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1.
The construction by Hall of a fundamental orthodox semigroup W
B
from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup S
B
that plays the role of W
B
for a class of semigroups having a band of idempotents B. Specifically, the semigroups we consider are weakly B-abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E(S)=B lies in our class. On the other hand, if a semigroup S lies in our class, then S is Ehresmann if and only if B is a semilattice.
The Hall semigroup W
B
is a subsemigroup of S
B
, as are the (weakly) idempotent connected semigroups V
B
and U
B
. We show how the structure of S
B
can be used to extract information relating to arbitrary weakly B-abundant semigroups with (C).
This work was carried out during a visit to Lisbon of the second author funded by the London Mathematical Society and while
the first author was a member of project POCTI/0143/2003 of CAUL financed by FCT and FEDER. 相似文献
2.
3.
ABSTRACT The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups. 相似文献
4.
Yanhui Wang 《Semigroup Forum》2014,89(3):518-545
An orthogroup is a completely regular orthodox semigroup. The main purpose of this paper is to find a representation of a (generalised) orthogroup with band of idempotents B in terms of a fundamental (generalised) orthogroup. The latter is a subsemigroup of the Hall semigroup W B (or of its generalisations V B ,U B and S B ). We proceed in the regular case by constructing a fundamental completely regular subsemigroup \(\overline{W_{B}}\) of W B , using two different methods. Our subsemigroup plays the role for orthogroups that W B plays for orthodox semigroups, in that it contains a representation of every orthogroup with band of idempotents B, with kernel of the representation being μ, the greatest congruence contained in \(\mathcal{H}\) . To develop an analogous theory for classes of generalised orthogroups, that is, to extend beyond the regular case, we replace \(\mathcal{H}\) by \(\widetilde{\mathcal{H}}_{B}\) . Generalised orthogroups are then classes of weakly B-superabundant semigroups with (C). We first consider those satisfying an idempotent connected condition (IC) or (WIC). We construct fundamental weakly B-superabundant subsemigroups \(\overline{V_{B}}\) (respectively, \(\overline{U_{B}}\) ) of V B (respectively, U B ) with (C) and (IC) (respectively, with (C) and (WIC)) such that any weakly B-superabundant semigroup with (C) and (IC) (respectively, with (C) and (WIC)) admits a representation to \(\overline{V_{B}}\) (respectively, \(\overline{U_{B}}\) ), with kernel of the respresentation being μ B , the greatest congruence contained in \(\widetilde{\mathcal{H}}_{B}\) . Finally, we remove the idempotent connected condition and find a representation for an arbitrary weakly B-superabundant semigroup with (C), making use of fresh technology, constructing a fundamental weakly B-superabundant subsemigroup \(\overline{S_{B}}\) of S B , with the appropriate universal properties. We note that our results are needed in a parallel paper to complete the representation of arbitrary weakly B-superabundant semigroups with (C) as spined products of superabundant Ehresmann semigroups and subsemigroups of S B . 相似文献
5.
Yevhen Zelenyuk 《Semigroup Forum》2001,63(3):457-465
A semigroup S is called an absolute coretract if for any continuous homomorphism f from a compact Hausdorff right topological semigroup T onto a compact Hausdorff right topological semigroup containing S algebraically there exists a homomorphism g \colon S→ T such that f\circ g=id
S
. The semigroup β\ben contains isomorphic copies of any countable absolute coretract. In this article we define a class C of semigroups of idempotents each of which is a decreasing chain of rectangular semigroups. It is proved that every semigroup
from C is an absolute coretract and every finite semigroup of idempotents, which is an absolute coretract, belongs to C .
July 25, 2000 相似文献
6.
Abstract. Weakly left ample semigroups are a class of semigroups that are (2,1) -subalgebras of semigroups of partial transformations, where the unary operation takes a transformation α to the identity map in the domain of α . It is known that there is a class of proper weakly left ample semigroups whose structure is determined by unipotent monoids acting on semilattices or categories. In
this paper we show that for every finite weakly left ample semigroup S , there is a finite proper weakly left ample semigroup
and an onto morphism from
to S which separates idempotents. In fact,
is actually a (2,1) -subalgebra of a symmetric inverse semigroup, that is, it is a left ample semigroup (formerly, left type A). 相似文献
7.
The concepts of ℒ*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the ℒ*-inverse
semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups.
We shall also provide a constructed example for the ℒ*-inverse semigroups by using the left wreath products. 相似文献
8.
We consider the class of weakly U-abundant semigroups satisfying the congruence condition (C) containing both the class of regular semigroups and the class of abundant semigroups as its subclasses. The class of weakly U-abundant semigroups with a medial projection satisfying the congruence condition (C) will be particularly studied. This kind of semigroups will be called medial weakly U-abundant semigroups. In this paper, we establish a structure theorem for such semigroups. It is proved that every medial weakly U-abundant semigroup can be expressed by some kind of bands and quasi-Ehresmann semigroups. Our theorem generalizes and enriches the structure theorem given by M. Loganathan in 1987 for regular semigroups with a medial idempotent. 相似文献
9.
As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated
by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in
on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W
U
and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known
result of El-Qallali and Fountain for type W semigroups.
This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation
of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123) 相似文献
10.
Bernd Billhardt 《Semigroup Forum》2005,70(2):243-251
A regular (inverse) semigroup S is called F-regular (F-inverse), if each class of the least group congruence S contains a greatest element with respect to the natural partial order on S. Such a semigroup is necessarily an E-unitary regular (hence orthodox) monoid. We show that each F-regular semigroup S is isomorphic to a well determined subsemigroup of a semidirect product of a band X by S/S, where X belongs to the band variety, generated by the band of idempotents ES of S. Our main result, Theorem 4, is the regular version of the corresponding fact for inverse semigroups, and might be useful to generalize further features of the theory of F-inverse semigroups to the F-regular case. 相似文献
11.
Xiang-Yun Xie 《Semigroup Forum》2001,63(2):180-190
In this paper, some characterizations that an ordered semigroup S is a band of weakly r-archimedean ordered subsemigroups of S are given by some relations on S . We prove that an ordered semigroup S is a band of weakly r -archimedean ordered subsemigroups if and only if S is regular band of weakly r -archimedean ordered subsemigroups. Finally, we obtain that a negative ordered semigroup S is a band of weakly r-archimedean ordered subsemigroups of S if and only if S is a band of r-archimedean ordered subsemigroups of S . As an application the corresponding results on semigroups without order can be obtained by moderate modifications.
August 27, 1999 相似文献
12.
We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion
for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark
that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of
techniques that may be of interest in their own right. 相似文献
13.
Isabel M. Araújo Mário J. J. Branco Vitor H. Fernandes Gracinda M. S. Gomes N. Ruškuc 《Semigroup Forum》2001,63(1):49-62
Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids,
semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band
Y of monoids S
α
(α∈ Y ) is finitely generated/presented if and only if Y is finite and all S
α
are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is
not finitely presented, but which is a disjoint union of two finitely presented subsemigroups.
January 21, 2000 相似文献
14.
Yingdan Ji 《代数通讯》2013,41(12):5149-5162
Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R0[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras. 相似文献
15.
Matrices of bisimple regular semigroups 总被引:1,自引:0,他引:1
Janet E. Mills 《Semigroup Forum》1983,26(1):117-123
A semigroup S is a matrix of subsemigroups Siμ, i ε I, μ ε M if the Siμ form a partition of S and SiμSjν≤Siν for all i, j in I, μ, ν in M. If all the Siμ are bisimple regular semigroups, then S is a bisimple regular semigroup. Properties of S are considered when the Siμ are bisimple and regular; for example, if S is orthodox then each element of S has an inverse in every component Siμ. 相似文献
16.
Gracinda M. S. Gomes 《Acta Mathematica Hungarica》2005,109(1-2):33-51
Summary We consider proper (idempotent pure) extensions of weakly left ample semigroups. These are extensions that are injective in
each <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\widetilde{\mathcal{R}}$-class.
A graph expansion of a weakly left ample semigroup S is shown to be such an extension of S. Using semigroupoids acted upon by weakly left ample semigroups, we prove that any weakly left ample semigroup which is a
proper extension of another such semigroup T is (2,1)-embeddable into a λ-semidirect product of a semilattice by T. Some known results, by O'Carroll, for idempotent pure extensions of inverse semigroups and, by Billhardt, for proper extensions
of left ample semigroups follow from this more general situation. 相似文献
17.
R. J. Warne 《Semigroup Forum》1997,54(1):271-277
An algebraA satisfiesTC (the term condition) if
for any
and anyn + 1-ary termp.TC algebras have been extensively studied. We previously determined the structure of allTC semigroups. We use this result to show that ifS is aTC semigroup thenS
E = {a ε S | ax is an idempotent for somex ε S} is an inflation ofS
Reg (the set of regular elements ofS) andS
Reg ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup. As a corollary of this result, we show thatS is a semisimpleTC semigroup iffS ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup. 相似文献
18.
IfS
n
andC
n
denote, respectively, the symmetric group and inverse semigroup onn symbols, thenS
n⊂Cn and a semigroupT⊂Cn isS
n
-normal ifα
−1
Tα ⊂Tfor every α∈S
n
. TheS
n
-normal semigroups are classified. 相似文献
19.
Mario Petrich 《Semigroup Forum》2005,71(3):366-388
On any regular semigroup S, the greatest idempotent pure congruence
τ the greatest idempotent separating congruence μ and the least
band congruence β are used to give the S-classification of regular semigroups as follows. These congruences generate a sublattice
Λ of the congruence lattice C(S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K- and T-relations
on C(S) to Λ. Such triples are characterized abstractly and form the objects of a category S whose morphisms are surjective K- and T-preserving homomorphisms subject to a mild condition. The class of regular semigroups
is made into a category S whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor
from S to S. The effect of the S-classification on Reilly semigroups and cryptogroups is discussed briefly. 相似文献
20.
We show that if (S(t))
t≧0 is a contraction semigroup on a closed convex subset of a uniformly convex Banach space, then every bounded and “asymptotically
isometric” almost-orbit of (S(t))
t≧0 is weakly almost periodic in the sense of Eberlein. As one consequence, results on the existence of almost periodic solutions
to the abstract Cauchy problem are obtained without the need fora priori compactness assumptions. As a further consequence, the known strong ergodic limit theorems for (almost-) orbits of nonlinear
contraction semigroups turn out to be special cases of Eberlein’s classical ergodic theorem for weakly almost periodic functions. 相似文献