CC BY
34
1Аи(к)1 > кзЛ(33)
к
где Д„(^ = (Лп-1 (k)Ь(k),...,Л(k)Ь(k),Ь(k)). Выберем спектр замкнутой системы в виде
= Фг = —г (г = 1,...,п). (34)
Л п +1
Поскольку согласно теореме Гершгорина [5] каждое собственное число ^ (k) матрицы Л(и) расположено в одном из кругов
п
3 = 1',зфг
для отделимости спектра (34) от спектра матрицы Л(и) достаточно выполнения условия
п
|а« (k)| > 1ац (k)l + 8 (г = 1,...,п; 6> 0). (35)
Поэтому будем предполагать, что параметр Л удовлетворяет неравенству
Л > max{ 1, 1/8}. (36)
Определим управление формулой
4^) = в*^^), (37)
где в^) является решением системы уравнений
в* (^<1^) = -1 (г = 1,..., п), (38)
<(k) = (Л(k) - Л^п)-^). (39)
Замкнутая управлением (37) система (30) примет вид
Хк+1 = Б^х^), (40)
где Б(Ь) = Л(^) + Ь^)в*^). Воспользуемся спектральным разложением
D(k) = ^2 XiMi(k), Mi(k) = di(k)g*(k), D(k)di(k) = Xidi(k), D* (k)gi(k) = \^(к),
d*(k)gi(k) =
i=l
1 при i = j,
0 при i = j.
Приращение функции Ляпунова V(x(k)) = |x(k)|2 имеет вид
V(x(k + 1)) — V(x(k)) = x*(k)P(k)xk,
где P(k) = D(k)D*(k) — In. Поэтому для отрицательности матрицы P(k) достаточно выполнения условия
sup ID(k)l < 1, (41)
k
которое будет иметь место, если справедливо соотношение
п
эир^ |М3(к)| < Л.
к - .. 3 = 1
Рассуждая так же, как в разделе 2, приходим к оценке
пп
впр^ |Мз(к)| < К9Л-»^ |(к)|.
к 3=1 1,3=1
Поэтому неравенство (41) будет выполнено, если
п л^+1 вир V \Пц{к)\ < -. (42)
к ,.л Кд
»,3 = 1
Сформулируем полученный результат.
Теорема 2. При выполнении условий (31), (32), (33), (35), (36) система (30) глобально экспоненциально устойчива, если управление определяется формулами (37), (38), (39), (34).
4. Заключение. Рассматривается непрерывная система, у которой элементы матрицы объекта управления и столбца распределения управления являются неупре-ждающими функционалами произвольной природы и ограничены. В частности, они могут быть нелинейными функциями времени и состояния системы. Предполагается, что определитель матрицы управляемости отделен от нуля. По выбранному постоянному спектру, расположенному в левой полуплоскости, строится линейная обратная связь, коэффициенты которой выражаются через элементы матрицы объекта управления и столбца распределения управления. Найдены условия, при которых замкнутая система глобально экспоненциально устойчива. Аналогичный результат получен для дискретной системы.
Литература
1. Isidory A. Nonlinear Control Systems. II. London: Springer, 1999.
2. Zak S. H. Systems and Control. Oxford: Oxford Univ. Press, 2002.
3. Гелиг А. Х., Леонов Г. А., Якубович В. А. Устойчивость нелинейных систем с неединственным состоянием равновесия. М.: Наука, 1978.
4. Yakubovich V. A., Leonov G. A., Gelig A. Kh. Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. New Jersey: World Scientific, 2004.
5. Гантмахер Ф. Р. Теория матриц. М.: Наука, 1967.
Статья поступила в редакцию 27 июня 2013 г.
YflK 512.53
BecTHHK Cn6ry. Cep. 1. 2013. Bun. 4
FINITE BASIS PROBLEM FOR THE DIRECT PRODUCT OF SOME J-TRIVIAL MONOID WITH GROUPS OF FINITE EXPONENT
Edmond W. H. Lee
Division of Math, Science, and Technology, Nova Southeastern University, Florida 33314, USA, PhD, Assistant Professor, edmond.lee@nova.edu
1. Introduction
For any set W of words over a countably infinite alphabet X, let S(W) denote the Rees quotient of the free monoid X* modulo the ideal of all words that are not factors of any word in W. Equivalently, S(W) can be treated as the monoid that consists of every factor of every word in W, together with a zero element 0, with binary operation • given
by
U v _ / UV, if UV is a factor of some word in W; | 0, otherwise.
The empty factor, more conveniently written as 1, is the unit element of the monoid S (W). If W _ {Wi,...,Wn}, then write S(W) _ S(Wi,..., Wn). Each S(W) is a J-trivial monoid with 0 and 1 as its only idempotent elements.
The class R of Rees quotient monoids of the form S(W) constitutes a significant source of examples in the study of the finite basis problem for semigroups and monoids. For instance, one of the first published examples of non-finitely based finite semigroups, due to Perkins [10] in the 1960s, is the monoid
S(xyxy, xyzyx, xzyxy, x2z)
of order 25. Recent investigations [3-8, 12] shed more light on the finite basis problem for monoids in R. In particular, results from Jackson [3] and Jackson and Sapir [5] demonstrate that with respect to the finite basis property, the class R behaves as irregularly as the class of all finite semigroups. Refer to the survey by Volkov [14] for more information on the finite basis problem for monoids in R and for finite semigroups in general.
The present article is concerned with the finite basis problem for the class of monoids. One of the aforementioned irregularities of the class R demonstrated by Jackson and Sapir is the non-closure of its finitely based finite monoids under the formation of direct products.
Theorem 1 (Jackson [4] and Jackson and Sapir [5]). There exist finitely based finite monoids Mi and M2 for which the direct product Mi x M2 is non-finitely based. Examples of these monoids include
• M1 _ S(xyhxty) and M2 _ S(xhytxy);
• Mi _ S(x2y2x, xyxyx, xy2x2) and M2 _ S(x2y2,xyxy,xy2x,yx3y);
• Mi _ S(xyzxy, xyzyx) and M2 _ S(xnyn) for any n > 2.
Remark 2 Although there exist examples of two finitely based finite semigroups with non-finitely based direct product [11, 13], none of these examples implies examples of monoids Mi and M2 that satisfy Theorem 1.
© Edmond W. H. Lee, 2013
One objective of the present article is to expand the list in Theorem 1 with new, simpler examples of monoids Mi and M2. This is achieved by a general result concerning the Rees quotient monoid S(xyx) of order seven.
Theorem 3 The direct product of the monoid S(xyx) with any noncommutative group of finite exponent is non-finitely based.
The monoid S(xyx) was first investigated in detail by Jackson [1, 2], who proved that it is a finitely based monoid that generates a semigroup variety with continuum many subvarieties. He later showed that the monoid variety generated by S(xyx) contains only five subvarieties [4].
Now the well-known theorem of Oates and Powell states that all finite groups are finitely based [9]. Therefore Theorem 3 provides new, easily described monoids Mi and M2 that satisfy Theorem 1; the simplest case occurs when Mi is the monoid S(xyx) and M2 is the symmetric group of order six. Note that most of the monoids in Theorem 1 are substantially larger:
• IS(xyhxty)l = |S (xhytxy)l = 21;
• IS(x2y2x, xyxyx, xy2x2)l = 22 and |S(x2y2,xyxy,xy2x,yx3y)| = 21;
• IS(xyzxy, xyzyx)l = 21 and |S(xnyn)l = (n + 1)2 + 1.
In contrast with Theorem 3, the direct product of the monoid S(xyx) with any commutative group is finitely based [7]. Therefore the finite basis problem for the direct product of S(xyx) with any group of finite exponent has a complete solution.
Corollary 4 For any group G of finite exponent, the direct product S(xyx) x G is finitely based if and only if G is commutative.
Notation and background material are given in Section 2. Some restrictions on identities satisfied by the monoid S(xyx) and by noncommutative groups are established in Section 3. The proof of Theorem 3 is then given in Section 4.
2. Preliminaries
Let X be a countably infinite alphabet throughout. For any subset Y of X, let Y+ and Y* denote the free semigroup and free monoid over Y, respectively. Elements of X are called letters and elements of X* are called words. For any word W,
• the number of occurrences of a letter x in W is denoted by occ(x, W);
• a letter x is simple in W if occ(x, W) = 1;
• the set of simple letters of W is denoted by sim (W);
• the content of W, denoted by con(W), is the set of letters occurring in W.
An identity is written as U « V where U,V G X +. An identity U « V is balanced if occ(x, U) = occ(x, V) for all x G X .A monoid M satisfies an identity U « V if, for any substitution <p : X —^ M, the elements U<p and V<p of M coincide. A set E of identities satisfied by a monoid M is a basis of M if E implies every identity satisfied by M .A monoid is finitely based if it possesses a finite basis.
For any word W and any set Y of letters, let W[Y] denote the word obtained from W by retaining the letters from Y. It is easily shown that if a monoid satisfies an identity U « V, then it also satisfies the identity U [Y ] ~ V [Y ] for any Y^X.
3. Restrictions on identities
For each n > 1 , define the word
Jn _ ( n(x* hi) ) y{ n(xizi)j y( JJ(tizi)
_ xihi • • • xnhnyxizi • • • xnznytizi • • • tnzn.
The words Jn were first employed by Jackson [4, proof of Lemma 5.5] to prove that the monoid S(xhxyty) is non-finitely based.
Lemma 5 (Jackson [4, Lemmas 3.3 and 4.4]). Let U « V be any identity satisfied by the monoid S(xyx). Then
(i) sim(U) _ sim(V) and con(U) _ con(V);
(ii) U[x, y] _ xy if and only if V[x, y] _ xy;
(iii) U [x, y] _ xyx if and only if V [x, y] _ xyx.
Lemma 6 Let Jn « W be any identity satisfied by the monoid S(xyx). Then
(i) W[xi, hj] _ xihjxi if and only if i < j;
(ii) W[ti, zj] _ zjtizj if and only if i < j;
(iii) W[xi,tj] _ xitjxi and W[hi,zj] _ zjhizj for any i and j;
(iv) W[hi, y] _ yhiy and W[y,U] _ yUy for any i. Proof. This follows from Lemma 5 (iii) because
(i) Jn[xi, hj] _ xihjxi when i < j and Jn[xi, hj] _ hjx2 when i > j;
(ii) Jn\ti, zj] _ zjtizj when i < j and Jn[ti, zj] _ zj^ti when i > j;
(iii) Jn[xi,tj] _ x2tj and Jn[hi,zj] _ hizj2 for any i and j;
(iv) Jn [hi, y] _ hiy2 and Jn[y,U] _ y2ti for any i. □
Lemma 7 Let U « V be any identity satisfied by any noncommutative group with U[x,y] _ xyxy. Then V[x, y] (j/ {x2y2,xy2x,yx2y}.
Proof. Any group that satisfies any of the following identities is commutative: xyxy « x2y2, xyxy « xy2x, and xyxy « yx2y. □
Lemma 8 Let Jn « W be any identity satisfied by a noncommutative group. Then
(i) W[xi, y] _ x2y2 and W[y, zi] _ y2z2 for any i;
(ii) W[xi, zj] _ xizjxizj when i < j;
(iii) W[xi, zj] _ x2z2 when i > j.
Proof. This follows from Lemma 7 because
(i) Jn[xi, y] _ xiyxiy and Jn[y, zi] _ yziyzi for any i;
(ii) Jn[xi, zj] _ x2zj when i < j;
(iii) Jn[xi, zj] _ xizjxizj when i > j. □
Lemma 9 Suppose that Jn « W is any balanced identity satisfied by the monoid S(xyx) x G, where G is any noncommutative group. Then Jn _ W.
Proof. Since Jn[hi,ti | 1 < i < n] _ hi • • • hnti • • • tn, it follows from Lemma 5 (ii) that W[hi,ti | 1 < i < n] _ hi • • • hnti • • • tn. Therefore by Lemma 6 (i)-(iii),
(a) W [xi, hi ,ti, zi | 1 < i < n] _ xi hi • • • xnhnUtizi • • • tnzn
for some U ( {xi, zi,..., xn, zn}+ such that occ(xi,U) _ occ(zi,U) _ 1 for all i. If U[xi, xi+i] _ xi+ixi for some i, then W[xi, xi+i] _ xix22+ixi, whence Lemma 7 is violated because Jn[xi,xi+i] _ xixi+ixixi+i. Therefore U[xi,xi+i] _ xixi+i for all i so that
(b) U[xi,.. .,xn]_ xi • • • xn. Similarly,
(c) U [zi, ...,zn]_ zi ••• zn.
Hence by (a)-(c) and Lemma 8 (ii)-(iii),
W[xu hi,ti,zi | 1 < i < n] = xihi • • • xnhnx\z\ • • • XnZntiZi • • • tnZn.
Since the identity Jn « W is balanced, occ(y, W) = 2. It now follows from Lemmas 6 (iv) and 8 (i) that W = Jn. □
4. Proof of Theorem 3
Suppose that G is any noncommutative group of finite exponent m. Let Jn denote the word obtained from Jn by replacing the first y with ym+1:
Jn = ( f[(xihi^j ym+1( f[(xiz^ y ( f[(tiZi)) .
Lemma 10 For each n > 1, the identity Jn « J'n is satisfied by the monoid S(xyx) x G.
Proof. Let p : X ^ S(xyx) be any substitution. Since the first and last occurrences of y in Jn and in Jn do not sandwich any simple letters, it is easily seen that if yp = 1, then Jnp and Jnp are not factors of xyx so that Jnp = 0 = J!rip in S(xyx). If yp = 1, then clearly Jnp = J!rip in S(xyx). Therefore the monoid S(xyx) satisfies the identity J « J'
Jn ~ Jn.
Since the group G has exponent m, it satisfies the identity ym+1 « y and so also the identity Jn « Jn. □
Seeking a contradiction, suppose that the monoid S(xyx) x G is finitely based, say with finite basis E of identities. Then there exists n > 1 such that each identity U « V in E involves at most n distinct letters, that is, |con(UV)| < n. It is shown in Lemma 11 below that no such identity can be used to convert the word Jn into a different word. Therefore the identity Jn « J'n is not implied by E and so is not satisfied by the monoid S(xyx) x G, contradicting Lemma 10.
Lemma 11 Let U « V be any identity satisfied by the monoid S(xyx) x G with |con(UV)| < n. Suppose that there exist words P,Q £ X* and an endomorphism p : X* ^ X* such that Jn = P(Up)Q. Then Jn = P(Vp)Q.
Proof. The lemma clearly holds if Up = 1, so assume that Up = 1. By Lemma 5
(i),
(a) sim(U) = sim(V) and con(U) = con(V).
Further, the word Up is a nonempty factor of Jn and occ(x,Jn) < 2 for all x £ X. Therefore
(b) if x £ con(U) and xp = 1, then occ(x, U) < 2. For the remainder of this proof, it is shown that
(t) if x £ con(U) and xp = 1, then occ(x, U) = occ(x, V). It follows that P(Up)Q « P(Vp)Q is a balanced identity satisfied by the monoid S(xyx) x G. Consequently, Jn = P(Up)Q = P(Vp)Q by Lemma 9.
Let x £ con(U) be such that xp = 1. If the letter x is simple in U, then occ(x, U) = 1 = occ(x, V) by (a) so that (t) holds. Therefore it suffices to further assume that the letter x is non-simple in U, whence occ(x, U) = 2 by (b). Thus U = UixU2xU3 for some U1, U2, U3 £ X* with x £ con(U1U2U3), and
Jn = P (U1p)(xp)(U2 p)(xp)(U3 p)Q.
Now it is easily seen that
(c) if Jn = • • • W • • • W • • • for some W G X +, then W G {x\,..., xn, y,z1,..., zn}. Therefore the factor xp of Jn coincides with one of x\,... ,xn,y, z\,..., zn. It follows that
(d) the factor U2p of Jn contains 2n distinct letters.
Since |con(UixU2xU3)| = |con(U)| < n, the word U2 contains fewer than n distinct letters. If each letter in U2 is mapped by p to either 1 or a single letter, then |con(U2p)| < |con(U)| < n violates (d). Therefore there exists some letter in U2, say z, such that zp is neither 1 nor a single letter, that is,
(e) the factor zp of Jn contains at least two letters.
Now write U2 = U2zU2' for some Щ, U2! G X* so that U = UlxU2zU2!xU3. If the letter z is non-simple in U, then
Jn = P (U<p)Q = P ••• zp ••• zp ••• Q,
which is impossible in view of (c) and (e). The letter z is thus simple in U, whence U[x, z] = xzx. Lemma 5 (iii) then implies that V[x, z] = xzx. Therefore occ(x, U) = 2 = occ(x, V) and (t) holds. □
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Статья поступила в редакцию 27 июня 2013 г.
