# An holomorphic study of Smarandache automorphic and cross inverse property loops.

By studying the holomorphic structure of automorphic inverse property quasigroups and loops[AIPQ and (AIPL)] and cross inverse property quasigroups and loops[CIPQ and (CIPL)], it is established that the holomorph of a loop is a Smarandache; AIPL, CIPL, Kloop, Bruck-loop or Kikkawa-loop if and only if its Smarandache automorphism group is trivial and the loop is itself is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop.

Keywords Smarandache loop, holomorph of loop, automorphic inverse property loop (AIPL), cross inverse property loop(CIPL), K-loop, Bruck-loop, Kikkawa-loop.

[section] 1. Introduction

1.1 Quasigroups and loops

Let L be a non-empty set. Define a binary operation (.) on L : If x * [member of] L for all x, y [member of] L, (L, *) is called a groupoid. If the system of equations

a*x=b and y*a=b

have unique solutions for x and y respectively, then (L,) is called a quasigroup. 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. Now, if there exists a unique element e [member of] L called the identity element such that for all x [member of] L, x * e = e * x = x, (L, *) is called a loop. To every loop (L, *) with automorphism group AU M (L, *), there corresponds another loop. Let the set H = (L, *) X AU M(L, *). If we define 'o' on H such that ([alpha], x) o ([beta], y) = ([alpha][beta], x[beta], * y) for all ([alpha], x), ([beta], y)[member of] H, then H(L, *) = (H, o) is a loop as shown in Bruck  and is called the Holomorph of (L, *). A loop (quasigroup) is a weak inverse property loop (quasigroup)[WIPL(WIPQ)] if and only if it obeys the identity

x[(yx).sup.P] = [y.sup.P] or [(xy).sub.[lambda]]X = [y.sup.[lambda]].

A loop (quasigroup) is a cross inverse property loop (quasigroup) [CIPL(CIPQ)] if and only if it obeys the identity

xy*[x.sup.p] = y or x*y[x.sup.P] = y or [x.sup.[lambda]] * (yx) = y or [x.sup[lambda]y * x = y.

A loop (quasigroup) is an automorphic inverse property loop (quasigroup) [AIPL (AIPQ) ] if and only if it obeys the identity

[(xy).sup.P]=[x.sup.p][y.sup.p] or [(xy).sup.[lambda]]=[x.sup.[lambda]][y.sup.[lambda]]

Consider (G, *) and (H, o) being two distinct groupoids(quasigroups, loops). Let A, B and C be three distinct non-equal bijective mappings, that maps G onto H. The triple [alpha] = (A, B, C) is called an isotopism of (G, *) onto (H, o) if and only if

xA o yB = (x * y)C [for all] x, y [member of] G.

The set SYM(G, *) = SY M(G) of all bijections in a groupoid (G, *) forms a group called the permutation (symmetric) group of the groupoid (G, *). If (G, *) (H, o), then the triple [alpha] = (A, B, C) of bijections on (G, *) is called an autotopism of the groupoid(quasigroup, loop) (G, *). Such triples form a group AUT(G, *) called the autotopism group of (G, *). Furthermore, if A = B = C, then A is called an automorphism of the groupoid(quasigroup, loop) (G, *). Such bijections form a group AU M(G, *) called the automorphism group of (G, *).

The left nucleus of L denoted by [N.sub.lambda] (L, *) = {a [member of] L : ax * y = a * xy [for all] x, y [member of] L}. The right nucleus of L denoted by [N.sub.p](L, *) = {a [member of] L : y * xa = yx * a [for all] x, y [member of] L}. The middle nucleus of L denoted by [N.sub.[micro] (L, *) = {a [member of] L : ya * x = y * ax [for all] x, y [member of] L}. The nucleus of L denoted by N(L, *) = [N.sub.[lambda]](L, *)[intersection] [N.sub.p](L, *) [intersection] [N.sub.[micro]] (L, *). The centrum of L denoted by C(L, *) = {a [member of] L : ax = xa [intersection] x [member of] L}. The center of L denoted by Z(L, *) = N(L, *) [intersection] C(L, *).

As observed by Osborn , a loop is a WIPL and an AIPL if and only if it is a CIPL. The past efforts of Artzy , ,  and , Belousov and Tzurkan  and recent studies of Keedwell , Keedwell and Shcherbacov , and  are of great significance in the study of WIPLs, AIPLs, CIPQs and CIPLs, their generalizations (1.e m-inverse loops and quasigroups, (r,s,t)-inverse quasigroups) and applications to cryptography. For more on loops and their properties, readers should check , , , ,  and .

Interestingly, Adeniran  and Robinson , Oyebo and Adeniran , Chiboka and Solarin , Bruck , Bruck and Paige , Robinson , Huthnance  and Adeniran  have respectively studied the holomorphs of Bol loops, central loops, conjugacy closed loops, inverse property loops, A-loops, extra loops, weak inverse property loops, Osborn loops and Bruck loops. Huthnance  showed that if (L, *) is a loop with holomorph (H, o), (L, -) is a WIPL if and only if (H, o) is a WIPL. The holomorphs of an AIPL and a CIPL are yet to be studied.

For the definitions of inverse property loop (IPL), Bol loop and A--loop readers can check earlier references on loop theory.

Here, a K--loop is an A--loop with the AIP, a Bruck loop is a Bol loop with the AIP and a Kikkawa loop is an A--loop with the IP and AIR

1.2 Smarandache quasigroups and loops

The study of Smarandache loops was initiated by W. B. Vasantha Kandasamy in 2002. In her book , she defined a Smarandache loop (S--loop) as a loop with at least a subloop which forms a subgroup under the binary operation of the loop. In , the present author defined a Smarandache quasigroup (S--quasigroup) to be a quasigroup with at least a non-trivial associative subquasigroup called a Smarandache subsemigroup (S--subsemigroup). Examples of Smarandache quasigroups are given in Muktibodh . In her book, she introduced over 75 Smarandache concepts on loops. In her first paper , on the study of Smarandache notions in algebraic structures, she introduced Smarandache : left(right) alternative loops, Bol loops, Moufang loops, and Bruck loops. But in , the present author introduced Smarandache inverse property loops (IPL), weak inverse property loops (WIPL), G--loops, conjugacy closed loops (CC--loop), central loops, extra loops, A--loops, K--loops, Bruck loops, Kikkawa loops, Burn loops and homogeneous loops.

A loop is called a Smarandache A--loop(SAL) if it has at least a non-trivial subloop that is a A--loop.

A loop is called a Smarandache K--loop(SKL) if it has at least a non-trivial subloop that is a K--loop.

A loop is called a Smarandache Bruck--loop(SBRL) if it has at least a non-trivial subloop that is a Bruck--loop.

A loop is called a Smarandache Kikkawa--loop(SKWL) if it has at least a non-trivial subloop that is a Kikkawa--loop.

If L is a S--groupoid with a S-subsemigroup H, then the set SSYM(L, *) = SSYM(L) of all bijections A in L such that A : H [right arrow] H forms a group called the Smarandache permutation(symmetric) group of the S--groupoid. In fact, SSYM(L) [less than or equal to] SY M(L).

The left Smarandache nucleus of L denoted by S[N.sub.[lambda]]A(L, *) = [N.sub.[lambda]] (L, *) [intersection] H. The right Smarandache nucleus of L denoted by S[N.sub.p](L, *) = [N.sub.p](L, *) [intersection] H. The middle Smarandache nucleus of L denoted by S[N.sub.[micro] (L, *) = [N.sub.p] (L, *) [intersection] H. The Smarandache nucleus of L denoted by SN(L, *) = N(L, *) [intersection] H. The Smarandache centrum of L denoted by SC(L, *) = C(L, *) [intersection] H. The Smarandache center of L denoted by SZ(L, *) = Z(L, -) [intersection] H.

Definition 1.1. Let (L, *) and (G, o) be two distinct groupoids that are isotopic under a triple (U, V, W). Now, if (L, *) and (G, o) are S--groupoids with S-subsemigroups L' and G' respectively such that A : L' [right arrow] G', where A [member of] {U, V, W}, then the isotopism (U, V, W): (L, *) [right arrow] (G, o) is called a Smarandache isotopism(S--isotopism).

Thus, if U = V = W, then U is called a Smarandache isomorphism, hence we write (L, *) [?] (G, o).

But if (L, *) = (G, o), then the autotopism (U, V, W) is called a Smarandache autotopism (S--autotopism) and they form a group SAUT (L, *) which will be called the Smarandache autotopism group of (L, *). Observe that SAUT(L, *) [less than or equal to] AUT(L, *). Furthermore, if U = V = W, then U is called a Smarandache automorphism of (L, *). Such Smarandache permutations form a group SAU M(L, *) called the Smarandache automorphism group(SAG) of (L, *).

Let L be a S--quasigroup with a S--subgroup G. Now, set [H.sub.S] = (G, *) x SAUM(L, *). If we define 'o' on [H.sub.S] such that ([alpha], x) o ([beta], y) ([alpha][beta], x[beta] * y) for all ([alpha], x), ([beta], y) [member of] [H.sub.s], then [H.sub.S](L, *) ([H.sub.S], o) is a quasigroup.

If in L, [s.sup.[lambda]] * s[alpha] [member of] SN(L) or s[alpha] * [s.sup.P] [member of] SN(L) [for all]s [member of] G and [alpha] [member of] SAU M(L, *), ([H.sub.S], o) is called a Smarandache Nuclear--holomorph of L, if [s.sup.[lambda]]. s[alpha] [member of] SC(L) or s[alpha] * [s.sup.P] [member of] SC(L) [for all] s [member of] G and [alpha] [member of] SAU M(L, *), ([H.sub.S], o) is called a Smarandache Centrum--holomorph of L hence a Smarandache Central--holomorph if [s.sup.[lambda]] * s[alpha] [member of] SZ(L) or s[alpha] * [s.sup.P] [member of] SZ(L) [for all] s [member of] G and [alpha] [member of] SAU M(L, *).

The aim of the present study is to investigate the holomorphic structure of Smarandache AIPLs and CIPLs(SCIPLs and SAIPLs) and use the results to draw conclusions for Smarandache K--loops(SKLs), Smarandache Bruck--loops(SBRLs) and Smarandache Kikkawaloops (SKWLs). This is done as follows.

1. The holomorphic structure of AIPQs(AIPLs) and CIPQs(CIPLs) are investigated. Necessary and sufficient conditions for the holomorph of a quasigroup(loop) to be an AIPQ(AIPL) or CIPQ(CIPL) are established. It is shown that if the holomorph of a quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL), then the holomorph is isomorphic to the quasigroup(loop). Hence, the holomorph of a quasigroup(loop) is an AIPQ(AIPL) or CIPQ(CIPL) if and only if its automorphism group is trivial and the quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL). Furthermore, it is discovered that if the holomorph of a quasigroup(loop) is a CIPQ(CIPL), then the quasigroup(loop) is a flexible unipotent CIPQ(flexible CIPL of exponent 2).

2. The holomorph of a loop is shown to be a SAIPL, SCIPL, SKL, SBRL or SKWL respectively if and only its SAG is trivial and the loop is a SAIPL, SCIPL, SKL, SBRL, SKWL respectively.

[section] 2. Main results

Theorem 2.1. Let (L,*) be a quasigroup(loop) with holomorph H(L). H(L) is an AIPQ(AIPL) if and only if

1. AUM(L) is an abelian group,

2. ([[beta].sup.-1], [alpha], I) [member of] AUT(L) [for all] [alpha], [beta] [member of] AUM(L) and

3. L is a AIPQ(AIPL).

Proof. A quasigroup(loop) is an automorphic inverse property loop(AIPL) if and only if it obeys the AIP identity.

Using either of the definitions of an AIPQ(AIPL), it can be shown that H(L) is a AIPQ(AIPL) if and only if AUM(L) is an abelian group and ([[beta].sup.-1], [J.sub.[rho]],[alpha] [J.sub.[rho]])[J.sub.[rho]] [member of] AUT(L) [for all] [alpha], [beta] [member of] AUM(L). L is isomorphic to a subquasigroup(subloop) of H(L), so L is a AIPQ(AIPL) which implies ([J.sub.[rho]], [J.sub.[rho]], [J.sub.[rho]]) [meber of] AUT(L). So, ([[beta].sup.-1], [alpha],I) [meber of] AUT(L) [for all] [alpha],[beta][meber of] AUM(L).

Corollary 2.1. Let (L, *) be a quasigroup(loop) with holomorph H(L). H(L) is a CIPQ(CIPL) if and only if

1. AUM(L) is an abelian group,

2. ([[beta].sup.-1],[alpha], I) [meber of] AUT(L) [for all] [alpha],[beta] [member of] AUM(L) and

3. Lisa CIPQ(CIPL).

Proof. A quasigroup(loop) is a CIPQ(CIPL) if and only if it is a WIPQ(WIPL) and an AIPQ(AIPL). L is a WIPQ(WIPL) if and only if H(L) is a WIPQ(WIPL).

If H(L) is a CIPQ(CIPL), then H(L) is both a WIPQ(WIPL) and a AIPQ(AIPL) which implies 1., 2., and 3. of Theorem 2.1. Hence, L is a CIPQ(CIPL). The converse follows by just doing the reverse.

Corollary 2.2. Let (L, .) be a quasigroup(loop) with holomorph H(L).

If H(L) is an AIPQ(AIPL) or CIPQ(CIPL), then H(L) [congruent to] L.

Proof. By 2. of Theorem 2.1, ([[beta].sup.-1],[alpha], I) [meber of] AUT(L) [for all] [alpha],[beta] [meber of] AUM(L) implies [chi] [[beta].sup.-1]*y [alpha] = [chi]* y which means [alpha], [beta] = I by substituting [chi] = e and y = e. Thus, AUM(L) = {I} and so H(L) [congruent to] L.

Theorem 2.2. The holomorph of a quasigroup(loop) L is a AIPQ(AIPL) or CIPQ(CIPL) if and only if AUM(L) = {I} and L is a AIPQ(AIPL) or CIPQ(CIPL).

Proof. This is established using Theorem 2.1, Corollary 2.1 and Corollary 2.2.

Theorem 2.3. Let (L, -) be a quasigroup(loop) with holomorph H(L). H(L) is a CIPQ(CIPL) if and only if AUM(L) is an abelian group and any of the following is true for all [chi], y [member of] L and [alpha], [beta], [member of] AUM(L).

1. ([chi][beta] * y) [[chi].sup.[rho]]= y [alpha].

2. [chi][beta] * y [chi][rho] = y [alpha].

3. ([chi][lamda][[alpha].sup.-1] [beta] [alpha] * y [alpha]) * [chi] = y.

4. [chi][lamda][[alpha].sup.-1] [beta][alpha] * (y [alpha] * [chi]) = y.

Proof. This is achieved by simply using the four equivalent identities that define a CIPQ(CIPL):

Corollary 2.3. Let (L, -) be a quasigroups(loop) with holomorph H(L).

If H(L) is a CIPQ(CIPL) then, the following are equivalent to each other

1. ([[beta].sup.-1][J.sub.[rho], [alpha][J.sub.[rho]], [J.sub.[rho])[member of] AUT(L) [for all] [alpha], [beta] [member of] AUM(L).

2. ([[beta].sup.-1] [J.sub.[lamda], [alpha] [J.sub.[lamda],[J.sub.[lamda])[member of] AUT(L)[for all] [alpha],[beta], [member of] AUM(L).

3. ([chi][beta]* y)[[chi].sup.][rho] = y [alpha].

4. [chi][beta]* y [[chi].sup.[rho]] = y [alpha].

5. ([[chi].sup.[lamda]][[alpha].sup.-1][beta][alpha] * y [alpha]) * x = y.

6. [[chi].sup.[lamda][alpha].sup.-1] [beta][alpha] * (y [alpha] * x) = y.

Hence, ([beta],[alpha], I), ([alpha], [beta], I), ([beta], I, [alpha]), (I, [alpha], [beta]) [member of] AUT(L) [for all] [alpha], [beta] [member of] AUM(L).

Proof. The equivalence of the six conditions follows from Theorem 2.3 and the proof of Theorem 2.1. The last part is simple.

Corollary 2.4. Let (L, *) be a quasigroup(loop) with holomorph H(L).

If H(L) is a CIPQ(CIPL) then, L is a flexible unipotent CIPQ(flexible CIPL of exponent 2).

Proof. It is observed that [J.sub.[rho] = [J.sub.[lamda] = I. Hence, the conclusion follows.

Remark. The holomorphic structure of loops such as extra loop, Bol-loop, C-loop, CCloop and A-loop have been found to be characterized by some special types of automorphisms such as

1. Nuclear automorphism(in the case of Bol-,CC- and extra loops),

2. central automorphism(in the case of central and A-loops).

By Theorem 2.1 and Corollary 2.1, the holomorphic structure of AIPLs and CIPLs is characterized by commutative automorphisms.

Theorem 2.4. The holomorph H(L) of a quasigroup(loop) L is a Smarandache AIPQ(AIPL) or CIPQ(CIPL) if and only if SAUM(L) = {I} and L is a Smarandache AIPQ(AIPL) or CIPQ(CIPL).

Proof. Let L be a quasigroup with holomorph H(L). If H(L) is a SAIPQ(SCIPQ), then there exists a S-subquasigroup HS(L) C H(L) such that Hs(L) is a AIPQ(CIPQ). Let Hs(L) G x SAUM(L) where G is the S-subquasigroup of L. From Theorem 2.2, it can be seen that Hs(L) is a AIPQ(CIPQ) if and only if SAUM(L) = {I} and G is a AIPQ(CIPQ). So the conclusion follows.

Corollary 2.5. The holomorph H(L) of a loop L is a SKL or SBRL or SKWL if and only if SAUM(L) = {I} and L is a SKL or SBRL or SKWL.

Proof. Let L be a loop with holomorph H(L). Consider the subloop [H.sub.s](L) of H(L) such that [H.sub.s](L) = G x SAUM(L) where G is the subloop of L.

1. Recall that by [Theorem 5.3, ], Hs(L) is an A-loop if and only if it is a Smarandache Central-holomorph of L and G is an A-loop. Combing this fact with Theorem 2.4, it can be concluded that: the holomorph H(L) of a loop L is a SKL if and only if SAUM(L) {I} and L is a SKL.

2. Recall that by  and , Hs(L) is a Bol loop if and only if it is a Smarandache Nuclearholomorph of L and G is a Bol-loop. Combing this fact with Theorem 2.4, it can be concluded that: the holomorph H(L) of a loop L is a SBRL if and only if SAUM(L) {I} and L is a SBRL.

3. Following the first reason in 1., and using Theorem 2.4, it can be concluded that: the holomorph H(L) of a loop L is a SKWL if and only if SAUM(L) = {I} and L is a SKWL.

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Temitope Gbolahan Jaiyeola

Department of Mathematics, Obafemi Awolowo University, Ile Ife, Nigeria.

Email: jaiyeolatemitope@yahoo.com/tjayeola@oauife.edu.ng
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