Non-commutative Grillet semigroups
Grillet semigroups are introduced. This class of semigroups contains regular semigroups and complete commutative semigroups (by Grillet’s terminology). Some structural theorems are proved.
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| Cite this: | Non-commutative Grillet semigroups / B.V. Novikov // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 298–307. — Бібліогр.: 11 назв. — англ. |
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| citation_txt | Non-commutative Grillet semigroups / B.V. Novikov // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 298–307. — Бібліогр.: 11 назв. — англ. |
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| description | Grillet semigroups are introduced. This class of semigroups contains regular semigroups and complete commutative semigroups (by Grillet’s terminology). Some structural theorems are proved.
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 17 (2014). Number 2, pp. 298 – 307
c© Journal “Algebra and Discrete Mathematics”
Non-commutative Grillet semigroups
B. V. Novikov
Abstract. Grillet semigroups are introduced. This class of
semigroups contains regular semigroups and complete commutative
semigroups (by Grillet’s terminology). Some structural theorems
are proved.
1. Introduction
For the class of finite commutative semigroups I.S.Ponizovsky [3,4]
introduced the notion of an elementary semigroup. He constructed for the
finite commutative semigroups an ideal series whose factors are elementary
semigroups, and used it for the study of matrix representations. Further
P.-A. Grillet [7, 8] generalized this construction and used it for a class
of commutative semigroups called by him complete. Non-commutative
analogues of elementary semigroups occur in many tasks, namely, a descrip-
tion of homomorphisms into commutative semigroups [2], constructing
of quasi-Frobenius algebras. Therefore it is rational to consider these
semigroups from a unified point of view.
In Section 2 we recall the Ponizovsky’s results, further we generalize
them to non-commutative semigroups (Section 3). In Section 4 we offer
an approach to investigation of elementary semigroups by bi-actions.
All non-defined here notions may be taken from [1].
2010 MSC: 20M10.
Key words and phrases: epigroup, elementary semigroup, complete commutative
semigroup.
B. V. Novikov 299
2. Ponizovsky’s construction
Recall one Ponizovsky’s result by Grillet’s monograph [7].
A commutative semigroup P is called elementary if it is an ideal
extension of a nilpotent semigroup N by a group G so that the unity of
G is a unity of P also (therefore P has 0 and 1; in particular, groups and
nilpotent semigroups are not elementary).
We need the following notations. By E(S) we shall denote the set
of all idempotents of the semigroup S. Define a partial order on the set
E(S) as follows: e 6 f for e, f ∈ E(S) if and only if ef = e.
Ponizovsky’s Theorem. Every finite commutative semigroup S with 0
and 1 has an ideal series
S = I0 ⊃ I1 ⊃ . . . ⊃ Im+1 = 0
whose factors Ik/Ik+1 (k 6 m) are elementary.
Proof. It will be made by induction on |S|. Let e be a minimal non-zero
idempotent in S. Put Im=Se and denote by H the subgroup of invertible
elements of Im. Obviously the complement M=Im \ H is an ideal of Im.
As E(Im) = {0, e} and |Im| < ∞, then for every element of M some
power of it is equal to zero; therefore the semigroup M is nilpotent.
It remains to use the induction to S/Im.
Grillet calls the semigroups Ik/Ik+1 of proved theorem Ponizovsky’s
factors; we shall call them briefly Π-factors.
It is obvious that the existence the zero and the unity in S is not
essential; it’s only then that the group and nilpotent semigroups were not
considered as Π-factors.
Ponizovsky applied this theorem to the investigations of semigroup
algebras. Namely, let F be a field, S be a finite commutative semigroup,
Pk = Ik/Ik+1 be its Π-factors. Then a (contracted) semigroup algebra
FS decomposes into a direct sum of ideals
FS = FP0 ⊕ . . . ⊕ FPm,
moreover, in view of the existence of unities in Π-factors, the algebra FS
is quasi-Frobenius if and only if the algebras FPk are quasi-Frobenius.
From this Ponizovsky obtained the necessary and sufficient conditions of
the quasi-Frobenius property of semigroup algebras.
300 Non-commutative Grillet semigroups
Grillet [7] generalized the Ponizovsky’s Theorem to a class of infinite
semigroups which were called by him complete and used this result for a
description of congruences of a free commutative semigroup.
Moreover, Grillet shown how to construct the commutative semigroup
by means of its Π-factors.
Another application of Ponizovsky’s Theorem may be noted.
Corollary ([7], Corollary IV.5.6). A finite commutative subdirectly inde-
composable semigroup is either a group, or a nilpotent semigroup, or an
elementary semigroup.
Thus, the Ponizovsky’s Theorem is an effective tool for investiga-
tion in commutative semigroups. Therefore we may think that its non-
commutative variant will be also used.
Some words about a structure of elementary semigroups. It is non-
trivial, and it must be investigated “modulo” groups and nilpotent semi-
groups, i.e. to reduce it to the problem of extension. Two approaches may
take place. An elementary semigroup P = G ∪ N is an ideal extension of
N by means of G. Such presentation are unique and visual, however it
does not give an acceptable solution of the problem of synthesis P from
N and G.
3. Non-commutative situation
Recall that a semigroup S is called an epigroup1 [5] if for any its
element a, some degree of it lies in a subgroup. Denote the unity of this
subgroup by ea. Obviously all the finite semigroups are epigroups.
We will need some well-known properties of epigroups.
1) An ideal extension is an epigroup if and only if the ideal
and the quotient semigroup are epigroups;
2) a 0-simple semigroup is an epigroup if and only if it is
completely 0-simple.
Throughout this section S will denote an epigroup with zero (of
course, this supposition does not restrict the generality of the results,
since we may add a zero to the semigroup S). The set of all idempotents
of the semigroup S we denote by E(S).
1This term belongs to L.N. Shevrin and it is a synonymous with English terms
“group-bounded semigroup” and “quasi-periodic semigroup”.
B. V. Novikov 301
Define a quasi-order � on the set E(S) as follows:
e � f ⇐⇒ SeS ⊆ SfS.
Clearly, the equivalence relation ≈ associated with the quasi-order � is
the Green relation J . A non-zero idempotent e ∈ E(S) we call an atom
if it is 0-minimal relatively to � (i.e. 0 6= f � e implies f ≈ e).
An epigroup S is called elementary if SeS = S for some atom e ∈ E(S).
The next statement yields a rough description of the structure of such
semigroup.
Proposition 1. A semigroup T is an elementary epigroup if and only if
the following conditions are satisfied:
a) TeT = T for some idempotent e ∈ E(T );
b) T contains such nil ideal K 6= T that the Rees quotient
semigroup T/K is completely 0-simple.
Proof. Let the conditions a) and b) are fulfilled. As K and T/K are
epigroup then T is an epigroup. Further, T is elementary since T/K is
completely 0-simple and therefore every its non-zero idempotent is an
atom.
Conversely, let an epigroup T be elementary, e be an atom, TeT = T
and K be the greatest its nil ideal. Let I be an ideal of T containing
K, I 6= T , a ∈ I. Then I is an epigroup and therefore ea ∈ I. Thus
TeaT ⊆ I ⊂ TeT , and hence ea = 0. Then I is a nil ideal which is
impossible. Therefore the semigroup T/K is 0-simple. In view of the
Property 2) of epigroups, it is completely 0-simple.
The proof shows that the ideal K is the nil radical of the semigroup
T . If it is nilpotent (this takes place, in particular, when T is finite), then
the description of the elementary semigroup is simpler.
Proposition 2. Let the nil radical K of the semigroup T be nilpotent.
Then T is an elementary epigroup if and only if T 2 = T and the semigroup
T/K is completely 0-simple.
Proof. The part “only if” is obvious. Conversely, let T/K is completely
0-simple, T 2 = T , B = T \ K and e = e2 ∈ B. Clearly, e is an atom and
B ⊆ BeB ⊆ TeT . Show that K \ K2 ⊂ TeT .
Let x ∈ K \ K2. As T 2 = T , then x = yz, and either y ∈ B or z ∈ B.
If y ∈ B ⊆ BeB then x ∈ BeBT ⊆ TeT . For z ∈ B we obtain x ∈ TeT
too.
302 Non-commutative Grillet semigroups
Finally, it follows from K \ K2 ⊂ TeT that Ki \ Ki+1 ⊆ Ki−1(K \
K2) ⊆ TeT for any i > 2 which yields
K =
⋃
i>1
(Ki \ Ki+1) ⊆ TeT.
A number of examples of elementary epigroups gives the next state-
ment.
Proposition 3. If e is an atom of an epigroup S then the semigroup
SeS is elementary.
Proof. In view of Property 1) of epigroups, the semigroup T = SeS is an
epigroup. As e ∈ SeS then
TeT = SeSeSeS ⊇ SeS = T,
hence TeT = T .
Show that e is an atom of T . Let 0 6= f ∈ E(T ). As f ∈ T = SeS,
then f � e in S, therefore e ≈ f in S. Thus, SfS = SeS and
TfT = SeSfSeS = SeSeSeS = T,
i.e. e ≈ f in T .
We want to extend the Grillet’s results [7, 8] to the non-commutative
case.
For e ∈ E(S) we put
L(e) =
⋃
f∈E(S), f≺e
SfS.
We call the set P (e) = SeS \ L(e) a Π-factor, and the quotient semigroup
P (e) = SeS/L(e) a Π-factor. Note that every Π-factor is non-empty:
really, in the opposite case we will have e ∈ SfS for some f ≺ e which is
impossible.
Proposition 4. Every Π-factor is an elementary epigroup.
Proof. In view of the Property 1) P (e) is an epigroup. As P (e)
⋃
L(e) =
SeS = (SeS)e(SeS) ⊆ P (e)eP (e)
⋃
L(e), then P (e) ⊆ P (e)eP (e), there-
fore P (e)eP (e) = P (e). It remains to note that e is an atom in P (e).
B. V. Novikov 303
We call an epigroup S a Grillet semigroup if it satisfies the next
condition:
(Gr) For every a ∈ S the set
M(a) = {e ∈ E(S) | a ∈ SeS}
is non-empty and has a least element ea relatively the quasi-
order � (i.e. ea � e for all e ∈ M(a)).
The commutative Grillet semigroups are exactly the semigroups called
complete by Grillet2.
The elementary and regular semigroups are Grillet semigroup. How-
ever, in contrast of the commutative situation, not every finite monoid is
a Grillet semigroup. For example, the monoid S1 obtained by the external
accession unity from the semigroup S = {0, a, e, f} where ea = af = a,
e2 = e, f2 = f, and other products are equal to zero. Indeed, in this case
M(a) = {e, f} but e and f are non-compatible with one another.
The next statement is a generalization of the Proposition IV.5.3 of [7]
to the non-commutative case.
Proposition 5. A Grillet semigroup is a disjoint union of its Π-factors.
Proof. If S is a Grillet semigroup then every its element a lies in P (ea);
Therefore S is covered by its Π-factors. Assume that a ∈ P (e) ∩ P (f)
for some e, f ∈ E(S). By the condition (Gr), there is an idempotent
g ∈ E(S) such that g � e, g � f and a ∈ SgS. But a 6∈ L(e), therefore
SgS = SeS and similarly SgS = SfS, hence P (e) = P (f).
Under some additional restrictions the converse is true.
Proposition 6. Let S satisfy the finiteness condition of the strictly
decreasing chains of idempotents e1 ≻ e2 ≻ . . .. If S is a disjunct union
of its Π-factors then it satisfies the condition (Gr).
Proof. Let a be arbitrary element of S. Then a ∈ P (e) for some e ∈ M(a).
Show that e is a least element of M(a).
Really, let e 6≈ f1 and f1 ∈ M(a), i.e. a ∈ Sf1S ∩P (e). As by condition
the Π-factors do not intersect, then a ∈ L(f1). Therefore a ∈ Sf2S for
some idempotent f2 ≺ f1. By repeating these considerations we obtain
an idempotent f3 ≺ f2 ≺ f1 such that a ∈ Sf3S and so on. Since this
chain is finite, we will obtain for some n a relation a ∈ P (fn). This yields
e ≈ fn ≺ f1.
2Warning: the original definition of complete semigroup given in the book [7] was
corrected in [8].
304 Non-commutative Grillet semigroups
4. Bi-orbits and elementary semigroups
We obtained a “rough” description of elementary semigroups. The
“thin” structure of these semigroups is much more complicated (it is seen
even in commutative case). The classical theory of ideal extension (with
using translations) works badly for nil-semigroup. In view of this we will
consider another approach to this problem.
Let T = B ∪ K be a finite elementary semigroup (we preserve denota-
tions of Section 3). Firstly, we restrict only with the case when B = G
is a group and its unity e is the unity of T . Secondly, it is easy to see
that G(Ki \ Ki+1)G ⊆ Ki \ Ki+1, therefore it seems reasonable on the
first step to consider the semigroups of view (G ∪ Ki)/Ki+1. Note that
Ki/Ki+1 is a semigroup with null multiplication, i.e. it has a very poor
algebraic structure.
Thus we come to the consideration of a class of semigroups satisfying
the following conditions.
a) T = G ∪ K is a finite semigroup, G is its subgroup;
b) the unity e ∈ G is also the unity of T ;
c) K2 = 0, K 6= 0.
We can look at K \ 0 as a set on which the group G operates from two
sides. So it is convenient to introduce the concept of “bi-action” and to
study its properties.
Definition 1. Let G, H be groups, X be a set. A set X is called (G, H)-
set if two actions are defined, namely, a left action G on X and a right
action H on X such that 1 · x = x · 1 = x and
∀ a ∈ G ∀ b ∈ H ∀ x ∈ X (ax)b = a(xb).
The mapping G × X × H −→ X is called a bi-action3.
(G, H)-set is called bi-orbit if
∀ x, y ∈ X ∃a ∈ G ∃b ∈ H y = axb.
In application to elementary semigroups we will deal with one group
G = H, however it is convenient to construct the theory for two group.
Here the set X is the set K \ 0 which is, clearly, a disjunct union of
bi-orbits. In view of this we will restrict with a more narrow class of
elementary semigroups adding the following condition to conditions a)–c):
3In the monograph [10] such sets X are called the unitary (G, H)-bi-acts and the
defined below bi-orbits are cyclic bi-acts.
B. V. Novikov 305
d) K \ 0 is a bi-orbit relatively the bi-action of the pair (G, G).
Thus, let X be a (G, H)-bi-orbit. Clearly, a bi-action is equivalent to the
usual transitive action of the group G × Hop on the set X (here Hop
is a group anti-isomorphic to H). Of course, any group anti-isomorphic
to the group H is isomorphic to H. However we are convenient to consider
the group Hop as a group with inverse multiplication x ∗ y = y · x.
Thus, the description of bi-orbits is reduced to the description of subgroups
of the group G × Hop.
The next statement is known as “Goursat Lemma” (algebraic).
Lemma 1. Let G, H be groups, A, B, C, D be subgroups of groups G and
H such that
G > A ⊲ B, H > C ⊲ D, A/B ∼= C/D.
Let ϕ be an isomorphism of A/B onto C/D. Then the subset
F = {(x, y) ∈ G × H | ∃a ∈ A, c ∈ C :
x ∈ aB, y ∈ cD, ϕ(aB) = cD} =
⋃
a∈A, c∈C
ϕ(aB)=cD
(aB × cD)
is a subgroup of the group G × H. Conversely, any subgroup of the group
G × H is of such view.
The proof of this lemma can be found, for example, in [11]; see also
exhaustible survey and the generalizations of Goursat Lemma in [6]. Note
that, in the denotations of Lemma, for given subgroup F < G × H we
have
A = {g ∈ G | ∃h ∈ H : (g, h) ∈ F},
B = {g ∈ G | (g, 1) ∈ F},
C = {h ∈ H | ∃g ∈ G : (g, h) ∈ F},
D = {h ∈ H | (1, h) ∈ F}.
Let P [resp., Q] be a system of representatives of left cosets of the
subgroup B in the group G [resp., right cosets of C in H].
Let X be a (G, H)-bi-orbit. Fix an arbitrary element x0 ∈ X. Then
X = Gx0H. Let F = {(g, h)|g ∈ G, h ∈ H, gx0h = x0}. It is not
difficult to check that F is a subgroup of the group G × Hop. In this
denotation A = {g ∈ G|∃h ∈ H gx0h = x0}, B = {g ∈ G|gx0 = x0},
306 Non-commutative Grillet semigroups
C = {h ∈ H|∃g ∈ G gx0h = x0}, D = {h ∈ H|x0h = x0}. Show that the
bi-orbit X can be identified with the set P × Q. Really, for all g ∈ G,
h ∈ H we have h = cq for some c ∈ C, q ∈ Q, ux0c = x0 for some
u ∈ G, gu−1 = pb for some p ∈ P, b ∈ B. Therefore gx0h = gx0cq =
gu−1ux0cq = gu−1x0q = pbx0q = px0q. Further, let px0q = p′x0q′ for
some p, p′ ∈ P, q, q′ ∈ Q. Then x0 = p−1p′x0q′q−1, hence q′q−1 ∈ C,
which implies q = q′. This yields p−1p′ ∈ B, therefore p = p′.
Corollary 2. In the denotations of Lemma 1,
[G × H : F ] = [G : A][H : D] = [G : B][H : C].
Corollary 3. In the introduced above denotations P × Q is a system of
representatives of the group F in the group G × H.
Introduce the denotations: G//B denotes the set of the left cosets of
B in G, C\\H is the set of the right cosets of C in H. In view of Lemma 1
and its corollaries, it is not difficult to obtain the following result.
Theorem 3. Let G and H be groups. Fix in G and H the subgroups such
that
G > A ⊲ B, H > C ⊲ D, A/B ∼= C/D.
Let P and Q be such that in Corollary 3, λ : A/B → C/D be an isomor-
phism. Define a bi-action on the set X = G//B × C\\H as follows:
u(xB, Cy)v = (uxB, Cyv) (u, x ∈ G, y, v ∈ H).
Then X is a (G, H)-bi-orbit. Moreover, any (G, H)-bi-orbit can be ob-
tained so.
Example. Consider the “non-proper” cases, i.e. when A, B, C, D coincide
with 1 or with G, H.
a) A = G, B = 1, C = H, D = 1. Then X = G × 1 and bi-action has
a view u · (g, 1) · v = (ugλ−1(v), 1).
b) A = B = C = D = 1. Then X = G × H, u · (x, y) · v = (ux, yv).
c) A = B = G, C = D = 1. Then X = 1×H, u ·(1, h) ·v = (1, λ(u)hv).
d) A = B = 1, C = D = H. Then X = G × 1, u · (g, 1) · v =
(ugλ−1(v), 1).
B. V. Novikov 307
Thus, the elementary semigroups can be described in case when
K2 = 0. Really, in this case K \0 is a union of bi-orbits, the multiplication
of elements of K by one another is trivial, and the multiplication of
elements of K by elements of G (on the left and on the right) is defined
from the description of bi-orbits.
References
[1] Clifford, A. H. and G. B. Preston, The Algebraic Theory of Semigroups, Math.
Surveys, no. 7, Amer. Math. Soc., Providence, RI, Vol. I, 1961, Vol. II, 1967.
[2] Ponizovsky I. S. On homomorhisms of semigroups onto commutative semigroups.
Siberian Math. J., 2(1961), N.5, 719-733 [in Russian].
[3] Ponizovsky I. S A remark on commutative semigroups. Dokl. Akad. Nauk SSSR,
142(1962), N.6, 1258-1260 [in Russian].
[4] Ponizovsky I. S. Matrix representations of finite commutative semigroups. Siberian
Math. J., V. 11, Iss. 5, pp. 816-822. [Translated from Sibirskii Matematicheskii
Zhurnal, Vol. 11, No. 5, pp. 1098-1106, September-October, 1970].
[5] Shevrin L. N. On the theory of epigroups, I: Math. sb., 1994, 185, 8, 129-160, II:
Math. sb., 1994, 185, 9, 153-176 [in Russian].
[6] Bauer K., Sen D., Zvengrowski P. A Generalized Goursat Lemma.
http://arxiv.org/abs/1109.0024 (2011).
[7] Grillet P. A. Semigroups. Marcel Dekker, 1995.
[8] Grillet P. A. Subcomplete commutative semigroups. Acta. Sci. Math. (Szeged),
67(2001), N3-4, 601-628.
[9] Novikov B. V. On quasi-Frobenius semigroup algebras. In: Groups, Rings and
Group Rings (Lect. Notes in Pure and Appl. Math, 247, 2006, 287-291).
[10] Kilp M., Knauer U., Mikhalev A.V. Monoids, Acts and Categories. W. de Gruyter,
N.Y. — Berlin, 2000.
[11] Wikipedia Goursat lemma. http://en.wikipedia.org
Contact information
B. V. Novikov Kharkov National University, Svobody sq., 4,
61077, Kharkov, Ukraine
E-Mail: novikov@univer.kharkov.ua
Received by the editors: 25.03.2014
and in final form 25.03.2014.
|
| id | nasplib_isofts_kiev_ua-123456789-153339 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-01T09:31:41Z |
| publishDate | 2014 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Novikov, B.V. 2019-06-14T03:23:58Z 2019-06-14T03:23:58Z 2014 Non-commutative Grillet semigroups / B.V. Novikov // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 298–307. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC:20M10. https://nasplib.isofts.kiev.ua/handle/123456789/153339 Grillet semigroups are introduced. This class of semigroups contains regular semigroups and complete commutative semigroups (by Grillet’s terminology). Some structural theorems are proved. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Non-commutative Grillet semigroups Article published earlier |
| spellingShingle | Non-commutative Grillet semigroups Novikov, B.V. |
| title | Non-commutative Grillet semigroups |
| title_full | Non-commutative Grillet semigroups |
| title_fullStr | Non-commutative Grillet semigroups |
| title_full_unstemmed | Non-commutative Grillet semigroups |
| title_short | Non-commutative Grillet semigroups |
| title_sort | non-commutative grillet semigroups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/153339 |
| work_keys_str_mv | AT novikovbv noncommutativegrilletsemigroups |