Semigroups, Proceedings of the International Conference in Braga, Portugal, 1999 (P. Smith, E. Giraldes and P. Martins, eds.), World Scientific, 2000, pp. 106–113
SEMIGROUPS SATISFYING SOME VARIABLE IDENTITIES ´ ´ AND TATJANA PETKOVIC ´ MIROSLAV CIRI C ´ University of Niˇs, Faculty of Philosophy, Cirila i Metodija 2, 18000 Niˇs, Yugoslavia E-mail:
[email protected], {mciric,tanjapet}@archimed.filfak.ni.ac.yu ´ STOJAN BOGDANOVIC University of Niˇs, Faculty of Economics, Trg VJ 11, 18000 Niˇs, Yugoslavia E-mail:
[email protected]
Putcha and Weissglass in [13] and [14] used variable identities to characterize periodic semigroups which are nilpotent extensions of unions of groups and semilattices of groups. In this paper they are used to describe periodic semigroups which are nil-extensions and retractive nil-extensions of unions of groups in the general and various special cases. The obtained results generalize those of Putcha and Weissglass, as well as the results of Bell [2] concerning rings satisfying vari´ c and Bogdanovi´ able semigroup identities and the results of Ciri´ c [7] concerning semigroups satisfying some ordinary identities.
1
Introduction and Preliminaries
An identity over an alphabet A is a pair of words from the free semigroup A+ which is usually written as a formal equality of these words. A semigroup S is said to satisfy a set of identities Σ over A if the kernel of each homomorphism from A+ into S contains Σ. But if the kernel of each homomorphism from A+ into S contains a non-trivial identity from Σ, then we say that S satisfies variabily Σ, or that it satisfies Σ as a variable identity. This is the same concept which was introduced by Putcha and Weissglass in [13] and [14], but the definition given here is closer to the definition of ordinary identities than the one of Putcha and Weissglass. In a way, this concept traces one’s origin to the concept of pseudo identities and pseudo varieties, introduced by Schein in the 1960’s (or disjunctive identities and varieties, as they were called in [12]). The related concepts, the so-called inclusive identities and collective identities, were studied in [1], [10], [11] and [12]. Putcha and Weissglass in [13] and [14] used variable identities to characterize periodic semigroups which are nilpotent extensions of unions of groups Supported by Grant 04M03B of RFNS through Math. Inst. SANU.
44
and semilattices of groups. In this paper they are used to describe periodic semigroups which are nil-extensions and retractive nil-extensions of unions of groups in the general and various special cases. The obtained results generalize those of Putcha and Weissglass, as well as the results of Bell [2] concerning ´ c and Bogrings satisfying variable semigroup identities and the results of Ciri´ danovi´c [7] concerning semigroups satisfying some ordinary identities. ´ c and Bogdanovi´c in [8] used variable identities to Note also that Ciri´ describe semigroups having some properties as hereditary ones. For more information about variable identities and various related concepts we refer to ´ c and Petkovi´c [6]. the survey paper by Bogdanovi´c, Ciri´ Throughout the paper, N denotes the set of all positive integers and for m, n ∈ N, m ≤ n, we set [m, n] = {i ∈ N | m ≤ i ≤ n}. For a semigroup S, E(S) denotes the set of all idempotents of S, and Gr(S) is the set of all group (completely regular) elements of S. For e ∈ E(S), Ge denotes the maximal subgroup of S with e as its identity, and Te = {x ∈ S | (∃n ∈ N) xn ∈ Ge }. A semigroup S with zero 0 is a nil-semigroup if for any a ∈ S there exists n ∈ N such that an = 0, and a nilpotent semigroup if there exists n ∈ N such that S n = {0}, and then S is also called n-nilpotent. Let a semigroup S be an ideal extension of a semigroup T . If the Rees factor semigroup S/T is a nilsemigroup (nilpotent, n-nilpotent) then S is called a nil-extension (nilpotent extension, n-nilpotent extension) of T , and if there exists a homomorphism ϕ : S → T such that aϕ = a, for each a ∈ T , then ϕ is called a retraction and S is a retractive extension of T . We shall use the following notations for classes of semigroups: notation UG LG RG G
class of semigroups
notation S N Nn
unions of groups left groups right groups groups
class of semigroups semilattices nil-semigroups n-nilpotent semigroups
Let X1 and X2 be two classes of semigroups. By X1 ◦X2 we denote the Mal’cev product of the classes X1 and X2 , i.e. the class of all semigroups S on which there exists a congruence % such that S/% ∈ X2 and any %-class which is a subsemigroup of S belongs to X1 . It is clear that X ◦ S is the class of all semilattices of semigroups from the class X . If X2 is a subclass of the class N of nil-semigroups, then X1 ◦ X2 is the class of all semigroups which are ideal (nil) extensions of semigroups from X1 by semigroups from X2 . In this case by X1 ~ X2 we denote the class of all semigroups which are retractive extensions of semigroups from X1 by semigroups from X2 . 45
The free semigroup over an alphabet A is denoted by A+ , and for n ∈ N, An = {x1 , x2 , . . . , xn }. For a word w ∈ A+ , |w| denotes the length of w, |x|w the number of appearances of the letter x in w, h(w) (t(w)) the first (last) letter of w (head and tail of w), and c(w) denotes the set of all letters which appear in w. To emphasize the fact that x1 , x2 , . . . , xn are all letters that appear in w we write w(x1 , x2 , . . . , xn ) instead of w. For w ∈ A+ such that |w| ≥ 2, h(2) (w) (t(2) (w)) denotes the prefix (suffix) of w of the length 2. If w ∈ A+ , x ∈ A, such that w = xv (w = vx) for v ∈ A+ and x ∈ / c(v), then we write x k w (x k w). Otherwise we write x ∦ w (x ∦ w). r
l
r
l
The next lemma, taken from [4], will be used in the further work. Lemma 1 Let S be nil-extension of a semigroup K which is a union of groups. If there exists a retraction ϕ of S onto K, then it is unique and has the following representation: xϕ = xe,
where e ∈ E(S) such that x ∈ Te .
According to the well-known Munn’s lemma, there exists at most one e ∈ E(S) such that x ∈ Te , and then we also have xe = ex, so we can write ex instead of xe in the above representation for ϕ. For undefined notions and notations we refer to the book by Howie [9]. 2
The Main Results
Variable identities that are studied here consist of some particular kinds of identities. The first kind are the identities over An of the form x1 u(x2 , . . . , xn−1 )xn = w(x1 , x2 , . . . , xn ),
(1)
with n ≥ 3, having some of the following properties: (A1) for a fixed i ∈ [1, n], xi appears once on one side of (1) and at least twice on another side; (B1) |w| 6= |u| + 2; (C1.1) x1 ∦ w;
(C1.2) h(2) (w) = x21 ;
(C1.3) h(w) 6= x1 ;
(D1.2) t(2) (w) = x2n ;
(D1.3) t(w) 6= xn .
l
(D1.1) xn ∦ w; r
We also deal with identities over An of the form x1 u(x2 , . . . , xn ) = v(x1 , . . . , xn−1 )xn , with n ≥ 3, having some of the following properties: 46
(2)
(A2) for a fixed i ∈ [1, n], xi appears once on one side of (2) and at least twice on another side; (B2) |u| = 6 |v|; (C2.2) h(2) (v) = x21 ;
(C2.3) h(v) 6= x1 ;
(2)
(D2.3) t(u) 6= xn .
(D2.2) t
(u) =
x2n ;
The third considered kind of identities are the ones over An of the form x1 x2 · · · xn = w(x1 , x2 , . . . , xn ),
(3)
with n ≥ 2, having some of the following properties: (A3) for a fixed i ∈ [1, n], xi appears once on one side of (3) and at least twice on another side; (B3) |w| ≥ n + 1; (C3.1) x1 ∦ w;
(C3.2) h(2) (w) = x21 ;
(C3.3) h(w) 6= x1 ;
(D1.2) t(2) (w) = x2n ;
(D1.3) t(w) 6= xn .
l
(D3.1) xn ∦ w; r
The main result of the paper is the following theorem. Theorem 1 A semigroup S satisfies a variable identity consisting of all identities of a form F having properties P if and only if S is a periodic semigroup from a class C, where F, P and C are given by the following table: F 1.1.1. 1.1.3. 1.3.1. 1.2.2. 2.2.2. 1.2.3. 2.2.3. 1.3.2. 2.3.2. 1.3.3. 2.3.3.
(1) (1) (1) (1) (2) (1) (2) (1) (2) (1) (2)
P
C
(A1), (B1), (C1.1), (D1.1) (A1), (B1), (C1.1), (D1.3) (A1), (B1), (C1.3), (D1.1) (B1), (C1.2), (D1.2) (B2), (C2.2), (D2.2) (B1), (C1.2), (D1.3) (B2), (C2.2), (D2.3) (B1), (C1.3), (D1.2) (B2), (C2.3), (D2.2) (A1), (B1), (C1.3), (D1.3) (A2), (B2), (C2.3), (D2.3) 47
UG ◦ N (LG ◦ S) ◦ N (RG ◦ S) ◦ N UG ~ N (LG ◦ S) ~ N (RG ◦ S) ~ N (G ◦ S) ~ N
F 3.1.1. 3.1.3. 3.3.1. 3.2.2. 3.2.3. 3.3.2. 3.3.3.
(3) (3) (3) (3) (3) (3) (3)
P
C
(A3), (B3), (C3.1), (D3.1) (A3), (B3), (C3.1), (D3.3) (A3), (B3), (C3.3), (D3.1) (B3), (C3.2), (D3.2) (B3), (C3.2), (D3.3) (B3), (C3.3), (D3.2) (B3), (C3.3), (D3.3)
UG ◦ Nn (LG ◦ S) ◦ Nn (RG ◦ S) ◦ Nn UG ~ Nn (LG ◦ S) ~ Nn (RG ◦ S) ~ Nn (G ◦ S) ~ Nn
Proof of 1.1.1 Let S satisfy the variable identity consisting of all identities of the form (1) having the properties (A1), (B1), (C1.1) and (D1.1). By (B1) it follows that S is periodic. Let x ∈ S, e ∈ E(S). First we have that some of the mentioned identities lies in the kernel of the homomorphism ϕ : A+ n → S determined by x1 ϕ = xe and xj ϕ = e, for j ∈ [2, n]. By this it follows that xe = wϕ =
(xe)|x1 |w e(xe)|x1 |w
if h(w) = x1 . if h(w) = 6 x1
In the first case, if h(w) = x1 , by (C1.1) it follows |x1 |w ≥ 2, and then xe ∈ Gr(S), which was to be proved. Consider the second case: h(w) 6= x1 . In this case we have that xe = e(xe)|x1 |w = ee(xe)|x1 |w = exe. Next we have that an identity of the form (1) with the properties (A1), (B1), (C1.1) and (D1.1) lies in the kernel of the homomorphism ψ : A+ n → S determined by xi ψ = xe and xj ψ = e, for j ∈ [1, n], j 6= i, where i ∈ [1, n] is the one fixed in (A1). By this it follows that er xe = es (xe)t , for some integers r, s ≥ 0 and t ≥ 2, and since xe = exe, then xe = (xe)t , so xe ∈ Gr(S). This proves that Gr(S) is a left ideal of S. In the same way we prove that it is a right ideal, and since S is periodic, then S is a nil-extension of Gr(S) ∈ UG. Conversely, let S be periodic and a nil-extension of a semigroup K ∈ UG, and let x, a, y ∈ S. Then ak ∈ K, for some k ∈ N, whence xak y ∈ K. Thus, xak y belongs to some periodic subgroup of K, so xak y = (xak y)m+1 , for some m ∈ N. This means that S |=v {x1 xk2 x3 = (x1 xk2 x3 )m | k, m ∈ N}. 48
Proof of 1.1.3 Let S satisfy the variable identity consisting of all identities of the form (1) having the properties (A1), (B1), (C1.1) and (D1.3). Since (D1.3) implies (D1.1), then by 1.1 it follows that S is a nil-extension of a semigroup K ∈ UG. To prove that K ∈ LG ◦ S, it is enough to prove that E(K) is a left regular band. Let e, f ∈ E(K) and let the homomorphism ϕ : A+ n → S be determined by xn ϕ = f
and xj ϕ = e, for j ∈ [1, n − 1].
Then the kernel of ϕ contains an identity of the form (1) having the properties (A1), (B1), (C1.1) and (D1.3), whence it follows that ef = (ef )k e, for some k ∈ N. Thus, ef = ef e, which was to be proved. Conversely, let S be periodic and a nil-extension of a semigroup K ∈ LG ◦ S. Then E(S) = E(K) is a left regular band and by Theorem 2.2 of [3], S is a semilattice Y of semigroups Sα , α ∈ Y , such that any Sα is a nil-extension of a left group Kα . Moreover, K is a semilattice Y of left groups Kα , α ∈ Y . Let x, a, y ∈ S. Then there exists k ∈ N such that xk ∈ E(S) and ak ∈ K. On the other hand, xak y, xak yxk ∈ Sα ∩ K = Kα , for some α ∈ Y , and since Kα is left simple, then xak y ∈ Kα xak yxk , i.e. xak y = sxak yxk , for some s ∈ S. Finally, since xk is an idempotent, then xak y = sxak yxk = sxak yxk xk = xak yxk . Thus, S |=v {x1 xk2 x3 = x1 xk2 x3 xk1 | k ∈ N}. The proof of 1.3.1 will be omitted, because it is dual to the proof of 1.1.3. Proof of 1.2.2 and 2.2.2. Let S satisfy the variable identity consisting of all identities of the form (1) with (B1), (C1.2) and (D1.2). By (B1) we have that S is periodic. Let x, a, y ∈ S, and let ϕ : A+ n → S be the homomorphism determined by x1 ϕ = x,
xn ϕ = y
and xj ϕ = a, for j ∈ [2, n − 1].
Then the kernel of ϕ contains an identity of the form (1) with (B1), (C1.2) and (D1.2), whence it follows that xa|u| y = x2 sy 2 , for some s ∈ S, i.e. xan y ∈ x2 Sy 2 , for some n ∈ N. Therefore, by Theorem 1 of [4] we have that S is a retractive nil-extension of a semigroup K ∈ U G. Let S satisfy the variable identity consisting of all identities of the form (2) with (B2), (C2.2) and (D2.2). In this case we have a proof similar to the previous one. By (B2) we have that S is periodic. Let x, a, y ∈ S. Considering the homomorphism ϕ : A+ n → S defined by x1 ϕ = x and xj ϕ = a, for j ∈ [2, n], 49
we have that xak = x2 s, for some k ∈ N and s ∈ S, and considering the homomorphism ψ : A+ n → S defined by xn ψ = y
and xj ϕ = a, for j ∈ [1, n − 1],
we obtain that am y = ty 2 , for some m ∈ N and t ∈ S. Therefore, xak+m y = x2 sty 2 , and as in the previous case we conclude that S ∈ U G ~ N . Conversely, let S be periodic and a retractive nil-extension of K ∈ UG, let ϕ be the retraction of S onto K, and let x, a, y ∈ S. Then there exists n ∈ N such that xk , y k ∈ E(S) and ak ∈ K. Moreover, by Lemma 1 it follows that xϕ = xxk = xk+1 and yϕ = yy k = y k+1 , whence (xϕ)ak (yϕ) = xk+1 ak y k+1 k k xa y = (xa y)ϕ = (xϕ)ak y = xk+1 ak y . k xa (yϕ) = xak y k+1 = | k ∈ N} and S |=v {x1 xk2 xk+1 Therefore, S |=v {x1 xk2 x3 = xk+1 xk2 xk+1 3 3 1 k+1 k x1 x2 x3 | k ∈ N}. Proof of 1.2.3 and 2.2.3. Let S satisfy the variable identity consisting of all identities of the form (1) with (B1), (C1.2) and (D1.3). Since (C1.2) implies (A1) and (C1.1), then by 1.1.3 we have that S is a nil-extension of a semigroup K ∈ LG ◦ S. We prove that for each x ∈ S and e ∈ E(S) the following conditions hold xe ∈ xm Se, for each m ∈ N, ex ∈ eSxm , for each m ∈ N.
(4) (5)
It is clear that (4) holds for m = 1. Let m ∈ N such that xe = xm se, for some s ∈ S and consider the homomorphism ϕ : A+ n → S determined by x1 φ = xm
and xj φ = se, for j ∈ [2, n].
Since the kernel of φ contains an identity of the form (1) with (B1), (C1.2) and (D1.3), we have that xm (se)k = x2m te, for some k ∈ N and t ∈ S. We also have that se ∈ K = Gr(S), so seH(se)k , where H is the Green’s relation on S, whence se = (se)k p, for some p ∈ S. Now we have that xe = xm se = xm see = xm (se)k pe = x2m tepe ∈ xm+1 Se. Therefore, by induction we conclude that (4) holds for each m ∈ N. On the other hand, considering the homomorphism ψ : A+ n → S determined by xn ψ = ex and xj ψ = e, for j ∈ [1, n − 1], 50
we obtain that ex = (ex)k e, for some k ∈ N, whence ex = exe. Let m ∈ N. Then exH(ex)m , so ex = s(ex)m , for some s ∈ S, whence it follows that ex = eex = es(ex)m = esexm ∈ eSxm , since ex = exe. Therefore, we have proved (5). Using (4) and (5) we prove that the mapping ϕ : S → K defined by xϕ = xe, where e ∈ E(S) such that x ∈ Te , is a retraction of S onto K. It is enough to show that is is a homomorphism. Let x, y ∈ S and let x ∈ Te , y ∈ Tf and xy ∈ Tg , for some e, f, g ∈ E(S). By (4) and (5) it follows that yg = f yg, xf = exf , exy = exyg and ey = eyf , whence (xy)ϕ = xyg = xf yg = exf yg = exyg = exy = xey = xeyf = (xϕ)(yϕ). Further, let S satisfy the variable identity consisting of all identities of the form (2) with (B2), (C2.2) and (D2.3), and let x ∈ S and e ∈ E(S). Using the same methodology we obtain that xe = (xe)k , for some k ∈ N, k ≥ 2, and ex = exe, whence it follows that ex = exe = e(xe)k = (ex)k e = (ex)k . Thus, Gr(S) is an ideal of S and S is a nil-extension of Gr(S). In the same way as in the previous case we prove that Gr(S) ∈ LG ◦ S and S is a retractive extension of Gr(S). Conversely, let S be periodic and a retractive nil-extension of a semigroup K ∈ LG ◦ S, let ϕ be the retraction of S onto K, and let x, a, y ∈ S. As in 1.1.3 we obtain that there exists k ∈ N such that xk , ak ∈ E(S) and xak y = xak yxk = xak yak , whence (xak yxk )ϕ = (xϕ)ak yxk = xk+1 ak yxk k k xa y = (xa y)ϕ = . (xϕ)ak y = xk+1 ak y Therefore, S |=v {x1 xk2 x3 = xk+1 xk2 x3 xk1 | k ∈ N} and S |=v {x1 xk2 x3 xk2 = 1 k+1 k x1 x2 x3 | k ∈ N}. The proofs of 1.3.2 and 2.3.2 will be omitted, because they are dual to the proofs of 1.2.3 and 2.2.3. Proof of 1.3.3 and 2.3.3. Let S satisfy the variable identity consisting of all identities of the form (1) with (A1), (B1), (C1.3) and (D1.3). Since (C1.3) implies (C1.1) and (D1.3) implies (D1.1), then by 1.1.3 and 1.3.1 it follows that S is a nil-extension of K ∈ LG ◦ S ∩ RG ◦ S = G ◦ S. By Theorem 3 of [5], (G ◦ S) ◦ N = (G ◦ S) ~ N . Let S satisfy the variable identity consisting of all identities of the form (2) with (A2), (B2), (C2.3) and (D2.3). As in the previous proofs, by (B2) we have that S is periodic, using (C2.3) and (D2.3) we prove that the idempotents of S are central, i.e. ex = xe, for all e ∈ E(S), x ∈ S, and by (A2) we obtain 51
that ex = (ex)k = (xe)k = xe, for some k ∈ N, k ≥ 2, and conclude that Gr(S) is an ideal of S. Therefore, S ∈ (G ◦ S) ◦ N = (G ◦ S) ~ N . Conversely, let S be periodic and a retractive nil-extension of a semigroup K ∈ G ◦ S, let ϕ be the retraction of S onto K, and let x, a, y ∈ S. Then there exists k ∈ N such that ak , xk , y k ∈ E(S) = E(K), and then xak y ∈ K, xϕ = xk+1 , yϕ = y k+1 . Since the idempotents of S are central, then (xϕ)ak (yϕ) = xk xak yy k = y k xak yxk k k . xa y = (xa y)ϕ = xak (yϕ) = xa2k y k+1 = xak y k+1 ak (xϕ)ak y = xk+1 a2k y = ak xk+1 ak y Hence, S |=v {x1 xk2 x3 = xk3 x1 xk2 x3 xk1 | k ∈ N} and S |=v {x1 xk2 xk+1 xk2 = 3 k k+1 k x2 x1 x2 x3 | k ∈ N}. The proofs of 3.1.1–3.3.3 will be omitted because they are immediate consequences of the previous theorems and the following one: Theorem 2 Let n ∈ N, n ≥ 2, and let S be a nil-semigroup satisfying the variable identity of the form (3) having the property (B3). Then S is nnilpotent. Proof Let us prove that any nilpotent subsemigroup Q of S is n-nilpotent. Let Q be a k-nilpotent semigroup. By the hypothesis, for arbitrary a1 , a2 , . . . , an there exists a word w1 = w1 (x1 , . . . , xn ) such that |w1 | ≥ n + 1 and a1 a2 · · · an = w1 (a1 , a2 , . . . , an ). This proves that Qn = Qn+1 , whence we obtain that Qn = Qm , for any m ∈ N, m ≥ n, and hence, Qn = Qk = {0}. Therefore, the set of indices of nilpotency of all nilpotent subsemigroup of S is bounded, so by Theorem 3 of [15] we have that S is nilpotent, and hence, it is n-nilpotent. The theorems characterizing nilpotent and nil-extensions of bands, left regular bands and semilattices, and their retractive analogues, are very similar to the previous ones, so they will be omitted. We only note that the variable identities describing these semigroups consist of the corresponding identities from the above theorems, having an additional property: (A1–3)∗ for a fixed i ∈ [1, n], xi appears once on one side of (1) (resp. (2), (3) ), and exactly twice on another side. This condition forces all subgroups of a semigroup satisfying it to be oneelement. 52
Remark that the conditions (Ci.j) and (Di.k) in the claim i.j.k are necessary. Namely, adding to the variable identity from the claim i.j.k an identity which does not satisfy (Ci.j) or (Di.k) we leave the class of semigroups from ´ c and i.j.k. This is an immediate consequence of the results given by Ciri´ Bogdanovi´c in [7] (see also [6]). References 1. J. Almeida, On power varieties of semigroups, J. Algebra 120 (1989), 1–17. 2. H. E. Bell, A commutativity study for periodic rings, Pacific J. Math. 70 (1977), 29–36. 3. S. Bogdanovi´c, Semigroups of Galbiati-Veronesi , Algebra and Logic, Zagreb, 1984, Novi Sad, 1985, 9–20. ´ c, Retractive nil-extensions of regular semi4. S. Bogdanovi´c and M. Ciri´ groups, Proc. Japan Acad. Ser. A, 68 (1992), no. 6, 126–130. ´ c, Semigroups of Galbiati-Veronesi IV (Bands 5. S. Bogdanovi´c and M. Ciri´ of nil-extensions of groups, Facta Univ. (Niˇs), Ser. Math. Inform. 7 (1992), 23–35. ´ c and T. Petkovi´c, Uniformly π-regular rings and 6. S. Bogdanovi´c, M. Ciri´ semigroups: A survey, Topics from Contemporary Mathematics, Zborn. Rad. Mat. Inst. SANU 9 (17) (1999), 1–79. ´ c and S. Bogdanovi´c, Nil-extensions of unions of groups induced 7. M. Ciri´ by identities, Semigroup Forum 48 (1994), 303–311. ´ c and S. Bogdanovi´c, The five-element Brandt semigroup as a 8. M. Ciri´ forbidden divisor , Semigroup Forum (to appear). 9. J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, Oxford: Clarendon Press, 1995. 10. E. S. Lyapin, Atoms of the lattice of inclusive varieties of semigroups, Sibir. Mat. Zhurn. 16 (1975), no. 6, 1224–1230 (in Russian). 11. E. S. Lyapin, Identities valid globally in semigroups, Semigroup Forum 24 (1982), 263–269 12. G. Mashevitzky, On a finite basis theorem for universal positive formulas, Algebra Universalis 35, (1996), 124–140. 13. M. S. Putcha and J. Weissglass, Semigroups satisfying variable identities, Semigroup Forum 3 (1971), 64–67. 14. M. S. Putcha and J. Weissglass, Semigroups satisfying variable identities II, Trans. Amer. Math. Soc. 168 (1972), 113–119. 15. L. N. Shevrin, Semigroups all of whose subsemigroups are nilpotent, Sibir. Mat. Zhurn. 11 (1961), no. 6, 936–942 (in Russian). 53
