A particular case of extended isotopisms: Santilli's isotopisms

A particular case of extended isotopisms: Santilli's isotopisms. Raúl M. Falcón Ganfornina and Juan Núñez Valdés Dpto. Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Aptdo 1160. 41080-Sevilla (España). E-mails: r[email protected] and [email protected] Abstract Due to a mathematical necessity, it has been proved that it is convenient to give a new interpretation of the multiplicity of Santilli's isounity b I=b I(x, v, t, µ, ρ, ...) of any Santilli's isotopism, as a family of classical Bruck's isotopisms. In this paper we prove that every Santilli's isotopism is indeed an extended isotopism. 1 Hadronic Journal, Vol. 29, No. 3 (2006), pp. 285 - 298. 1 Introduction A partial algebra is a nonempty set S endowed with binary operations which have as domain a subset D of S×S . S is an algebraic structure if all these binary operations have domain D=S×S . A quasigroup [1] is an algebraic structure Q endowed with a product · , such that if any two of the three symbols a, b, c in the equation a·b=c are given as elements of Q , the third one is uniquely determined as an element of Q . It is equivalent to say that Q is endowed with left / and right \ division. A loop is a quasigroup with an unit element. Two quasigroups (Q1,·) and (Q2,◦) are isotopic [2] if there are three bijections α, β, γ from Q2 to Q1 , such that: γ(a◦b) = α(a)·β(b), for all a, b ∈Q2.(1) The triple Θ = (α, β, γ) is called an isotopism from (Q1,·) to (Q2,◦) and it is denoted (Q1,·)Θ= (Q2,◦) . If Q1=Q2 and α=β=γ , the isotopism is indeed an isomorphism . If Q1=Q2 and · ≡ ◦ , Θ is called an autotopism . Let us also recall that the concept of isotopism can be extended in the same way as in other algebraic structures as a triple of bijections of such a structure, which veries an equality analogous to (1). If we consider the multiplication table of a quasigroup, we obtain a Latin square. A Latin square , L , of order n , is a n×n array with elements chosen from a set of n symbols N={x1, ..., xn} , such that each symbol occurs precisely once in each row and each column. The set of Latin squares of order n is denoted by LS(n) . A partial Latin square , P , of order n , is a n×n array with elements chosen from a set of n symbols, such that each symbol occurs at most once in each row and in each column. The set of partial Latin squares of order n is denoted as PLS(n) . The algebraic structure 2 having a partial Latin square as its multiplication table is called a partial quasigroup . From now on, we will consider N={0,1, ..., n −1} . So, if L= (lij) , the orthogonal array representation of L is the set of n2 triples {(i, j, lij) : 0 ≤ i, j ≤n−1} . An isotopism of a Latin square L is a triple Θ = (α, β, γ)∈ In=Sn×Sn×Sn , where Sn is the symmetric group on N and so, α, β and γ are respectively, permutations of rows, columns and symbols of L . The resulting square LΘ is also a Latin square and it is said to be isotopic to L . In particular, if L= (lij) , then LΘ={(i, j, γ−1(lα(i)β(j)): 0 ≤i, j ≤n−1} . The set of all Latin squares isotopic to L is called the isotopy class of L . Fixed another Latin square L′∈LS(n) , the set of all isotopisms from L to L′ is denoted by U(L, L′) = {Θ∈ In:LΘ=L′} . An isotopism which maps L to itself is an autotopism . The stabilizer subgroup of L in In is its autotopism group , U(L) = {Θ∈ In:LΘ=L} . Given P∈PLS(n) , contained in L , an isotopism of P is a triple Θ=(α, β, γ)∈ In , where γ(∅) = ∅ . Given F⊆ U(L) , it is dened the extended autotopy [6] PF= ∪Θ∈FPΘ∈PLS(n) . The set F is called an extended autotopism from P to PΘ . If (Q1,·) and (Q2,◦) are, respectively, the partial quasigroups associated to P and PF , then it is denoted (Q1,·)F= (Q2,◦) . This concept of extended autotopism can be analogously dened in any multiplication table. So, it can be considered not only in partial quasigroups but in any partial algebra. By the other way, Santilli proposed in 1978 [7] a possible model of isotopism, which he called Santilli's isotopism , which allows to construct the named isostructures , based on an isounit I . For a historical vision of the development of the isotheory and a wide bibliography related to it, the reader can consult [3]. Particularly, Santilli's isotopic model is based on the generalization of the initial unit: I→b I=b I(x, v, t, µ, ρ, ...) , where x, v, t, µ, ρ, ... are variables with values in xed domains X, V, T, M, R, ... , as coordinates, 3 velocity, temperature, density, etc. So, xed any mathematical structure E , endowed with a binary law × with unit element I∈E , this model considers a set V⊇E , endowed with an associative binary law ∗ with unit element I , where the restriction of the law ∗ to E coincides with × , b I∈V for all factors x, v, t, µ, ρ, ... (where here the domain of the coordinate factor is X=E ) and T=T(x, v, t, µ, ρ, ...) = b I−I . V, T and b I are respectively called general set, isotopic element and isounity of the isotopism. So, it is dened an algebraic structure b E , endowed with the law b × with unit b I as: E→b E:x→bx=x∗b I;bab ×b b=ba∗T∗b b=d a∗b, for all a, b ∈b E. (2) From a mathematical point of view, it has been proved [4] that it can be dicult to obtain explicitly the general set V . By the other way, when the isounit b I has more than an unique value (that is, if b I depends on external factors), it has been also seen that calculus and notations can be arduous. In this paper, we will see that these diculties can be avoided by reinterpreting Santilli's isotopisms as a family of classical Bruck's isotopisms. In this way, the multiplicity of the isounity b I can be displaced to a multiplicity of isotopisms. So, in the second section of this paper we compare Santilli's isotopisms with the classical Bruck's ones. To do it, we generalize the concept of extended autotopism to extended isotopism and we prove that every Santilli's isotopism is indeed an extended isotopism. In the third section we give a concrete example of this result, by working with quasigroups. 4 2 Comparison between Santilli's and Bruck's isotopisms Fixed two Latin squares L, L′∈LS(n) , the concept of extended autotopism can be easily generalized to the set U(L, L′)⊆ In : Denition 2.1. Fixed a partial Latin square P∈PLS(n) contained in L and given F⊆ U(L, L′) , we will dene the extended isotopy of P associated to F as: PF=∪ Θ∈F PΘ∈PLS(n). The set F is called an extended isotopism from P to PΘ . We will denote (Q1,·)F= (Q2,◦) if (Q1,·) and (Q2,◦) are, respectively, the partial quasigroups associated to P and PF . It is easy to see that PF is indeed contained in L′ . This concept of extended isotopism can be generalized to any multiplication table and so, to any algebraic structure. In particular, we will use it to prove that every algebraic structure obtained by means of a Santilli's isotopism is an extended isotopism of the corresponding initial algebraic structure. In our study, we will consider the Santilli's isounit b I of the Santilli's isotopism (2) as the subset: b I={b I(x, v, t, µ, ρ, ...) : x∈X, v ∈V, t ∈T, µ ∈M, ρ ∈R, ...} ⊆ V. In this way, |b I| denotes the cardinal of b I . Lemma 2.2. If |b I|= 1 , then the Santilli's isotopism (2) is indeed an isomorphism. Proof. 5 It is sucient to take α:E→b E , such that α(x) = x∗b I . The associativity of ∗ and the existence of the inverse T=b I−I imply that α is a bijection. ✷ When |b I|>1 , we must explicit what elements of b I are used in the denition of b × in (2). In this way, all the possible values of the variable factors of which b I depends were denoted by F in [5], that is: F={f= (x, v, t, µ, ρ, ...) : x∈E, v ∈V, t ∈T, µ ∈M, ρ ∈R, ...}. So: b I={b I(f) : f∈F} ⊆ V. Now, xed f∈F , let us denote: bxf=x∗b I(f), for all x∈V; b Ef={bxf:x∈E}. Therefore: b E={bxf:x∈E, f ∈F}=∪ f∈Fb Ef. By the other way, xed any map: Φ : F×F→F: (f1, f2)→Φ(f1, f2), it was dened in [5] the law b × of (2) in b E as: baf1b ∗b bf2={[ x∗yΦ(fs,ft):bxfs=baf1 and byft=b bf2}x, y ∈E fs, ft∈F , for all a, b ∈E. However, we are not interested in algebraic multi-structures, that is, the set of the right side of the previous equality must be of cardinal one. For this reason, we must choose Φ in such a way that this set is unitary. We will say then that Φ is compatible with (E, ∗) . In the case in which in the 6 previous equality we can swap all the elements a, b, x, y in the complete set V , we will say that Φ is compatible with (V, ∗) and we can extend the law b ∗ from b E to V . Therefore, let us suppose from now on that Φ is compatible with (V, ∗) and let us x f1, f2∈F . We obtain then that: baf1b ∗b bf2=d a∗bΦ(f1,f2), for all a, b ∈V. (3) Now, if T(f1) = b I(f1)−I and T(f2) = b I(f2)−I , let us observe that (3) is equivalent to: baf1b ∗b bf2=((baf1∗T(f1)) ∗(b bf2∗T(f2)))∗b I(Φ(f1, f2)) . Finally, if T(Φ(f1, f2)) = b I(Φ(f1, f2))−I , the previous equality is equivalent to: (baf1b ∗b bf2)∗T(Φ(f1, f2)) = (baf1∗T(f1)) ∗(b bf2∗T(f2)). Let us now dene the bijections on V , αf1, βf2 and γΦ(f1,f2) , such that: αf1(bxf1) = bxf1∗T(f1) = x, βf2(bxf2) = bxf2∗T(f2) = x, γΦ(f1,f2)(bxΦ(f1,f2)) = bxΦ(f1,f2)∗T(Φ(f1, f2)) = x, So, we nally obtain that: γΦ(f1,f2)(baf1b ∗b bf2)=αf1(baf1)∗βf2(b bf2), for all a, b ∈V. (3′) That is, the triple Θ1,2= (αf1, βf2, γΦ(f1,f2)) is a classical isotopism from (V, ∗) to (V, b ∗) . In a similar way, we can obtain a classical isotopism Θi,j from (V, ∗) to (V, b ∗) starting from any (fi, fj)∈F×F . So, we can dene the family: F={Θi,j = (αfi, βfj, γΦ(fi,fj)) : (fi, fj)∈F×F}. 7 Particularly: Lemma 2.3. It is veried that: F⊆ U ((V, ∗),(V,b ∗)) . Proof. The result is immediate by keeping in mind the associativity of ∗ and the existence in any case of the isotopic element as the inverse of the corresponding isounity. ✷ Now, let us consider (E, ∗) as a partial substructure of (V, ∗) in the following sense: a∗b=   a×b , if (a, b)∈E×E, ∅ , if (a, b)∈ E×E. We obtain then the following: Proposition 2.4. They are veried: a) (E, ∗)Θ is a partial substructure of (V, b ∗) for all Θ∈F . b) (E, ∗)F= ( b E,b ∗) is an algebraic substructure of (V, b ∗) . In this way, (b E,b ∗) is indeed the extended isotopy of (E, ∗) associated to F . Proof Fixed f1, f2∈F and Θf1,f2= (αf1, βf2, γΦ(f1,f2))∈F , we can consider (E, ∗)Θ as a partial substructure of (V, b ∗) in the following sense: xb ∗y=   γ−1 Φ(f1,f2)(αf1(x)∗βf2(y)) , if x∈b Ef1 and y∈b Ef2, ∅ , otherwise. 8 By the other way, xed x, y ∈b E=∪f∈Fb Ef⊆V , we can nd a, b ∈E and fi, fj∈F , such that αfi(a) = x and βfj(b) = y . Particularly, the triple Θfi,fj= (αfi, βfj, γΦ(fi,fj))∈F allows to dene the product xb ∗y∈b E . Therefore, the law b ∗ is dened in b E×b E and (E, ∗)F= ( b E,b ∗) is an algebraic substructure of (V, b ∗) . ✷ So, as a consequence of the previous construction, the following result holds: Theorem 2.5. Every Santilli's isotopism is an extended isotopism. ✷ 3 An example of Santilli's isotopism Let us consider E={0,1} , V={0,1,2,3} and the laws × in E and ∗ in V , given by the followings multiplication tables: × 0 1 0 0 1 1 1 0 ∗ 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 So, (E, ∗) = (E, ×) . Now, let us consider: F={f1, f2}, b I(f1) = 0; b I(f2) = 2. 9