Dense ideal extensions of strict regular semigroups
If V is an existence variety of strict regular semigroups, then every semigroup in V has, within V, a maximal dense ideal extension.
Gespeichert in:
| Veröffentlicht in: | Algebra and Discrete Mathematics |
|---|---|
| Datum: | 2006 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2006
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/157391 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Dense ideal extensions of strict regular semigroups / F.J. Pastijn, L. Oliveira // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 67–80. — Бібліогр.: 24 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-157391 |
|---|---|
| record_format |
dspace |
| spelling |
Pastijn, F.J. Oliveira, L. 2019-06-20T03:12:14Z 2019-06-20T03:12:14Z 2006 Dense ideal extensions of strict regular semigroups / F.J. Pastijn, L. Oliveira // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 67–80. — Бібліогр.: 24 назв. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/157391 If V is an existence variety of strict regular semigroups, then every semigroup in V has, within V, a maximal dense ideal extension. The second author was partially supported by FCT through Centro de Matem´atica da Universidade do Porto en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Dense ideal extensions of strict regular semigroups Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Dense ideal extensions of strict regular semigroups |
| spellingShingle |
Dense ideal extensions of strict regular semigroups Pastijn, F.J. Oliveira, L. |
| title_short |
Dense ideal extensions of strict regular semigroups |
| title_full |
Dense ideal extensions of strict regular semigroups |
| title_fullStr |
Dense ideal extensions of strict regular semigroups |
| title_full_unstemmed |
Dense ideal extensions of strict regular semigroups |
| title_sort |
dense ideal extensions of strict regular semigroups |
| author |
Pastijn, F.J. Oliveira, L. |
| author_facet |
Pastijn, F.J. Oliveira, L. |
| publishDate |
2006 |
| language |
English |
| container_title |
Algebra and Discrete Mathematics |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
If V is an existence variety of strict regular semigroups, then every semigroup in V has, within V, a maximal dense
ideal extension.
|
| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/157391 |
| citation_txt |
Dense ideal extensions of strict regular semigroups / F.J. Pastijn, L. Oliveira // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 67–80. — Бібліогр.: 24 назв. — англ. |
| work_keys_str_mv |
AT pastijnfj denseidealextensionsofstrictregularsemigroups AT oliveiral denseidealextensionsofstrictregularsemigroups |
| first_indexed |
2025-11-24T11:38:39Z |
| last_indexed |
2025-11-24T11:38:39Z |
| _version_ |
1850845786980483072 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2006). pp. 67 – 80
c© Journal “Algebra and Discrete Mathematics”
Dense ideal extensions of strict regular
semigroups
F. J. Pastijn and L. Oliveira
Communicated by L. Marki
Abstract. If V is an existence variety of strict regular semi-
groups, then every semigroup in V has, within V, a maximal dense
ideal extension.
1. Introduction
For a general background on semigroup theory we refer to [4], [8], [9],
[18], [19]. We shall assume that the reader has some familiarity with
pseudosemilattices [13]. It will be useful to review some basic results
first.
Result 1.1. Let S be a regular semigroup and a, b ∈ S such that Ja ≤ Jb.
Then there exists c ∈ Da such that c ≤ b.
Proof. There exist s, t ∈ S1 such that a = sbt. Let e and f be idempo-
tents of S such that eR bt and f L seb. Then a = sbtR sebL ebf ≤ eb ≤
b.
A regular semigroup S is called completely semisimple if every
principal factor of S is completely simple or completely 0-simple [12].
Applying Result 1.1 twice we easily prove that if S is a regular semigroup
and a ∈ S such that Da 6= Ja, then Da contains distinct comparable
idempotents. >From this observation it follows that a regular semigroup
S is completely semisimple if and only if no D-class of S contains distinct
The second author was partially supported by FCT through Centro de Matemática
da Universidade do Porto
68 Dense ideal extensions of strict regular semigroups
comparable idempotents. It should also be clear now that for a comple-
tely semisimple semigroup the Green relations J and D coincide.
In view of Result 1.1 it is now natural to introduce the following.
A regular semigroup S is said to be strict if for all a, b ∈ S such that
Ja ≤ Jb there exists a unique c ∈ Da such that c ≤ b. From what we have
just seen it follows that every strict regular semigroup S is completely se-
misimple and in particular, J = D in S. Strict regular semigroups have
been investigated thoroughly in the past four decades and the present
paper is one more contribution. We shall explain below what we intend
to do.
For a regular semigroup S in general there is a natural way to intro-
duce a quasi-order on S/D and a partial order on S/J . When dealing
with a strict regular semigroup S, we have that J = D for S and we shall
have no qualms referring to the poset I = S/D. In this circumstance, if
Dα is the D-class corresponding to α ∈ I, then we refer to (Dα, α ∈ I)
as the poset of D-classes of S. If Dα is a D-class of the strict regular
semigroup S, let D0
α be the set Dα with an extra 0 adjoined. If α is not
the least element of I = S/D then we interpret D0
α as a principal factor
of S and otherwise D0
α is the completely simple semigroup Dα with a
zero adjoined. Then the direct product
∏
α∈I D
0
α has a largest maximal
ideal given by
M = {(aα, α ∈ I) | aα = 0 for some α ∈ I}.
The Rees quotient (
∏
α∈I D
0
α)/M will be denoted by
∏
0
α∈I D
0
α and will
be called the 0-direct product of the D0
α, α ∈ I. One verifies that∏
0
α∈I D
0
α is a completely 0-simple semigroup.
The following key result finds its origin in [11].
Result 1.2. (i) Every strict regular semigroup is isomorphic to a sub-
direct product of its principal factors.
(ii) A regular semigroup is strict if and only if it is isomorphic to a
subdirect product of completely simple and/or completely 0-simple
semigroups.
A class of regular semigroups is called an existence variety if it is
closed for the taking of homomorphic images, regular subsemigroups and
direct products. The study of existence varieties was initiated in [5] and
[10]. A regular semigroup S is said to be locally inverse [locally a
Clifford semigroup] if for every idempotent e of S, eSe is an inverse
[Clifford] semigroup. The class SR of all strict regular semigroups is
contained in the class LI of all locally inverse semigroups, and both classes
are existence varieties [5]. In fact, as observed in [5]
F. J. Pastijn, L. Oliveira 69
Result 1.3. A regular semigroup is strict if and only if it is locally a
Clifford semigroup.
Naturally, if S is a regular semigroup, then the existence variety gen-
erated by S is the smallest existence variety containing S.
Proposition 1.4. An existence variety consists of strict regular semi-
groups only, if and only if it is generated by a completely simple or by a
completely 0-simple semigroup.
Proof. Every existence variety V of locally inverse semigroups is gen-
erated by a single locally inverse semigroup, namely the so-called bifree
object in V on a countably infinite set of variables ([1], [24]). If V consists
of completely simple semigroups only then this bifree object is completely
simple, and V is generated by this completely simple semigroup.
We shall from now on assume that V is an existence variety consisting
of strict regular semigroups only, but that not all members of V are com-
pletely simple. Let S be a strict regular semigroup which generates V.
Then S is not completely simple and, if (Dα, α ∈ I) is the poset of D-
classes of S, then by Result 1.2 the 0-direct product
∏
0
α∈I D
0
α generates
V. This semigroup
∏
0
α∈I D
0
α is completely 0-simple.
The converse part is obvious.
As an example, the existence variety consisting of the strict regular
semigroups S for which eSe is a semilattice for every idempotent e of
S (see [13], [14]) is generated by a 5-element completely 0-simple semi-
group (see [5], [6]) and there are precisely 12 existence varieties properly
contained in this one. This existence variety is precisely the class of all
strict regular semigroups that are also combinatorial. For an account
of the lattice of existence varieties each consisting of strict inverse semi-
groups, we refer to Section XII.4 of [19]. An existence variety consists of
normal bands of groups only, if and only if it is generated by a comple-
tely simple semigroup or by a completely simple semigroup with a zero
adjoined. We refer to [21] for further details about the lattice of existence
varieties consisting of strict regular semigroups. Since this lattice is of the
power of the continuum we retain from Proposition 1.4 that the concept
of existence varieties is a powerful tool for classifying completely 0-simple
semigroups.
If V is an existence variety, S, T ∈ V and ϕ : S −→ T an injective
homomorphism such that Sϕ is an ideal of T , then we say that ϕ : S −→
T is an ideal extension of S within V. An ideal extension ϕ : S −→ T
within V is called dense if whenever ψ : T −→ U is a homomorphism such
that ϕψ : S −→ U is an ideal extension within V, then ψ is injective.
70 Dense ideal extensions of strict regular semigroups
A dense ideal extension ϕ : S −→ T within V is called maximal if
whenever ψ : T −→ U is a homomorphism such that ϕψ : S −→ U is a
dense ideal extension within V, then ψ is an isomorphism of T onto U .
The construction of ideal extensions within V simplifies considerably if
for every S ∈ V, there exists a maximal dense ideal extension of S within
V. We refer to [15], [16], [17], [18] for a background on ideal extensions
and dense ideal extensions, in general, and within the existence variety
of locally inverse semigroups in particular. The main goal of this paper
will be to show that there always exist maximal dense ideal extensions
within every existence variety consisting of strict regular semigroups.
For any regular semigroup S the set of order ideals of S (with respect
to the natural partial order) forms a subsemigroup O(S) of the power
semigroup on S, and the mapping τS : S −→ O(S) which associates with
a ∈ S the principal order ideal (a] is a faithful representation of S if and
only if S is locally inverse [15]. If T (S) denotes the regular part of the
idealizer of SτS in O(S), then τS : S −→ T (S) is a maximal dense ideal
extension of S within LI [15]. From [20] it follows that if S is a normal
band of groups, then τS : S −→ T (S) is a maximal dense ideal extension
of S within the existence variety generated by S. In [20] Petrich uses
threads to great advantage, and we shall follow suit in Section 2. Our
main result features in Section 3.
2. Strict regular semigroups of threads
Let S be a locally inverse semigroup and O(S) the semigroup of order
ideals of S. For H,H ′ ∈ O(S) we shall say that H and H ′ are pairwise
inverse threads of S if H = HH ′H, H ′ = H ′HH ′, and, HH ′ and H ′H
are subsemilattices of S. Further, H ∈ O(S) is called a thread of S if
for some H ′ ∈ O(S), H and H ′ are pairwise inverse threads. The set of
all threads of S is denoted by C(S). Threads were used for inverse semi-
groups in [23], for normal bands of groups in [20] and for locally inverse
semigroups in general in [17].
Given a locally inverse semigroup S, C(S) need not constitute a sub-
semigroup of O(S), even in the case where S is a completely 0-simple
semigroup (see Corollary 2.7 of [17]). But C(S) forms a subsemigroup of
O(S) if S is an inverse semigroup [23], or a normal band of groups [20] or
a straight locally inverse semigroup [17]. For a locally inverse semigroup
S in general we have SτS ⊆ T (S) ⊆ C(S) ⊆ O(S) [15], [17].
If S ∈ SR is a strict regular semigroup, then we let CSR(S) consist
of the H ∈ C(S) such that H intersects every D-class of S in at most
one element. If a ∈ S, then we have from the definition of SR that (a] ∈
CSR(S) and therefore SτS ⊆ CSR(S) ⊆ C(S) ⊆ O(S). The remaining
F. J. Pastijn, L. Oliveira 71
theorem of this section states that CSR(S) is the largest strict regular
subsemigroup of O(S) which is contained in C(S) and contains SτS .
We collect some auxiliary results from [15] and [17].
Result 2.1 (Lemmas 3.2 and 3.11 of [15], Result 2.1 of [17]). Let S be
a locally inverse semigroup.
(i) If H ∈ C(S) then no distinct elements of H are L- or R-related in
S.
(ii) For F ∈ C(S), F is an idempotent of O(S) if and only if it is a
subsemilattice of S.
(iii) Let H and H ′ be pairwise inverse threads and E = HH ′. Then for
every a ∈ H there exists a unique inverse a′ of a which belongs to
H ′ and the mapping H −→ H ′, a −→ a′ is an order isomorphism.
For every e ∈ E there exist unique h ∈ H and h′ ∈ H ′ such that
hR eLh′ and then h and h′ are pairwise inverse elements of S with
e = hh′.
Result 2.2 (Lemma 3.3 of [15]). Let S be a locally inverse semigroup
and E,F ∈ C(S) idempotents of O(S). Put G = {e ∧ f | e ∈ E, f ∈ F}.
Then
(i) G ∈ O(S) and G is a subpseudosemilattice of (E(S),∧),
(ii) G = {g ∈ E(S) | eR gL f for some e ∈ E, f ∈ F},
(iii) EG = G = GF .
Lemma 2.3. Let S be a locally inverse semigroup and (H,H ′), (K,K ′)
pairs of pairwise inverse threads of S. Let E = KK ′, F = H ′H and
G = {e ∧ f | e ∈ E, f ∈ F}. If G is an idempotent of C(S), then
GK ⊆ K, HG ⊆ H, HGK = HK, and, HK and K ′GH ′ are pairwise
inverse threads of S.
Proof. We assume that G is an idempotent of C(S) and thus by Result
2.1, G is a subsemilattice and an order ideal of S.
Let gb ∈ GK for some g ∈ G and b ∈ K. By Result 2.1 there exists a
unique inverse b′ of b in K ′. We let h = bb′∧ g and see that gb = ghb. By
Result 2.2 there exists f ∈ F such that gL f , thus h = bb′ ∧ f . Therefore
h ∈ G since bb′ ∈ E. Since h = hg in the semilattice G it follows that
h ≤ g and therefore also that gb = ghb = hb ≤ b, whence gb ∈ K since
K is an order ideal. We proved that GK ⊆ K, and in a dual way we can
prove that HG ⊆ H.
72 Dense ideal extensions of strict regular semigroups
>From the foregoing we have that K ′GK ⊆ K ′K and so K ′GK is
a subsemilattice and an order ideal of S by Result 2.1. We show in a
dual way that the same is true for HGH ′. Furthermore by Result 2.2,
KK ′G = G = GH ′H. Thus
(HGK)(K ′GH ′) = HGH ′,
(K ′GH ′)(HGK) = K ′GK,
(HGK)(K ′GH ′)(HGK) = HGK,
(K ′GH ′)(HGK)(K ′GH ′) = K ′GH ′,
and we see that HGK and K ′GH ′ are pairwise inverse threads of S.
It remains to show that HK = HGK. From HG ⊆ H we already
have that HGK ⊆ HK. Let a ∈ H, b ∈ K and, using Result 2.1, let a′
and b′ be the inverses of a and b in H ′ and K ′, respectively. Put f = a′a,
e = bb′, thus e ∧ f ∈ G. Then ab = a(e ∧ f)b ∈ HGK and we may
conclude that HK = HGK.
>From here on we confine ourselves to strict regular semigroups.
Lemma 2.4. Let S be a strict regular semigroup. For H ∈ C(S) the
following are equivalent:
(i) H ∈ CSR(S),
(ii) if (H,H ′) is some [any] pair of pairwise inverse threads of S, then
H ′ ∈ CSR(S),
(iii) if (H,H ′) is some [any] pair of pairwise inverse threads of S, then
HH ′ ∈ CSR(S).
Proof. Assume that H ∈ CSR(S) and (H,H ′) is any pair of pairwise
inverse threads of S. Let D be a D-class of S and c, d ∈ H ′ ∩ D. By
Result 2.1 there exist a, b ∈ H such that a and b are inverses of c and
d, respectively. Then a, b ∈ H ∩D and since H intersects D in at most
one element we need to have a = b. Therefore acR ad and caL da, and
since HH ′ and H ′H are semilattices, we need to have that ac = ad and
ca = da. Therefore cH d and thus c = d by Result 2.1. We conclude that
H ′ ∈ CSR(S). We proved that (i) and (ii) are equivalent statements.
Assume that H ∈ CSR(S) and let H ′ ∈ C(S) such that H and H ′
are pairwise inverse threads. Then HH ′ is a subsemilattice and an order
ideal of S by Result 2.1 and H ′ ∈ CSR(S) by the above. Let D be a
D-class of S and e, f ∈ HH ′ ∩ D. By Result 2.1 there exist a, b ∈ H,
a′, b′ ∈ H ′ such that a′ is an inverse of a, b′ is an inverse of b, aR eL a′,
F. J. Pastijn, L. Oliveira 73
bR f L b′ and aa′ = e, bb′ = f . Since eD f , so also aD b and a′D b′, thus
a = b and a′ = b′ since H,H ′ ∈ CSR(S). Therefore e = f . We proved
that the equivalent conditions (i) and (ii) each imply (iii).
Assume that H and H ′ are pairwise inverse threads of S such that
HH ′ ∈ CSR(S). Let a, b ∈ H such that aD b. By Result 2.1 there exist
a′, b′ ∈ H ′ such that a′ and b′ are inverses of a and b respectively. Then
aa′R aD bR bb′ with aa′, bb′ ∈ HH ′, whence aa′ = bb′ and so aR b. By
Result 2.1 it follows that a = b. Therefore H ∈ CSR(S). We proved that
(iii) implies (i).
Theorem 2.5. If S is a strict regular semigroup, then CSR(S) is the
largest strict regular subsemigroup of O(S) which is contained in C(S)
and contains SτS .
Proof. For the strict regular semigroup S we let E and F be idempotents
of CSR(S). In particular, by Result 2.1, E and F are both subsemilattices
and order ideals of S. We put G = {e ∧ f | e ∈ E, f ∈ F}. Assume that
g1, g2 ∈ G such that g1 D g2 in S. By Result 2.2 there exist e1, e2 ∈ E
and f1, f2 ∈ F such that e1 R g1 L f1 and e2 R g2 L f2. Then e1 D e2 and
f1 D f2, hence e1 = e2, f1 = f2 since E,F ∈ CSR(S), and so g1 = g2. By
Result 2.2, G is an order ideal of S which is also a subpseudosemilattice
of (E(S),∧) and by the foregoing no two distinct elements of G can be
L- or R-related. It follows that G is a subsemilattice and an order ideal
of S, and thus an idempotent of C(S), by Result 2.1. From the above
it follows that moreover, G intersects every D-class of S in at most one
element and so we conclude that G is an idempotent of CSR(S).
We now let H,K ∈ CSR(S) and we let (H,H ′), (K,K ′) be pairs of
pairwise inverse threads of S. We put E = KK ′, F = H ′H and G = {e∧
f | e ∈ E, f ∈ F}. By Lemma 2.4 and the above, H,H ′,K,K ′, E, F,G ∈
CSR(S). By Lemma 2.3, HK and K ′GH ′ are pairwise inverse threads
of S, and using Result 2.2 and Lemma 2.3, (HK)(K ′GH ′) ⊆ HH ′,
where HH ′ is an idempotent of CSR(S) by Lemma 2.4. It follows that
(HK)(K ′GH ′) is an idempotent of CSR(S) and so again by Lemma 2.4,
HK and K ′GH ′ are pairwise inverse threads which belong to CSR(S).
It follows that CSR(S) is a regular subsemigroup of O(S) which is con-
tained in C(S). By Proposition 2.4 of [16], CSR(S) is a locally inverse
semigroup.
Let E be an idempotent of CSR(S) and H, H ′ pairwise inverse ele-
ments in the inverse semigroup E CSR(S)E. In particular H and H ′ are
pairwise inverse threads. We put F = HH ′ and G = H ′H. In view of
Result 1.3 we need to show that E CSR(S)E is a Clifford semigroup. In
order to show that E CSR(S)E is a Clifford semigroup, it now suffices
74 Dense ideal extensions of strict regular semigroups
to show that F = G. It follows from Proposition 2.3 of [17] that the
semilattices F and G are order ideals of the semilattice E.
For f ∈ F , f = hh′ for some pairwise inverse elements h ∈ H and
h′ ∈ H ′ by Result 2.1. Putting g = h′h ∈ H ′H = G we thus have f D g
in S. Since f, g ∈ E and E ∈ CSR(S) it follows that f = g. Using
symmetry we conclude that F = G. Therefore E CSR(S)E is a Clifford
semigroup, and we conclude that CSR(S) is a strict regular semigroup.
Let C be a strict regular subsemigroup of O(S) which is contained in
C(S) and contains SτS . Let E be an idempotent of C. By Result 2.1,
E is a subsemilattice and an order ideal of S. Let e, f ∈ E and eD f
in S. Then (e] and (f ] are D-related idempotents of SτS and thus also
of C, and (e] ≤ E, (f ] ≤ E by Proposition 2.3 of [17]. It follows that
(e] = (f ] and thus e = f . We conclude that E ∈ CSR(S). By Lemma
2.4, C ⊆ CSR(S).
3. Dense ideal extensions within SR
For any strict regular semigroup S we define TSR(S) = T (S) ∩ CSR(S).
Then clearly SτS ⊆ TSR(S) and so τS : S −→ TSR(S) is an embedding.
In this section we shall show that this embedding gives rise to a maximal
dense ideal extension of S, not only within SR, but in fact also within
the existence variety generated by S.
Theorem 3.1. Let S be a strict regular semigroup and T a regular se-
migroup such that SτS ⊆ T ⊆ TSR(S). Then τS : S −→ T is a dense
ideal extension of S within SR. Conversely, every dense ideal extension
of S within SR is equivalent to a unique ideal extension τS : S −→ T
obtained in this way.
Proof. Let T be a regular semigroup such that SτS ⊆ T ⊆ TSR(S). Then
by Theorem 2.5, T is a strict regular semigroup and in view of Theorem
4.6 of [15], τS : S −→ T is a dense ideal extension within SR.
Again by Theorem 4.6 of [15] every dense ideal extension of S within
SR is equivalent to a unique dense ideal extension τS : S −→ T where
T ⊆ T (S). Here of course T itself must be a strict regular semigroup, and
T ⊆ C(S) ⊆ O(S) [15], [17]. Therefore by Theorem 2.5, T ⊆ CSR(S).
Hence T ⊆ TSR(S).
Theorem 3.2. Let S be a strict regular semigroup. Then τS : S −→
TSR(S) is a maximal dense ideal extension within SR.
Proof. Let H ∈ TSR(S) and H ′ ∈ T (S) an inverse of H in T (S). Then
H ′ ∈ C(S) is an inverse of H in CSR(S) by Lemma 2.4, whence H and H ′
F. J. Pastijn, L. Oliveira 75
are pairwise inverse elements in TSR(S). It follows that TSR(S) is a regu-
lar semigroup. Further, since TSR(S) ⊆ CSR(S) we have from Theorem
2.5 that TSR(S) is a strict regular semigroup. Therefore by Theorem 3.1,
τS : S −→ TSR(S) is a dense ideal extension within SR. We proceed to
show that this ideal extension is a maximal dense ideal extension within
SR.
Let ψ : TSR(S) −→ U be a homomorphism such that τSψ : S −→ U
is a dense ideal extension within SR. Then U is a strict regular semi-
group. We need to show that ψ is an isomorphism. By Theorem 3.1 there
exists a unique strict regular semigroup T such that SτS ⊆ T ⊆ TSR(S)
such that the dense ideal extensions τS : S −→ T and τSψ : S −→ U are
equivalent. By Corollary 4.5 of [15],
ζ : U −→ T (S), u −→ ((u] ∩ SτSψ)(τSψ)−1
is the unique homomorphism of U into T (S) which extends (τSψ)−1τS ,
and ζ is injective since τSψ : S −→ U is a dense ideal extension. Clearly
then, the ideal extensions τS : S −→ Uζ and τSψ : S −→ U are equiva-
lent, whence T = Uζ.
It remains to show that T = TSR(S) and that ψ and ζ are pairwise
inverse isomorphisms. The proof for this follows the same lines as in the
proof of Theorem 4.7 of [15].
We shall need the following result from [17].
Result 3.3 (Proposition 2.3 of [17]). Let S be a locally inverse semigroup
and T a regular subsemigroup of O(S) such that T ⊆ C(S). Then
(i) HRK in T if and only if there exists a bijection ϕ : H −→ K such
that hRhϕ for every h ∈ H,
(ii) H LK in T if and only if there exists a bijection ϕ : H −→ K such
that hLhϕ for every h ∈ H.
With the notation of the introduction we have
Lemma 3.4. Let S be a strict regular semigroup and (Dα, α ∈ I) the
poset of D-classes of S. Then every principal factor of CSR(S) [TSR(S)]
can be isomorphically embedded into the completely 0-simple semigroup∏
0
α∈I D
0
α.
Proof. >From Theorems 2.5 and 3.2 we know that CSR(S) and TSR(S)
are strict regular semigroups and thus in particular, each principal factor
of CSR(S) and of TSR(S) is a completely simple or completely 0-simple
76 Dense ideal extensions of strict regular semigroups
semigroup. Since TSR(S) is a regular subsemigroup of CSR(S), each prin-
cipal factor of TSR(S) is a subsemigroup of a principal factor of CSR(S).
Thus it suffices to prove the statement of the lemma for CSR(S) only.
Let D be a D-class of CSR(S) and let H,K ∈ D. By Result 3.3 and
Theorem 2.5 it follows that
{α ∈ I | Dα ∩H 6= ∅} = {α ∈ I | Dα ∩K 6= ∅}.
This justifies the notation
ID = {α ∈ I | Dα ∩H 6= ∅}
where H is some [any] element of D. By Result 1.1, ID is an order ideal
of I.
We have that |ID| = 1 if and only if D consists of the (a], where
a ∈ Dγ , with γ being the least element of I. If this is the case then
D ∼= Dγ is completely simple and D can be isomorphically embedded
into
∏
0
α∈I D
0
α.
We shall henceforth assume that |ID| ≥ 2. Then D cannot be the
least element in the poset of D-classes of CSR(S) and so the completely
0-simple semigroup D0 is a principal factor of CSR(S). We consider the
mapping
ϕ : D0 −→
∏
0
α∈ID
D0
α, H −→ (aα, α ∈ ID), H ∈ D
0 −→ 0,
where for every α ∈ ID, Dα ∩ H = {aα}. The mapping ϕ is obviously
injective. We need to show that ϕ is a homomorphism. Since bothD0 and∏
0
α∈ID
D0
α are completely 0-simple it suffices to show that for H,K ∈ D,
HK 6= 0 if and only if (Hϕ)(Kϕ) 6= 0, and if this is the case, then
(HK)ϕ = (Hϕ)(Kϕ).
In the following we shall chooseH,K ∈ D and we putHϕ = (aα , α ∈
ID) and Kϕ = (bα , α ∈ ID). Assume that HK 6= 0 in the completely
0-simple semigroup D0. There exists an idempotent G ∈ D such that
H LGRK. By Result 2.1, G is a subsemilattice and an order ideal
of S. We put Gϕ = (gα, α ∈ ID), and using Result 3.3 we have that
aα L gα R bα for every α ∈ ID. It follows that (Hϕ)(Kϕ) = (aαbα, α ∈
ID) 6= 0, where for every α ∈ ID, aαbα ∈ Dα ∩HK. From this it follows
that (HK)ϕ = (Hϕ)(Kϕ).
Assume that conversely (Hϕ)(Kϕ) 6= 0. Then aαbα 6= 0 for every α ∈
ID, and so there exists an idempotent gα ∈ Dα such that aα L gα R bα for
every α ∈ ID. We put G = {gα | α ∈ ID}. Let E and F be idempotents
of D such that E LH and F RK in D. Then Eϕ = (eα, α ∈ ID),
F. J. Pastijn, L. Oliveira 77
Fϕ = (fα, α ∈ ID) and by Result 3.3, eα L gα R fα for every α ∈ ID.
By Result 2.2, G = {f ∧ e | e ∈ E, f ∈ F}, and by Result 3.3, G is an
idempotent of CSR(S). Again by Result 3.3, H LGRK in D, whence
HK 6= 0 in D0.
It should be obvious that
∏
0
α∈ID
D0
α can be embedded into
∏
0
α∈I D
0
α.
Hence the required result follows.
Corollary 3.5. If S is a strict regular semigroup, then the existence
varieties each generated by S, TSR(S) and CSR(S) coincide.
Proof. If S is a completely simple semigroup then S ∼= CSR(S) ∼= TSR(S)
and the statement is obvious. Otherwise, as in the proof of Proposition
1.4 and with the notation of Lemma 3.4, S and
∏
0
α∈I D
0
α generate the
same existence variety. Since CSR(S) is a strict regular semigroup which
is not completely simple, the existence variety generated by CSR(S) is
contained in the existence variety generated by
∏
0
α∈I D
0
α by Lemma 3.4.
Since S ∼= SτS ⊆ TSR(S) ⊆ CSR(S) we may now conclude that the exis-
tence varieties each generated by S, TSR(S) and CSR(S) coincide.
Theorem 3.6. Let V be an existence variety of strict regular semi-
groups. Then for every S ∈ V, τS : S −→ TSR(S) is a maximal dense
ideal extension within V.
Proof. The statement follows from Theorem 3.2 and Corollary 3.5.
4. Problems
>From what we have seen, the following question comes naturally: char-
acterize the existence varieties V of locally inverse semigroups which sat-
isfy the property that for every S ∈ V has within V a maximal dense
ideal extension. At this time we only know of the following existence va-
rieties which satisfy this pleasant property: (i) the existence variety LI of
locally inverse semigroups [15], (ii) the (existence) variety I of all inverse
semigroups [22] and (iii) every existence variety contained in SR. There
are undoubtedly more existence varieties satisfying our request, and our
general question may well turn out to be a tool for locating interesting
existence varieties.
For a locally inverse semigroup S, the ideal extension τS : S −→ T (S)
is a maximal dense ideal extension of S within LI [15]. Another natural
question to ask is which existence varieties V of locally inverse semigroup
satisfy the stronger condition:
S ∈ V =⇒ T (S) ∈ V . (4.1)
78 Dense ideal extensions of strict regular semigroups
If S is a normal band of groups, then we immediately conclude that
C(S) = CSR(S), and therefore T (S) = TSR(S). It now follows from
Theorem 3.6 that every existence variety of normal bands of groups sat-
isfies this condition.
Every existence variety V satisfying condition (4.1) and contain-
ing the 5-element combinatorial Brandt semigroup must contain I (see
e.g. [19]). Recall that the existence variety NBG of all normal bands of
groups is the largest existence variety of locally inverse semigroups not
containing the 5-element combinatorial Brandt semigroup [5]. Let A2 be
the 5-element completely 0-simple semigroup which generates the exis-
tence variety of all combinatorial strict regular semigroups. In a future
paper we shall show that LI is the only existence variety of locally in-
verse semigroups with property (4.1) and containing A2. Any existence
variety not containing A2 must be an existence variety of E-solid regular
semigroups [6], that is, an existence variety of regular semigroups whose
idempotents generate a completely regular semigroup. Thus, any other
existence variety of locally inverse semigroups with property (4.1) must
be an existence variety of E-solid locally inverse semigroups.
Let ESLI be the class of all E-solid locally inverse semigroups. Thus
ESLI consists of all regular semigroups for which the idempotents gen-
erate a normal band of groups. We shall denote by GI the subclass of
ESLI of all generalized inverse semigroups, that is, of all regular semi-
groups whose idempotents form a normal band. Let also LGI [RGI] be
the class of all left [right] generalized inverse semigroups, that is, of all
regular semigroups whose idempotents form a left [right] normal band.
The four classes of locally inverse semigroups introduced above are ex-
amples of existence varieties of regular semigroups [5].
We know from [22] that I satisfies (4.1). From both Lemma 3.6 and
Theorem 3.9 of [15] we conclude that if S is a locally inverse semigroup,
then the structure of the pseudosemilattice of idempotents of T (S) de-
pends only on the structure of the pseudosemilattice E(S), and not on
the entire structure of S. Thus, the existence varieties ESLI, RGI and
LGI also satisfy (4.1) due to results stated at the end of [2]. If S is a
generalized inverse semigroup and E,F ∈ T (S) are idempotents, then
EF = E ∧ F since E(S) is a normal band for the semigroup operation,
and thus E(T (S)) is a subband of T (S), whence E(T (S)) is a normal
subband of T (S) and T (S) is a generalized inverse semigroup. Therefore,
the existence variety GI of all generalized inverse semigroups is another
existence variety with property (4.1). In conclusion
Proposition 4.1. Any existence variety
V ∈ {I, RGI, LGI, GI, ESLI, LI} ∪ [T, NBG]
F. J. Pastijn, L. Oliveira 79
satisfies condition (4.1), where T denotes the existence variety of all triv-
ial semigroups. Any other existence variety of locally inverse semigroups
satisfying condition (4.1) must be contained in [GI, ESLI].
The question of which existence varieties from [GI, ESLI] have the
property (4.1) is still open. Let V be an existence variety of completely
simple semigroups containing the rectangular bands. Then V ∨ I ∈
[GI, ESLI]. Note also that ESLI is the join of I with the existence
variety of all completely simple semigroups [3]. Maybe a first approach
to answer this open question is to see under which conditions V∨ I satis-
fies (4.1). It is not clear that the join of existence varieties satisfying (4.1)
is another existence variety satisfying (4.1), and, in fact, this may well be
not true. Even if we are able to characterize under which conditions V∨I
has property (4.1), most certainly this will not solve this open question
completely. In [3] an example is given of an existence variety of E-solid
locally inverse semigroups which is not the join of an existence variety of
completely simple semigroups with an existence variety of inverse semi-
groups. Although this example does not belong to [GI, ESLI], it is a
strong indication that there exist existence varieties in [GI, ESLI] which
cannot be written as V ∨ I with V an existence variety of completely
simple semigroups.
References
[1] Auinger, K., The bifree locally inverse semigroup on a set, J. Algebra 166 (1994),
630–650.
[2] Auinger, K., On the lattice of existence varieties of locally inverse semigroups,
Canad. Math. Bull. 37 (1994), 13–20.
[3] Auinger, K., J. Doyle, and P. R. Jones, On existence varieties of locally inverse
semigroups, Math. Proc. Camb. Philos. Soc. 115 (1994), 197–217.
[4] Clifford, A. H., and G. B. Preston, The Algebraic Theory of Semigroups, I, Ameri-
can Mathematical Society, Providence, 1961; II, American Mathematical Society,
Providence, 1967.
[5] Hall, T. E., Identities for existence varieties of regular semigroups, Bull. Austral.
Math. Soc. 40 (1989), 59–77.
[6] Hall, T. E., Regular semigroups: amalgamation and the lattice of existence vari-
eties, Algebra Universalis 29 (1991), 79–108.
[7] Hall, T. E., The concept of variety for regular semigroups, in “Proceedings of the
Monash Conference on Semigroups Theory”, T. E. Hall, P. R. Jones, J. Meakin
eds., World Scientific, Singapore, 1991; 101–116.
[8] Higgins, P. M., Techniques of Semigroup Theory, Oxford Science Publications,
Oxford University Press, Oxford, 1992.
[9] Howie, J. M., Fundamentals of Semigroup Theory, Clarendon Press, London,
1995.
80 Dense ideal extensions of strict regular semigroups
[10] Kaďourek, J., and M. Szendrei, A new approach in the theory of orthodox semi-
groups, Semigroup Forum 40 (1990), 257–296.
[11] Lallement, G., Demi-groupes réguliers, Ann. Mat. Pura Appl. 77 (1967), 47–130.
[12] Munn, W. D., Semigroups satisfying minimal conditions, Proc. Glasgow Math.
Assoc. (3) (1957), 145–152.
[13] Nambooripad, K. S. S., Pseudo-semilattices and biordered sets, I, Simon Stevin
55 (1981), 103–110; II, Simon Stevin 56 (1982), 143–159; III, Simon Stevin 56
(1982), 239–256.
[14] Pastijn, F. J., Regular locally testable semigroups as semigroups of quasi-ideals,
Acta Mathematica Acad. Sci. Hung. 36 (1980), 161–166.
[15] Pastijn, F. J., and L. Oliveira, Maximal dense ideal extensions of locally inverse
semigroups, Semigroup Forum 72 (2006), 441–458.
[16] Pastijn, F. J., and L. Oliveira, Ideal extensions of locally inverse semigroups,
accepted in Studia Sci. Math. Hungarica.
[17] Pastijn, F. J., and L. Oliveira, Rees matrix covers and the translational hull of a
locally inverse semigroup, preprint.
[18] Petrich, M., Introduction to Semigroups, Merrill, Columbus, 1973.
[19] Petrich, M., Inverse Semigroups, Wiley, New York, 1984.
[20] Petrich, M., The translational hull of a normal cryptogroup, Math. Slovaca 44
(1994), 245–262.
[21] Petrich, M., Bases for existence varieties of strict regular semigroups. Bull. Belg.
Math. Soc. 8 (2001), 411–450.
[22] Ponizovskǐı, I. S., Zamechanie ob inversnykh polugruppakh, Uspekhi Mat. Nauk
20 (1965), 147–148.
[23] Schein, B. M., Completions, translational hulls and ideal extensions of inverse
semigroups, Czechoslovak Math. J. 23 (98) (1973), 575–610.
[24] Yeh, Y. T., The existence of e-free objects in e-varieties of regular semigroups,
Int. J. Algebra Comput. 2 (1992), 471–484.
Contact information
F. J. Pastijn Department of Mathematics, Statistics and
Computer Science, Marquette University,
Milwaukee, WI 53201-1881, USA
E-Mail: francisp@mscs.mu.edu
L. Oliveira Departamento de Matemática Pura, Facul-
dade de Ciências da Universidade do Porto,
R. Campo Alegre 687, 4169-007 Porto, Por-
tugal
E-Mail: loliveir@fc.up.pt
Received by the editors: 11.05.2003
and in final form 29.03.2006.
|