CC BY
5
УДК 519.21
Algebraic Sets with Fully Characteristic Radicals
Mohammad Shahryari*
Faculty of Mathematical Sciences University of Tabriz 29 Bahman Blvd, Tabriz, 5166616471
Iran
Received 26.10.2016, received in revised form 26.11.2016, accepted 06.03.2017 We obtain a necessary and sufficient condition for an algebraic set in a group to have a fully characteristic radical. As a result, we see that if the radical of a system of equation S over a group G is fully characteristic, then there exists a class X of subgroups of G such that elements of S are identities of X.
Keywords: algebraic structures, equations, algebraic set, radical ideal, fully invariant congruence, fully
characteristic subgroup.
DOI: 10.17516/1997-1397-2017-10-3-293-297.
In this article, our notations are the same as [2-5] and [6]. The reader should review these references for a complete account of the universal algebraic geometry. We fix an algebraic language L and we denote equations in the both forms p « q and (p, q). For an algebra A of type L and any system of equations S C AtL(xi,... ,xn), we denote the corresponding radical ideal by RadA(S). Recall that
RadA(S) = {(p « q) e Atc(x1, ...,xn): Va(S ) C VA(p « q)},
where VA(S) and VA(p « q) are the corresponding algebraic sets of S and p « q, respectively. In other words, RadA(S) is the set of all atomic logical consequences of S in A. In general, one can not give a deductive description of RadA(S), because it depends on the axiomatizablity of the prevariety generated by A, [7]. In this direction, any good description of the radicals is important from the universal algebraic geometric point of view. In this article, we give a necessary and sufficient condition for RadA(S) to be fully characteristic congruence (invariant under all endomorphisms). We apply our main result to obtain connections between radicals, identities, coordinate algebras and relatively free algebras. Although most of the results can be formulate in the general frame of arbitrary algebraic structures, we mainly focus on groups in what follows. As a summary, we give here some results in the case of coefficient free algebraic geometry of groups.
Let G be a group and E C Gn be an algebraic set (with no coefficients). Then the radical Rad(E) is a fully characteristic (equivalently verbal) subgroup of the free group Fn, if and only if, there exists a family {Ki} of n-generator subgroups of G such that E = |Ji Kn. As a result, we will show that if Rad^(S) is a verbal subgroup of Fn, then there exists a family X of n-generator subgroups of G such that Rad^(S) is exactly the set of all group identities valid in X. We also see that under this conditions, there exists a variety W of groups, such that the n-generator relatively free group in W is the coordinate group of S.
1. Main result
Suppose L is an algebraic language and V is a variety of algebras of type L. For a finite set X = {xi,..., xn} of variables, FV(n) denotes the relatively free algebra of V generated by X.
* mshahryari@tabrizu.ac.ir © Siberian Federal University. All rights reserved
A congruence R in FV(n) is called fully characteristic (or fully invariant), if for all endomorphism a : FV(n) ^ FV(n) and (p,q) G R, we have (a(p),a(q)) G R. The following theorem concerns the situation the radicals of algebraic sets in which are fully characteristic.
Theorem 1. Let A G V be an algebra and E C An be an algebraic set. Then Rad(E) is fully characteristic congruence of FV(n), if and only if E = |Ji Kn for a family [Ki] of n-generator subalgebras of A.
Proof. Let [Ki] be a family of n-generator subalgebras of A and E = |Ji Kn be algebraic. Let a : FV(n) ^ FV(n) be an endomorphism and (p, q) G Rad(E). Suppose
a(xi) vi(x1i • • • i xn)
and a = (ai,..., an) G E. Hence, there is an index i such that ai,... ,an G Ki. Note that we have
a(p)(a) = p(vi,. .. ,vn)(a) = p(vi(a), .. . ,Vn(a)).
Since Ki is a subalgebra, so for any j we have Vj (a) G Ki, and therefore
(vi(a),...,vn(a)) G Kn C E.
This shows that
p(vi(a), .. ., Vn(a)) = q(vi(a), .. .,Vn(a)) = a(q)(a),
so, we have (a(p),a(q)) G Rad(E). Conversely, suppose that Rad(E) is fully characteristic and v1,... ,vn G FV(n) are arbitrary. Consider the endomorphism a(xi) = vi. For any (p,q) G Rad(E), we have (a(p), a(q)) G Rad(E), so for any a G E, the equality
p(vi(a),. . .,vn(a)) = q(vi(a),. . .,vn(a))
holds. This shows that
(vi(a),..., vn(a)) G VA(Rad(E)) = E. Therefore, for arbitrary vi,...,vn G FV(n) and a G E, we have
(vi(a),...,vn(a)) G E. Let K(a) be the subalgebra generated by ai,... ,an. We have
(vi(a),...,vn(a)) G K(a)n,
and hence
E = U K (a)n.
a
The proof is now completed. □
As a result, we see that for an arbitrary system of equations S, the radical RadA(S) is fully characteristic, if and only if, there exists a family [Ki] of n-generator subalgebras of A, such that
Va(S ) = y Kn. i
This shows that RadA(S) = p|i Rad(Kn). Since for an arbitrary algebra K, the radical Rad(Kn) is the set of identities of K with variables in X, so we have the following corollary.
Corollary 1. Let S be a system of equations in the variety V and A G V. If Rad^(S) is a fully characteristic, then there is a family of n-generator subalgebras of A, say X, such that
RadA(S ) = idx(n).
2. Application to groups
The first application of our main result is about algebraic sets in nilpotent groups the coefficient-free radicals in which are characteristic in the free group Fn. It is well-known that the classes of fully characteristic and verbal subgroups of Fn are the same. But it is a very hard problem to determine the structure of characteristic subgroups in free groups. In [8], Vaughan-Lee proved that if C is a characteristic subgroup of Fn and Yn(Fn) C C, then C is fully characteristic. We can use this result of Vaughan-Lee together with our main result to prove the following.
Theorem 2. Let G be a nilpotent group of class at most n and A C Gn be an algebraic set. Then Rad(A) is a characteristic subgroup of Fn, if and only if A = |Ji Kn for some family {Ki} of n-generator subgroups of G.
Proof. Let Rad(A) be a characteristic subgroup. Suppose w e jn(Fn). Since G is nilpotent of class at most n, so for all (ai,...,an) e A, we have w(ai,...,an) = 1. This shows that w e Rad(A). Therefore, we have Yn(Fn) C Rad(A) and hence the radical is a fully characteristic subgroup. Now the assertion follows from our main result. □
We are now going to consider the case of G-groups. Let G be an arbitrary group and V be the variety of G-groups (for the basic notions of G-groups, see [1]). Note that in this case we have FV (n) = G[X], where G[X] denotes the free product G * Fn. It is shown in [9] that the fully characteristic subgroups of G[X] are exactly the G-verbal subgroups. Recall that for an arbitrary set W of group words with coefficients from G, variables Ti,..., Tm and arbitrary G-group H, the G-verbal subgroup corresponding to W is the subgroup W(H) of H, generated by the set
{w(hi,... ,hm) : hi,...,hm e H,w e W}.
Corollary 2. Let S C G[X] and RadG(S) be a G-verbal subgroup of G[X]. Then all elements of S are G-identities of G.
Proof. Let E = VG(S). Since Rad(E) is G-verbal, so E = |Ji Kn, where every Ki is a G-subgroup of G. But, the only G-subgroup of G is G itself. So, we have E = Gn and hence any element of S is a G-identity of G. □
Suppose H is a G-group. Recall that the coordinate group of a set E e Hn is defined as
G[X ]
r(E) =
Rad(E) '
Similarly, for a system of equations S, we define the coordinate group rH(S) to be r(VH(S)). One of the main problems of the algebraic geometry over groups is the investigation of the structure of this coordinate group. We show that if RadH (S) is a G-verbal subgroup, then there exists a variety W of G-groups such that rH(S) = FW(n), the n-generator relatively free group of W.
Corollary 3. Let H be a G-group and S be a system of G-equations such that VH(S) = |Ji Kn for some family of n-generator G-subgroups of H. Then there is a variety W such that rH (S) = FW (n). The converse is also true.
Proof. We proved that RadH (S) is a G-verbal subgroup and hence there is a set W of G-group words, such that
RadH (S) = W (G[X ]).
Let W be the variety of G-groups defined by W. Then, we have
r (S) G[X] G[X] ( )
Th (S) = RMS) = W (G[X ]) Fw(n).
Conversely, suppose there is a variety W such that rH (S) = FW (n). Let W be the set of all n-variable G-identities valid in W. Note that W = W(G[X]) and
F GX ] Fw (n) = W.
This shows that RadH(S) = W(G[X]) is a G-verbal subgroup of G[X] and hence VH(S) has a decomposition VH (S) = |Ji Kn for some family of n-generator G-subgroups of H. □
There are some examples of equational Noetherian relatively free groups: ordinary free groups, free elements of the varieties of the form var(H) in which H is equational Noetherian and free solvable groups of given fixed derived lengths. As another application, we obtain a sufficient condition for a variety to have equational Noetherian relatively free elements.
Corollary 4. Let H be a G-equational Noetherian group and W be a set of G-group words such that the G-verbal subgroup W(G[X]) is the radical of some subset of Hn. Let W be the variety of G-groups defined by W. Then FW(n) is G-equational Noetherian.
As a final point, note that one can ask about conditions under which the radical of a system of equations is a characteristic subgroup. It seems that this is more complicated but interesting problem.
References
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Алгебраические множества с вполне характеристическими радикалами
Мохаммад Шахриари
Факультет математических наук Университет Табриза Бахман бульвар, 29, Табриз, 5166616471
Иран
Получено необходимое и достаточное условие того, чтобы алгебраическое множество в группе имело вполне характеристический радикал. В результате показано, что если радикал системы уравнений Я над группой О является вполне характеристическим, то существует такой класс X подгрупп в О, что элементы из Я — тождества X.
Ключевые слова: алгебраические структуры, уравнения, алгебраическое множество, радикальный идеал, вполне инвариантная конгруэнция, вполне характеристическая подгруппа.
