Smarandache inverse property loop in which all its, -principal isotopes are Smarandache, -principal isotopes is universal i and only i it is a Smarandache Mouan loop in which all its, -principal isotopes are Smarandache, -principal isotopes. Proo Let (G, ) be a SIPL with a SIP-subloop (S, ). I (G, ) is an arbitrary, -principal 12

13 isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, (G, ) is a universal SIPL i and only i every isotope (H, ) is a SIPL. (H, ) is a SIPL i and only i it has at least a SIP-subloop (S, ). By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) to be (S, ) which is already a SIP-subloop in (H, ). So, (S, ) is also a SIPsubloop in (G, ). As shown in [24], (S, ) and its, -isotope(smarandache, -isotope) (S, ) are SIP-subloops i and only i (S, ) is a Mouan subloop(i.e a SM-subloop). So, (G, ) is SML. Conversely, i (G, ) is SML, then there exists a SM-subloop (S, ) in (G, ). I (G, ) is an arbitrary, -principal isotope o (G, ), then by Lemma 2.1, (S, ) is a subloop o (G, ) i (S, ) is a Smarandache, -principal isotope o (S, ). Let us choose all (S, ) in this manner. So, By Theorem 2.1, or any isotope (H, ) o (G, ), there exists a (G, ) such that (H, ) = (G, ). So we can now choose the isomorphic imae o (S, ) to be (S, ) which is a SMsubloop in (H, ) usin the same reasonin in Theorem 3.3. So, (S, ) is a SM-subloop in (G, ). Mouan loops have the inverse property(ip), hence, (S, ) and (S, ) are SIPsubloops in (G, ) and (G, ) respectively. Thence, (G, ) and (G, ) are SMLs. Thereore, (G, ) is a universal SIPL. Corollary 3.1 I a Smarandache let(riht) inverse property loop is universal then ( ) (R R 1, L ρ λr 1 R ρl 1, R 1 R ρ) (R ρl 1 L λr 1, L L 1, L 1 λ L λ) is an autotopism o an SLIP(SRIP)-subloop o the SLIPL(SRIPL) such that and are S- Proo This ollows by Theorem 3.6 and Theorem 3.1. Corollary 3.2 I a Smarandache inverse property loop is universal then (R L 1 L λr 1, L R 1 R ρl 1, L 1 L λr 1 R ρ), (R L 1 L λr 1, L R 1 R ρl 1, R 1 R ρl 1 L λ), (R L 1 L λr 1 R ρr 1, L L 1 λ, L 1 L λ), (R R 1 ρ, L R 1 R ρl 1 L λl 1, R 1 R ρ), (R L 1 L λr 1, L λr 1 R ρl 1 λ, R 1 R ρl 1 L λ), (R ρl 1 L λr 1 ρ, L R 1 R ρl 1, L 1 L λr 1 R ρ) are autotopisms o an SIP-subloop o the SIPL such that and are S- Proo This ollows rom Theorem 3.7 and Theorem

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