Computation of isotopisms of algebras over finite fields by means of graph invariants

arXiv:1609.01061v1 [math.RA] 5 Sep 2016 Computation of isotopisms of algebras over finite fields by means of graph invariants O. J. Falc´on1, R. M. Falc´on2, J. N´u˜nez1, A. M. Pacheco3, M. T. Villar1 1Department of Geometry and Topology. University of Seville, Spain. 2Department of Applied Mathematics I. University of Seville, Spain. 3Department of Quantitative Methods. Loyola University Andalusia, Spain. Abstract In this paper we define a pair of faithful functors that map isomorphic and isotopic finite-dimensional algebras over finite fields to isomorphic graphs. These functors reduce the cost of computation that is usually required to determine whether two algebras are isomorphic. In order to illustrate their efficiency, we determine explicitly the classification of twoand threedimensional partial quasigroup rings. Keywords: Graph theory, finite field, isomorphism, Latin square. 2000 MSC: 05C25, 05C30, 05B15. 1. Introduction Graph invariants constitute an interesting tool in Chemistry, Communication or Engineering [8, 16, 19]. In Mathematics, one of the topics for which graph invariants have revealed to play an important role is the classical problem of deciding whether two algebras are isomorphic. This problem is usually dealt with by computing the reduced Gr¨obner basis of the system of polynomial equations that is uniquely related to the structure constants of both algebras. This computation is, however, very sensitive to the number of variables [12] and gives rise to distinct problems of computation time and memory usage even for low-dimensional algebras [9, 13]. This paper deals with Graph Theory in order to reduce this cost of computation. Email addresses: oscf[email protected]s (O. J. Falc´on1), [email protected] (R. M. Falc´on2), [email protected] (J. N´u˜nez1), [email protected] (A. M. Pacheco3), [email protected] (M. T. Villar1) Preprint submitted to Journal of Computational and Applied MathematicsSeptember 6, 2016 Graph invariants have been proposed in the last years as an efficient alternative to study isomorphisms of distinct types of algebras [2, 4, 15]. Nevertheless, the problem of identifying a functor that relates the category of algebras with that of graphs remains still open. Based on a proposal of McKay et al. [17] for identifying isotopisms of Latin squares with isomorphisms of vertex-colored graphs, we describe in Section 3 a pair of graphs that enable us to find faithful functors between finite-dimensional algebras over finite fields and these types of graphs. These functors map isomorphic and isotopic algebras to isomorphic graphs. Reciprocally, any pair of isomorphic graphs is uniquely related to a pair of algebras so that there exists a multiplicative map between them. The main advantage of our proposal, apart from the reduction of the mentioned cost of computation, is the feasibility of studying the possible isomorphism between two given finitedimensional algebras defined over the same field, whatever the types of both algebras are. As an illustrative example, we focus in Section 4 on the classification of partial quasigroup rings according to the known isotopism classes of partial Latin squares on which they are based. 2. Preliminaries In this section we expose some basic concepts and results on Graph Theory, isotopisms of algebras, partial Latin squares and Computational Algebraic Geometry that we use throughout the paper. For more details about these topics we refer, respectively, to the manuscripts [14, 1, 7, 5]. 2.1. Graph Theory Agraph is a pair G= (V, E) formed by a set Vof vertices and a set E of 2-subsets of Vcalled edges. Two vertices defining an edge are said to be adjacent. The degree of a vertex vis the number d(v) of edges containing v. The graph Gis vertex-colored if there exists a partition of Vinto color sets. The color of a vertex vis denoted as color(v). An isomorphism between two vertex-colored graphs Gand G′is any bijective map fbetween their sets of vertices that preserves collinearity and color sets, that is, such that it maps edges to edges and color(f(v)) = color(v), for all vertex vin G. 2.2. Isotopisms of algebras Two algebras Aand A′over a field Kare said to be isotopic if there exist three non-singular linear transformations f,gand hfrom Ato A′such that f(u)g(v) = h(uv), for all u, v ∈A. The triple (f, g, h) is an isotopism between Aand A′. If f=g=h, then this constitutes an isomorphism. 2 The structure constants of an n-dimensional algebra Aover a field K of basis {e1,...,en}are the numbers ck ij ∈Ksuch that eiej=Pn k=1 ck ijek, for all i, j ≤n. If all of them are zeros, then Ais abelian. In particular, the n-dimensional abelian algebra is not isotopic to any other n-dimensional algebra. The left annihilator of a vector subspace Sof the algebra Ais the set AnnA−(S) = {u∈A|uv = 0,for all v∈S}. Its right annihilator is the set AnnA+(S) = {u∈A|vu = 0,for all v∈S}. The intersection of both sets is the annihilator AnnA(S). Lemma 1. Let (f, g, h)be an isotopism between two n-dimensional algebras Aand A′, and let Sbe a vector subspace of A. Then, a) f(AnnA−(S)) = AnnA′−(g(S)). b) g(AnnA+(S)) = AnnA′+(f(S)). c) f(AnnA−(S)) ∩g(AnnA+(S)) = AnnA′(f(S)∩g(S)). Proof. Let us prove assertion (a). Assertion (b) follows similarly and assertion (c) is a consequence of (a) and (b). Let u∈g(S) and v∈f(AnnA−(S)). Then, vu =f(f−1(v))g(g−1(u)) = h(f−1(v)g−1(u)) = h(0) = 0, because g−1(u)∈Sand f−1(v)∈AnnA−(S). Hence, f(AnnA−(S)) ⊆AnnA′−(g(S)). Now, let u∈AnnA′−(g(S)) and v∈S. From the regularity of f, we have that h(f−1(u)v) = ug(v) = 0. The regularity of hinvolves that f−1(u)v= 0. Thus, u∈f(AnnA−(S)) and hence, AnnA′−(g(S)) ⊆f(AnnA−(S)). The derived algebra of Ais the subalgebra A2={uv |u, v ∈A} ⊆ A. Lemma 2. Let (f, g, h)be an isotopism between two n-dimensional algebras Aand A′. Then, h(A2) = A′2. Proof. The regularity of fand ginvolves that f(A) = g(A) = A′and hence, A′2=f(A)g(A) = h(A2). Let ·be a partial binary operation over the set [n] = {1,...,n}. The pair ([n],·) is called a partial magma of order n. It is isotopic to a partial magma ([n],◦) if there exist three permutations α,βand γin the symmetric group Snsuch that α(i)◦β(j) = γ(i·j), for all i, j ≤nsuch that i·jexists. If α=β=γ, then the partial magmas are said to be isomorphic. The triple (α, β, γ) is an isotopism of partial magmas (an isomorphism if α=β=γ). 3 Apartial magma algebra A·based on a partial magma ([n],·) is an ndimensional algebra over a field Ksuch that there exists a basis {e1,...,en} satisfying that, if i·jexists for some pair of elements i, j ≤n, then eiej= cijei·jfor some non-zero structure constant cij ∈K\{0}. If all the structure constants are equal to 1, then this is called a partial magma ring. Lemma 3. Two partial magma rings are isotopic (isomorphic, respectively) if their respective partial magmas on which they are based are isotopic (isomorphic, respectively). Proof. Let A·and A◦be two partial magma rings based, respectively, on two isotopic partial magmas ([n],·) and ([n],◦). Let {e1,...,en}and {e′ 1,...,e′ n} be the respective bases of these two algebras and let (f, g, h) be an isotopism between their corresponding partial magmas. For each α∈ {f, g, h}, let us define the map α(ei) = e′ α(i). Then, f(ei)g(ej) = e′ f(i)e′ g(j)=e′ f(i)◦g(j)= e′ h(i·j)=h(ei·j) = h(eiej). From linearity, the triple (f, g, h) determines an isotopism between A·and A◦. If f=g=h, then this constitutes an isomorphism. The reciprocal of Lemma 3 is not true in general. Thus, for instance, the two partial magmas ([2],·) and ([2],◦) that are respectively described by the non-zero products 1 ·1 = 1 and 1 ◦1 = 1 = 2 ◦1 are not isotopic. Nevertheless, the partial magma rings A·and A◦, with respective bases {e1, e2}and {e′ 1, e′ 2}, are isotopic by means of the isotopism (f, Id,Id), where the linear transformation fis described by f(e1) = e′ 1and f(e2) = e′ 2−e′ 1. 2.3. Partial Latin squares Apartial quasigroup is a partial magma ([n],·) such that if the equations ix =jand yi =j, with i, j ∈[n], have solutions for xand yin [n], then these solutions are unique. The concepts of partial quasigroup algebras and partial quasigroup rings arise similarly to those of partial magma algebras and rings. Lemma 3 also holds analogously for partial quasigroup rings. Every partial quasigroup of order nconstitutes the multiplication table of apartial Latin square of order n, that is, an n×narray in which each cell is either empty or contains one element chosen from the set [n], such that each symbol occurs at most once in each row and in each column. Every isotopism of a partial quasigroup is uniquely related to a permutation of the rows, columns and symbols of the corresponding partial Latin square. The distribution of partial Latin squares into isotopism classes is known for order up six [10, 11]. In this paper we make use of graph invariants to 4 study which ones of the known non-isotopic classes of partial Latin squares of order n≤3 give rise to isotopic classes of partial quasigroup rings over the finite fields F2and F3. In this regard, it is straightforwardly verified that there exists only two one-dimensional partial quasigroup rings: the abelian and that one described by the product e1e1=e1. They constitute distinct isotopism classes. Let L= (lij) be a partial Latin square of order nwithout empty cells (that is, a Latin square). McKay et al. [17] defined the vertex-colored graph G(L) with n2+ 3nvertices {ri|i≤n} ∪ {ci|i≤n} ∪ {si|i≤n} ∪ {tij | i, j ≤n}, where each of the four subsets (related to the rows (ri), columns (ci), symbols (si) and cells (tij) of the Latin square L) has a different color, and 3n2edges {ritij, cjtij, slij tij |i, j ≤n}} (see Figure 1, where we have used distinct styles (◦,N,◮,◭and •) to represent the colors of the vertices). Two Latin squares L1and L2of the same order are isotopic if and only if the graphs G(L1) and G(L2) are isomorphic (see Theorem 6 in [17]). 1 2 2 1 ≡ Figure 1: Graph related to a Latin square of order 2. 2.4. Computational Algebraic Geometry Let K[X] be a multivariate polynomial ring over a field K. The algebraic set defined by an ideal Iof K[X] is the set V(I) of common zeros of all the polynomials in I. If this set is finite, then the ideal Iis zero-dimensional. This is radical if every polynomial f∈K[X] belongs to Iwhenever there exists a natural number msuch that fm∈I. The largest monomial of a polynomial in Iwith respect to a given monomial term ordering is its leading monomial. The ideal generated by all the leading monomials of Iis its initial ideal. A standard monomial of Iis any monomial that is not contained in its initial ideal. Regardless of the monomial term ordering, if the ideal I is zero-dimensional and radical, then the number of standard monomials in Icoincides with the Krull dimension of the quotient ring K[X]/I and with 5 the number of points of the algebraic set V(I). This is computed from the reduced Gr¨obner basis of the ideal. Specifically, a Gr¨obner basis of the ideal Iis any subset Gof polynomials in Iwhose leading monomials generate its initial ideal. This is reduced if all its polynomials are monic and no monomial of a polynomial in Gis generated by the leading monomials of the rest of polynomials in the basis. There exists only one reduced Gr¨obner basis, which can always be computed from Buchberger’s algorithm [3]. The computation that is required to this end is extremely sensitive to the number of variables. Theorem 1 ([12], Proposition 4.1.1).Let Fqbe a finite field, with qa power prime. The complexity time that Buchberger’s algorithm requires to compute the reduced Gr¨obner bases of an ideal hp1,...,pm, pq 1−p1,...,pq m−pmidefined over a polynomial ring Fq[x1,...,xn], where p1,...,pmare polynomials given in sparse form and have longest length l, is qO(n)+O(m2l). Here, sparsity refers to the number of monomials. Gr¨obner bases can be used to determine the isomorphisms and isotopisms between two n-dimensional algebras Aand A′over a finite field Fq, with qa prime power, respective basis {e1,...,en}and {e′ 1,...,e′ n}, and respective structure constants ck ij and c′k ij. To this end, let us define the sets of variables Fn={fij |i, j ≤n},Gn={gij |i, j ≤n}and Hn={hij |i, j ≤n}. These variables play the respective role of the entries in the regular matrices related to a possible isotopism (f, g, h) between the algebras Aand A′. Here, α(ei) = Pn j=1 αije′ j, for each α∈ {f, g, h}. From the coefficients of each basis vector emin the expression f(ei)g(ej) = h(eiej), we have that n X k,l=1 fikgjlc′m kl = n X s=1 cs ijhsm,for all i, j, m ≤n. Theorem 2. The next two assertions hold. a) The isotopism group between the algebras Aand A′is identified with the algebraic set of the ideal IIsot A,A′of Fq[Fn∪Gn∪Hn], which is defined as h n X k,l=1 fikgjlc′m kl − n X s=1 cs ijhsm |i, j, m ≤ni+hdet(M)q−1−1|M∈ {F, G, H} i, where F,Gand Hdenote, respectively, the matrices of entries in Fn,Gn and Hn. Besides, |V(IIsot A,A′)|= dimFq(Fq[Fn∪Gn∪Hn]/IIsot A,A′). 6 b) The isomorphism group between the algebras Aand A′is identified with the algebraic set of the ideal IIsom A,A′of Fq[Fn], which is defined as h n X k,l=1 fikfjlc′m kl − n X s=1 cs ijfsm |i, j, m ≤ni+hdet(F)q−1−1i, where Fdenotes the matrix of entries in Fn. Besides, |V(IIsom A,A′)|= dimFq(Fq[Fn]/IIsom A,A′). Proof. Let us prove the second assertion, being analogous the reasoning for assertion (a). The generators of the ideal IIsom A,A′involve each zero (f11,..., fnn) of its algebraic set to constitute the entries of the regular matrix of an isomorphism fbetween the algebras Aand A′. The result follows from the fact of being this ideal zero-dimensional and radical. Particularly, the ideal IIsom A,A′is zero-dimensional because its algebraic set is a finite subset of Fn2 q. Besides, from Proposition 2.7 of [5], the ideal Iis also radical, because, for each i, j ≤n, the unique monic generator of I∩Fq[fij] is the polynomial (fij)q−fij, which is intrinsically included in each ideal of Fq[Fn] and is square-free. Corollary 1. The complexity times that Buchberger’s algorithm requires to compute the reduced Gr¨obner bases of the ideals IIsot A,A′and IIsom A,A′in Theorem 2 are, respectively, qO(3n2)+O(n6n!) and qO(n2)+O(n6n!). Proof. We prove the result for the second ideal, being analogous the reasoning for the first one. The result follows straightforwardly from Theorem 1 once we observe that all the generators of the ideal in Theorem 2 are sparse in Fq[Fn]. More specifically, the number of variables is n2, the number of generators of the ideal under consideration that are not of the form (fij)q−fij is n3+ 1 and the maximal length of these generators is n!. Theorem 2 has been implemented as a procedure called isoAlg in the open computer algebra system for polynomial computations Singular [6]. This has been included in the library GraphAlg.lib, which is available online at http://personales.us.es/raufalgan/LS/GraphAlg.lib. Let us illustrate the use of this procedure with an example related to the distribution of the set P2(F2) of two-dimensional partial quasigroup rings over the finite field F2into isotopism and isomorphism classes. All the computations that are exposed throughout this paper are implemented in a system with an Intel Core i7-2600, with a 3.4 GHz processor and 16 GB of RAM. 7 Example 1. Let us consider the pair of partial quasigroup rings in P2(F2) that are respectively related to the partial Latin squares 1 2 2and 1 2 2 1 These two partial Latin squares are not isotopic because isotopisms preserve the number of filled cells. Nevertheless, their related partial quasigroup rings over F2, with respective bases {e1, e2}and {e′ 1, e′ 2}, and which are respectively described by the products (e1e1=e1, e1e2=e2=e2e1.and (e′ 1e′ 1=e′ 1=e′ 2e′ 2, e′ 1e′ 2=e′ 2=e′ 2e′ 1. are isotopic. Specifically, by implementing the procedure isoAlg, our system computes in 0seconds the existence of four isotopisms between these two partial quasigroup rings. One of this isotopisms is, for instance, the isomorphism fsuch that f(e1) = e′ 1and f(e2) = e′ 1+e′ 2. The procedure isoAlg also ensures us the existence of fas the unique possible isomorphism. ⊳ In practice, in those cases in which the run time required for the computations involved in Theorem 2 becomes excessive, it is recommendable to eliminate the generators of the corresponding ideal that are referred to the determinants of the matrices F,Gand H. This reduces the complexity time in Corollary 1 to qO(3n2)+O(n8) and qO(n2)+O(n8), respectively, and gives enough information to analyze a case study on which base the possible isomorphisms and isotopisms between two given algebras, whatever the base field is. The next example illustrates this fact by focusing on the possible isotopisms that there exist over any field between the two partial quasigroup rings that appear in Example 1. Example 2. The implementation of the procedure isoAlg enables us to ensure that, whatever the base field is, the reduced Gr¨obner basis of the ideal IIsot A,A′in Theorem 2 related to the isotopism group between the two partial quasigroup rings of Example 1 holds that 2h3 22 = 0 and h2 21 +h2 22 = 0. If the characteristic of the base field is not two, then h21 =h22 = 0. This involves Hto be singular and hence, these two partial quasigroup rings are not isotopic. Otherwise, it is straightforwardly verified that the linear transformation fthat is indicated in Example 1 constitutes an isomorphism between both rings for every base field of characteristic two. ⊳ 8 3. Description of faithful functors between algebras and graphs Based on the proposal of McKay et al. [17] for Latin squares, we describe now a pair of graphs that are uniquely related to a finite-dimensional algebra Aover a finite field K. Firstly, we define the vertex-colored graph G1(A) with four maximal monochromatic subsets RA={ru|u∈A\AnnA−(A)}, CA={cu|u∈A\AnnA+(A)},SA={su|u∈A2\ {0}} and TA={tu,v | u, v ∈A, uv 6= 0}, and edges {rutu,v, cvtu,v, suvtu,v |u, v ∈A, uv 6= 0}. From this graph we also define the vertex-colored graph G2(A) by adding the edges {rucu,|u∈A\AnnA(A)} ∪ {cusu|u∈A2\AnnA+(A)} ∪ {rusu|u∈ A2\AnnA−(A)}. As an illustrative example, Figure 2 shows the two graphs that are related to any n-dimensional algebra over the finite field F2, with basis {e1,...,en}, that is described as e1e2=e2e1=e1. G1G2 Figure 2: Graphs related to the algebra e1e2=e2e1=e1over F2. Lemma 4. The next assertions hold. a) If the algebra Ais abelian, then G1(A)and G2(A)have no vertices. b) The graph G1(A)does not contain triangles. c) In both graphs G1(A)and G2(A), •The number of vertices is |A\AnnA−(A)|+|A\AnnA+(A)|+|A2|+|{(u, v)∈A×A|uv 6= 0}| − 1. •The degree of the vertex tu,v is d(tu,v) = 3,for all u, v ∈Asuch that uv 6= 0. 9 Partial Latin square Vertices Edges Partial Latin square Vertices Edges Partial Latin square Vertices Edges 100 000 000 (4,4,1,16) 48 100 010 002 (7,7,3,34) 120 031 302 (7,7,7,42) 126 120 000 000 (4,6,3,24) 72 120 001 002 (7,7,3,36) 108 120 210 301 (7,7,7,42) 126 123 000 000 (4,7,7,28) 84 120 200 002 (7,7,3,36) 108 120 213 001 (7,7,7,42) 126 100 200 000 (6,4,3,24) 72 120 200 001 (7,7,3,38) 114 120 213 300 (7,7,7,42) 126 100 010 000 (6,6,1,24) 72 120 210 001 (7,7,3,38) 114 120 001 312 (7,7,7,43) 129 100 020 000 (6,6,3,28) 84 120 201 010 (7,7,3,40) 120 120 201 302 (7,7,7,43) 129 120 200 000 (6,6,3,32) 96 120 201 012 (7,7,3,40) 120 120 231 300 (7,7,7,43) 129 120 210 000 (6,6,3,32) 96 100 020 003 (7,7,7,37) 111 123 231 312 (7,7,7,43) 129 120 000 300 (6,6,6,32) 96 120 002 003 (7,7,7,38) 114 120 003 312 (7,7,7,44) 132 120 000 310 (6,6,6,36) 108 120 002 300 (7,7,7,38) 114 120 013 301 (7,7,7,44) 132 120 001 000 (6,7,3,32) 96 120 003 300 (7,7,7,38) 114 120 013 302 (7,7,7,44) 132 120 012 000 (6,7,3,36) 108 120 001 300 (7,7,7,39) 117 120 200 312 (7,7,7,44) 132 120 003 000 (6,7,7,34) 102 120 200 003 (7,7,7,40) 120 120 203 301 (7,7,7,44) 132 120 000 302 (6,7,7,36) 108 120 200 302 (7,7,7,40) 120 123 210 301 (7,7,7,44) 132 123 200 000 (6,7,7,36) 108 120 210 003 (7,7,7,40) 120 123 031 310 (7,7,7,45) 135 120 013 000 (6,7,7,38) 114 123 010 001 (7,7,7,40) 120 123 200 312 (7,7,7,45) 135 123 210 000 (6,7,7,38) 114 123 200 300 (7,7,7,40) 120 123 230 310 (7,7,7,45) 135 123 230 000 (6,7,7,40) 120 120 001 302 (7,7,7,41) 123 123 012 230 (7,7,7,46) 138 123 231 000 (6,7,7,40) 120 120 001 310 (7,7,7,41) 123 123 210 031 (7,7,7,46) 138 100 200 300 (7,4,7,28) 84 120 201 300 (7,7,7,41) 123 123 201 312 (7,7,7,46) 138 100 200 010 (7,6,3,32) 96 123 200 010 (7,7,7,41) 123 120 200 010 (7,6,3,36) 108 120 003 310 (7,7,7,42) 126 100 200 030 (7,6,7,34) 102 120 010 301 (7,7,7,42) 126 120 030 300 (7,6,7,36) 108 120 010 302 (7,7,7,42) 126 120 200 300 (7,6,7,36) 108 120 012 300 (7,7,7,42) 126 120 010 300 (7,6,7,38) 114 120 013 300 (7,7,7,42) 126 120 210 300 (7,6,7,38) 114 120 200 013 (7,7,7,42) 126 120 230 300 (7,6,7,40) 120 120 203 001 (7,7,7,42) 126 120 230 310 (7,6,7,40) 120 120 203 300 (7,7,7,42) 126 100 010 001 (7,7,1,28) 84 123 010 300 (7,7,7,42) 126 Table 5: Invariants of the graph G1related to non-abelian partial algebras in P3(F2). 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