American Journal of Modern Physics
Volume 4, Issue 5-1, October 2015, Pages: 47-51

Santilli Autotopisms of Partial Groups

Raúl M. Falcón1, Juan Núñez2

1Department of Applied Mathematics I, University of Seville, Seville, Spain

2Department of Geometry and Topology, University of Seville, Seville, Spain

(R. M. Falcón)
(J. Núñez)

Raúl M. Falcón, Juan Núñez. Santilli Autotopisms of Partial Groups. American Journal of Modern Physics. Special Issue: Issue I: Foundations of Hadronic Mathematics. Vol. 4, No. 5-1, 2015, pp. 47-51. doi: 10.11648/j.ajmp.s.2015040501.16

Abstract: This paper deals with those partial groups that contain a given Santilli isotopism in their autotopism group. A classification of these autotopisms is explicitly determined for partial groups of order n ≤ 4.

Keywords: Partial Group, Isotopism, Classification

Contents

1. Introduction

In 1942, Albert [1] introduced the concept of isotopy of algebras: Two algebras (A1, ·) and (A2, *) over a field K are said to be isotopic if there exist three regular linear transformations f, g and h from A1 to A2 such that

f(u) * g(v) = h(u · v), for all u, v ÎA1.              (1)

The algebra A2 is then said to be isotopic to the algebra A1, or, equivalently, A2 is an isotope of A1. The triple Ɵ = (f, g, h) is an isotopy or isotopism between both algebras A1 and A2. If f = g = h, then this is indeed an isomorphism. If the elements of A1 and A2 coincide, then the isotopism Ɵ is said to be principal if h is the trivial transformation Id, that is, if h(u)=Id(u)=u, for all uÎ A1. In this case, the algebra A2 is said to be a principal isotope of A1. In his original paper, Albert proposed the question as to whether a principal isotope of a Lie algebra is Lie. In this regard, he proved that a principal isotope A2 of a Lie algebra A1 with respect to a principal isotopism (f, g, Id) is a Lie algebra if and only if, for all u, v, w Î A1, it is verified that

f(u) · g(v) = - f(v) · g(u).                  (2)

f(f(u)·g(v))·g(w) - f(f(u)·g(w))·g(v) – f(u)·g(f(v)·g(w))=0.(3)

In 1944, Bruck [2] introduced the concept of isotopically simple algebra as a simple algebra such that all their isotopic algebras are simple. He proved in particular that the Lie algebra of order n · (n–1)/2, consisting of all skew-symmetric matrices over any subfield of the field of all reals, under the Lie product [u, v] = u · v – v · u, is isotopically simple. Further, the Lie algebra of order n · (n – 1) consisting of all skew-Hermitian matrices in any field R(i) (where R is a subfield of the reals and i2= -1), under the multiplication [u, v]=u · v – v · u, is an isotopically simple algebra over R.

More recently, in 1978, Santilli [3] generalized the associative product u · v between Hermitian generators of the universal enveloping associative algebra by considering the new product

u * v = u · T · v                                   (4)

where T is a positive-definite operator called the isotopic element, which can indeed depend on distinct factors

T = T(x,x’,x’’,…,µ,τ)                            (5)

The product

[u, v] = u * v – v * u                         (6)

preserves the Lie axioms and is called the Lie-isotopic product. The application to Lie’s theory (enveloping algebras, Lie algebras and Lie groups) that emerges from this new product is the so-called Lie-Santilli isotheory (see [3, pp. 287-290 and 329-374] and also [4-9]).

In the development of the isotheory, Santilli extended the unit of the ground field to the generalized unit or isounit

I = I(x,x’,x’’,…,µ,τ) =T -1                     (7)

He defined then the isonumbers

u = u * I(x,x’,x’’,…,µ,τ), for all u Î A.          (8)

and the isoproduct

[u, v] = u * v – v * u                       (9)

This isoproduct constitutes the Lie product of an isomorphic Lie algebra of A whenever the isounit Î is constant. In any other case, this determines a generalization of the classical notion (2) of isotopism. In order to analyze this fact, the authors [10] reinterpreted in 2006 the dependence on distinct factors of the isounit Î as a family of classical Bruck’s isotopisms. This reinterpretation became clearer shortly after [11] once the attention was focused not on isotopisms of algebras, but on isotopisms of partial quasigroups.

The term quasigroup was introduced in 1937 by Haussmann and Ore [12] to denote a nonempty set Q endowed with a product ·, such that if any two of the three symbols u, v and w in the equation u · v = w are given as elements of Q, then the third is uniquely determined as an element of Q. Its order is the cardinality of the underlying set, that is, the number of elements of the quasigroup Q. This is said to be a loop if it contains a unit element, that is, there exists an element e Î Q such that e · u = u · e = u for all u Î Q. Every associative loop is indeed a group. The multiplication table of a quasigroup of order n is a Latin square of order n, that is, an n x n array with elements chosen from a set of n distinct symbols such that each symbol occurs precisely once in each row and each column (see Figure 1).

 2 3 4 1 3 4 1 2 4 1 2 3 1 2 3 4

Figure 1. Latin square of order 4.

A partial Latin square of order n is an n x n array with elements chosen from a set of n distinct symbols such that each symbol occurs at most once in each row and each column (see Figure 2). It constitutes the multiplication table of a finite partial quasigroup (Q, ·) of order n. Let u, v Î Q. The product u · v is then an element of Q or it is undefined. This last case is denoted as u · v = Ø. By abuse of notation, it is also considered that u · Ø = Ø · u = Ø, for all u Î Q and hence, the partial quasigroup is associative if (u · v) · w = u · (v · w), for all u, v, w Î Q. It is a partial loop if there exists an element e Î Q such that e · u = u · e Î {u, Ø} for all u Î Q and there does not exist an element e’ ≠ e such that e’ · u = u or u · e’ = u. Every associative partial loop constitutes a partial group.

 1 2 4 3 4 3

Figure 2. Partial Latin square of order 4.

In 1943-44, Albert [13, 14] together with Bruck [15] extended the definition of isotopy from algebras to quasigroups. Particularly, two quasigroups (Q1, ·) and (Q2, *) of the same order are said to be isotopic if there exist three bijections f, g and h between their sets of elements such that

f(u) * g(v) = h(u · v), for all u, v Î Q1.         (10)

The definition can be naturally extended to partial quasigroups once it is considered h(Ø) = Ø. The triple Ɵ = (f, g, h) is said to be an isotopism between Q1 and Q2 and it is denoted Q2 = Q1Ɵ. If Q2 = Q1, then the isotopism Ɵ is said to be an autotopism of Q1 and f, g and h are permutations of the elements of Q1. The set of autotopisms of a (partial) quasigroup constitutes, therefore, a group with the component-wise composition of permutations.

In 2007, the authors [11] introduced the concept of Santilli isotopism between partial quasigroups. Specifically, an isotopism Ɵ = (f, g, h) between two partial quasigroups (Q1, ·) and (Q2, *) is said to be a Santilli isotopism if there exist three elements if, ig and ih in Q1 such that

f(u)=u· if, g(u)=u· ig and h(u)=u· ih, for all uÎ P1(11)

The triple (if, ig, ih) is denoted by S(Ɵ,Q1). If Q2 = Q1, then the isotopism Ɵ is said to be a Santilli autotopism of Q1.

In [11], there were exposed several properties of the set of partial quasigroups having a Santilli autotopism that fixes at least one of the symbols. An exhaustive study of Santilli autotopisms is, however, currently required. This paper is established as a first contribution in this regard. In Section 2, some new general properties of the set of Santilli isotopisms of (associative) partial quasigroups, partial loops and partial groups are analyzed. In Section 3, a classification of the Santilli autotopisms of groups of order n ≤ 6 is explicitly given. Remark that, even if the number of quasigroups is known for order up to 11 [16, 17], that of partial quasigroups is only known for order up to four [18, 19].

2. Santilli Autotopisms

From now on, every partial quasigroup of order n is considered to be formed by the set of elements {1,…, n}. The set of isotopisms of partial quasigroups of order n is then denoted as In = Sn x Sn x Sn, where Sn is the symmetric group on {1,…, n}. The set of fixed symbols in a permutation π Î Sn is denoted as

Fix(π) = {u Î {1,…n}such that π(u)=u}.       (12)

Let Ɵ Î In and let SQ(Ɵ), SL(Ɵ), SAQ(Ɵ) and SG(Ɵ) be, respectively, the sets of partial quasigroups, partial loops, associative partial quasigroups and partial groups that have Ɵ as a Santilli autotopism. The next results are satisfied.

Lemma 2.1. Let Ɵ = (f, g, h) Î In and (Q, ·) Î SQ(Ɵ) be such that S(Ɵ,Q)= (if, ig, ih). Then, ih = g(if). As a consequence,

(i · if) · (j · ig) = (i · j) · (if · ig), for all i, j Î Q.        (13)

Proof. Given vÎ Q, let u Î Q be such that f(u)=v. Then, v · ih = h(v) = h(f(u)) = h(u· if ) = f(u) · g(if ) = v · g(if ) and the result holds from the fact that Q is a partial quasigroup and h(v) Î Q.

Proposition 2.2. Let Ɵ = (f, g, h) Î In and (Q, ·) Î SQ(Ɵ) be such that S(Ɵ,Q)= (if, ig, ih). If h = f, then if Î Fix(g).

Proof. The result follows straightforward from Lemma 2.1 and the fact of being h = f.

Lemma 2.3. Let Ɵ = (f, g, h) Î In. If there exist two permutations α, β Î {f, g, h} such that α(u0) = β(u0) for some u0 Î Q, then α = β.

Proof. Let (Q, ·) be a partial quasigroup in SQ(Ɵ) and let iα, iβ Î Q be such that α(u) = u · iα and β(u) = u · iβ for all u Î Q. Particularly, u0 · iα = α(u0) = β(u0) = u0 · iβ. This product is not undefined because α(u0) Î Q. Since Q is a partial quasigroup, it must be then iα = iβ and hence, α = β.

Proposition 2.4. Let Ɵ = (f, g, h) Î In be such that Fix(g) = Ø. Then, f(u) ≠ h(u) for all uÎ Q.

Proof. Let uÎ Q be such that f(u) = h(u). From Lemma 2.3 it must be f = h. Thus, from Lemma 2.1, it is if = ih = g(if ) and hence, if Î Fix(g), which is a contradiction.

Lemma 2.5. Let Ɵ = (f, g, h) Î In and (Q, ·) Î SQ(Ɵ) be such that S(Ɵ,Q)= (if, ig, ih). If there exists u0 Î Q such that hm(g(u0))= g(fm(u0)) for some positive integer m, then ig Î Fix(gm). As a consequence, if Fix(gm) = Ø for some positive integer m, then hm(g(u))≠ g(fm(u)), for all u Î Q.

Proof. Let m be such that hm(g(u0))= g(fm(u0)) for some u0 Î Q. It is then fm(u0) · gm(ig) = hm(u0 · ig) = hm(g(u0))= g(fm(u0))= fm(u0) · ig. This product is not undefined because hm(g(u0)) Î Q. Since Q is a partial quasigroup, it must be then ig Î Fix(gm). The consequence is immediate.

Lemma 2.6. Let Ɵ = (f, g, h) Î In be such that |Fix(f)| · |Fix(g)| · |Fix(h)| >0. Let (Q, ·) Î SQ(Ɵ) be such that S(Ɵ,Q)= (if, ig, ih). If there exist u0 Î Fix(f), w0 Î Fix(h) and α Î {f, g, h} such that α(u0) = w0, then iαÎ Fix(g). Further, if igÎ Fix(g), then g(u) Î Fix(h) for all uÎ Fix(f).

Proof. It is satisfied that u0 · iα = α(u0) = w0 = h(w0) = h(u0 · iα)=f(u0)·g(iα) = u0 · g(iα). Since w0 Î Q and Q is a quasigroup, it must be iα Î Fix(g). Let us suppose now that ig Î Fix(g) and let us consider an element u Î Fix(f). Then g(u) = u · ig = f(u) · g(ig) = h(u · ig)=h(g(u)) and hence, g(u) Î Fix(h).

The next three results deal with the set of partial loops SL(Ɵ) having a Santilli isotopism Ɵ in their autotopism group.

Proposition 2.7. Let Ɵ = (f, g, h) Î In and (Q, ·) Î SL(Ɵ) be a partial loop with unit element e. Then, S(Ɵ,Q) = (f(e), g(e), g(f(e))).

Proof. Let S(Ɵ,Q) = (if, ig, ih). The result follows straightforward from Lemma 2.1 and the fact that π(e) Î Q. Hence, π(e) = e · iπ = iπ, for all π Î {f, g}.

Lemma 2.8. Let Ɵ = (f, g, h) Î In. If there exists a permutation π Î {f, g, h} such that Fix(π) ≠ Ø, then π = Id.

Proof. Let (Q, ·) Î SL(Ɵ) and S(Ɵ,Q)= (if, ig, ih). Let π Î {f, g, h} and u0 Î Q be such that π(u0) = u0. Since u0 = u0 · iπ, the element iπ is the unit element of the partial loop. Let u Î Q. Since π(u) Î Q, it is π(u) = u · iπ = u and hence, π = Id.

Lemma 2.9. Let Ɵ = (f, g, h) Î In and (Q, ·) Î SL(Ɵ) be a partial loop with unit element e. If e Î Fix(fm) for some positive integer m, then hm = gm. Similarly, if e Î Fix(gm), then hm = fm.

Proof. Let us suppose that e Î Fix(fm) for some positive integer m. Let u Î Q. It is gm(u) = e · gm (u) = fm(e) · gm(u) = hm(e · u). Since gm(u) Î Q, it must be e · u = u and hence, gm(u) = hm(u). The last assertion follows analogously.

We focus now on the set SAQ(Ɵ) of associative partial quasigroups having a Santilli autotopism in their autotopism group.

Proposition 2.10. Let Ɵ = (f, g, h) Î In. If SAQ(Ɵ) ≠ Ø, then h = g º f.

Proof. Let (Q, ·) Î SAQ(Ɵ) and S(Ɵ,Q)= (if, ig, ih). From Lemma 2.1, we know that ih = g(if ). Hence, for all u Î Q, it is verified that h(u) = u · ih = u · g(if) = u · ( if · ig) = (u · if )· ig =g(f(u)).

Lemma 2.11. Let Ɵ = (f, g, h) Î In be such that SAQ(Ɵ) ≠ Ø and let m ≤ n be a positive integer. Then

a)  SAQ(Ɵ) Í SAQ(Ɵm).

b)  SAQ(Ɵ) = SAQ((f, g º fm, h º fm)).

Proof. Let (Q, ·) Î SAQ(Ɵ) be such that S(Ɵ,Q)= (if, ig, ih) and let m ≤ n be a positive integer. Then

1.   The isotopism Ɵm is an autotopism of (Q, ·) because hm(u · v) = hm-1( f(u) · g(v)) = … = fm(u) · gm(v), for all u, v Î Q. Since the quasigroup (Q, ·) is associative, this is indeed a Santilli autotopism for which S(Ɵm,Q)= (ifm, igm, ihm).

2.   The isotopism (f, g º fm, h º fm) is an autotopism of (Q, ·) because h(fm(u · v)) = h((u · v) · ifm) = h(u · (v · ifm)) = h(u · fm(v)) = f(u) · g(fm(v)), for all u, v Î Q. Since the quasigroup (Q, ·) is associative, this is indeed a Santilli autotopism for which S((f, g º fm, h º fm),Q) = (ifm, ifm · ig, ifm · ih). Hence, SAQ(Ɵ) Í SAQ((f, g º fm, h º fm)).

Let us consider now an associative partial quasigroup (Q’,*)Î SAQ((f, g º fm, h º fm)) such that S((f, g º fm, h º fm),Q’) = (i1, i2, i3). It is then verified that Ɵ is a Santilli autotopism of (Q’, *) because, since fn=Id, it is h(u * v) = h(fn (u * v)) = h(fm(fn-m(u * v)) = h(fm(u * fn-m(v)) = f(u) * g(fm(fn-m(v)) = f(u) * g(fn(v)) = f(u) * g(v), for all u, v Î Q’. Further, S(Ɵ,Q’) = (i1, i2*i1n-m, i3*i1n-m). Hence, SAQ((f, g º fm, h º fm)) Í SAQ(Ɵ).

In general, given a positive integer m ≤ n, it is not true that SAQ(Ɵm) Í SAQ(Ɵ). Thus, for instance, the isotopism Ɵ = ((1234), (1234), (13)(24)) is a Santilli autotopism of the associative quasigroup whose multiplication table is the Latin square exposed in Figure 1. Nevertheless, even if the isotopism Ɵ2 = ((13)(24),(13)(24),Id) is a Santilli autotopism of the associative partial quasigroup whose multiplication table is exposed in Figure 3, this is not contained in SAQ(Ɵ).

 3 1 4 2 1 3 2 4

Figure 3. Partial Latin square of order 4.

Let us finish with a result about the set SG(Ɵ) of partial groups having a Santilli isotopism in their autotopism group.

Theorem 2.12. Let Ɵ = (f, g, h) Î In. If SG(Ɵ) ≠ Ø and Fix(f) ≠ Ø, then g = h and f = Id.

Proof. The result follows straightforward from Lemma 2.8 and Proposition 2.10.

3. Santilli Autotopisms of Partial Groups of Order n ≤ 4

The results that have been exposed in Section 2 can be taken into account in order to determine explicitly the set of Santilli isotopisms that are autotopisms of partial groups of a given order. To this end, we say that two isotopisms Ɵ1 = (f1, g1, h1) and Ɵ2 = (f2, g2, h2) in In are equivalent if f2 = f1 and there exists a positive integer m ≤ n such that g2 = g1 º f1mand h2 = h1 º f1m. From assertion (b) in Lemma 2.11, it is verified that SAQ(Ɵ1) = SAQ(Ɵ2). To be equivalent is then an equivalence relation in the set In. Let [Ɵ] denote the equivalence class of Ɵ Î In. We expose in Table 1 these equivalence classes for Santilli autotopisms of partial groups of order n ≤ 4. We focus on the case of non-trivial isotopisms, that is, those that do not coincide with (Id, Id, Id).

Table 1. Santilli autotopisms of partial groups.

We indicate for each class [Ɵ] in Table 1 the set SG(Ɵ) of partial groups that have all the isotopisms of the class in their corresponding autotopism group. The multiplication tables of the elements of these sets are described in Figures 4–12.

 1 2 2 1 2 1 1 2

Figure 4. Partial Latin squares related to A2.

 1 2 3 3 1 2 2 3 1 2 3 1 1 2 3 3 1 2 3 1 2 2 3 1 1 2 3

Figure 5. Partial Latin squares related to A3.

 1 2 3 4 4 1 2 3 3 4 1 2 2 3 4 1 2 3 4 1 1 2 3 4 4 1 2 3 3 4 1 2 3 4 1 2 2 3 4 1 1 2 3 4 4 1 2 3 4 1 2 3 3 4 1 2 2 3 4 1 1 2 3 4

Figure 6. Partial Latin squares related to A4.

 1 2 3 4 3 1 4 2 2 4 1 3 4 3 2 1 2 4 1 3 1 2 3 4 4 3 2 1 3 1 4 2 3 1 4 2 4 3 2 1 1 2 3 4 2 4 1 3 4 3 2 1 2 4 1 3 3 1 4 2 1 2 3 4

Figure 7. Partial Latin squares related to B4.

 1 2 3 4 2 1 4 3 4 3 1 2 3 4 2 1 2 1 4 3 1 2 3 4 3 4 2 1 4 3 1 2 3 4 2 1 4 3 1 2 1 2 3 4 2 1 4 3 4 3 1 2 3 4 2 1 2 1 4 3 1 2 3 4

Figure 8. Partial Latin squares related to C4.

 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 2 1 4 3 1 2 3 4 4 3 2 1 3 4 1 2 3 4 1 2 4 3 2 1 1 2 3 4 2 1 4 3 4 3 2 1 3 4 1 2 2 1 4 3 1 2 3 4

Figure 9. Partial Latin squares related to D4.

 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 3 4 4 3 3 4 4 3 4 3 3 4 4 3 3 4

Figure 10. Partial Latin squares related to E4.

 1 3 1 3 3 1 3 1 2 4 2 4 4 2 4 2 3 1 3 1 1 3 1 3 4 2 4 2 2 4 2 4

Figure 11. Partial Latin squares related to F4.

 1 4 1 4 4 1 4 1 2 3 2 3 3 2 3 2 3 2 3 2 2 3 2 3 4 1 4 1 1 4 1 4

Figure 12. Partial Latin squares related to G4.

Acknowledgements

The authors are very grateful to Prof. Santilli for his valuable help in our research on Santilli isotopisms and cordially congratulate him for his 80th birthday.

References

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