An. S¸t. Univ. Ovidius Constant¸a V ol. 16(2), 2008, 57–66Some isotopy-isomorphy conditions form-inverse quasigroups and loopsT`em´ıt´o.p´e.Gb´o.l´ah`an JA´IY´EO.L´AAbstractThis work presents a special type of middle isotopism under whichm-inverse quasigroups are isotopic invariant. Two distinct isotopy-isomorphy conditions for m-inverse loops are established. Only one ofthem characterizes isotopy-isomorphy in m-inverse loops while the otheris just a sufficient condition for isotopy-isomorphy for specially middleisotopic m-inverse quasigroup.1 IntroductionLet L be a non-empty set. Define a binary operation (·)onL :Ifx · y ∈ Lfor all x, y ∈ L,(L, ·) is called a groupoid. If the system of equations ;a · x = b and y · a = bhave unique solutions for x and y respectively, then (L, ·) is called a quasigroup.For each x ∈ L, the elements xρ= xJρ,xλ= xJλ∈ L such that xxρ= e andxλx = e are called the right, left inverses of x respectively. Now, if there existsa unique element e ∈ L called the identity element such that for all x ∈ L,x · e = e · x = x,(L, ·) is called a loop.Key Words: m-inverse quasigroups; m-inverse loops; Tm,isotopy.Mathematics Subject Classification: Primary 20NO5 ; Secondary 08A05Received: March, 2008Accepted: September, 20085758 T`em´ıt´o.p´e.Gb´o.l´ah`an Ja´ıy´eo.l´aKarklin’s and Karklin’ [10] introduced m-inverse loops. A loop is an m-inverse loop(m-IL) if and only if it obeys any of the equivalent conditions(xy)Jmρ· xJm+1ρ= yJmρand xJm+1λ· (yx)Jmλ= yJmλ.Keedwell and Shcherbacov [12] originally defined an m-inverse quasigroup(m-IQ) as a quasigroup that obeys the identity (xy)Jm· xJm+1= yJm, where Jis a permutation. For the sake of this present study, we shall take J = Jρandso m-IQs obey the equivalent identities that define m-ILs.m-IQs and m-ILs are generalizations of WIPLs and CIPLs, which corre-sponds to m = −1andm = 0 respectively. After the study of m-inverseloops by Keedwell and Shcherbacov [12], they have also generalized them toquasigroups called (r, s, t)-inverse quasigroups in [13] and [14]. Keedwell andShcherbacov [12] investigated the existence of m-inverse quasigroups and loopswith long inverse cycle such that m ≥ 1.They have been able to establish thatthe direct product of two m-inverse quasigroups is an m-inverse quasigroup.Consider (G, ·)and(H, ◦) two distinct groupoids (quasigroups, loops). LetA, B and C be three distinct non-equal bijective mappings, that map G ontoH. The triple α =(A, B, C) is called an isotopism of (G, ·)onto(H, ◦)ifxA ◦ yB =(x · y)C ∀ x, y ∈ G.• If α =(A, B, B), then the triple is called a left isotopism and thegroupoids(quasigroups, loops) are called left isotopes.• If α =(A, B, A), then the triple is called a right isotopism and thegroupoids(quasigroups, loops) are called right isotopes.• If α =(A, A, B), then the triple is called a middle isotopism and thegroupoids are called middle isotopes.If (G, ·)=(H, ◦), then the triple α =(A, B, C) of bijections on (G, ·)iscalled an autotopism of the groupoid(quasigroup, loop) (G, ·). Such triplesform a group AU T (G, ·) called the autotopism group of (G, ·). Furthermore,if A = B = C,thenA is called an automorphism of the groupoid (quasigroup,loop) (G, ·). Such bijections form a group AU M (G, ·) called the automorphismgroup of (G, ·).AsitwasobservedbyOsborn[15],aloopisaWIPLandanAIPLifand only if it is a CIPL. The past efforts of Artzy [1, 4, 3, 2], Belousov andTzurkan [5] and recent studies of Keedwell [11], Keedwell and Shcherbacov[12, 13, 14] are of great significance in the study of WIPLs, AIPLs, CIPQsand CIPLs, their generalizations(i.e m-inverse loops and quasigroups, (r,s,t)-inverse quasigroups) and applications to cryptography.Some Isotopy-Isomorphy Conditions 59The universality of WIPLs and CIPLs have been addressed by Osborn [15]and Artzy [2] respectively. Artzy showed that isotopic CIPLs are isomorphic.In 1970, Basarab [7] continued the work of Osborn since 1961 on universalWIPLs by studying isotopes of WIPLs that are also WIPLs after he hadstudied a class of WIPLs([6]) in 1967. Osborn [15], while investigating theuniversality of WIPLs, discovered that a universal WIPL (G, ·)obeystheidentityyx · (zEy· y)=(y · xz) · y ∀ x, y, z ∈ G (1)where Ey= LyLyλ = R−1yρR−1y= LyRyL−1yR−1y.Eight years after Osborn’s [15] 1960 work on WIPL, in 1968, Huthnance Jr.[9] studied the theory of generalized Moufang loops. He named a loop thatobeys (1) a generalized Moufang loop and later on in the same thesis, he calledthem M-loops. On the other hand, he called a universal WIPL an Osborn loopand the same definition was adopted by Chiboka [8].Moreover, it can be seen that neither WIPLs nor CIPLs have been shownto be isotopic invariant. In fact, it is yet to be shown that there exists a specialtype of isotopism(e.g left, right or middle isotopism) under which the WIPs orCIPs are isotopic invariant. Aside this, there has never been any investigationinto the isotopy of m-inverse quasigroups and loops.The aim of the present study is to present a special type of middle isotop-ism under which m-inverse quasigroups are isotopic invariant. Two distinctisotopy-isomorphy conditions for m-inverse loops are established. Only oneof them characterizes isotopy-isomorphy in m-inverse loops while the other isjust a sufficient condition for isotopy-isomorphy for specially middle isotopicm-inverse quasigroup.2 PreliminariesDefinition 2.1 Let L beaquasigroupandm ∈ Z. A mapping α ∈ SY M(L)where SY M(L) is the gr oup of all bijections on L which obeys the identityxρm=[(xα)ρm]α is called an n-weak right inverse permutation. Their set isrepresented by S(ρ,m)(L). Here, xρm= xJmρand xλm= xJmλ.Similarly, if α obeys the identity xλm=[(xα)λm]α it is called an m-weakleft inverse permutation. Their set is denoted by S(λ,m)(L)If α satisfies both, it is called a weak inverse p ermutation. Their set isdenoted by Sm(L).It can be shown that α ∈ SY M(L)isanm-weak right inverse if and only ifit is an m-weak left inverse permutation. So, Sm(L)=S(ρ,m)(L)=S(λ,m)(L).And thus, α is called an m-weak inverse permutation.60 T`em´ıt´o.p´e.Gb´o.l´ah`an Ja´ıy´eo.l´aRemark 2.1 Every permutation of order 2 that preserves the right(left) in-verse of each element in an m-inverse quasigroup is an m-weak right(left)inverse p ermutation.Throughout, we shall employ the use of the