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The notion of an inverse transversal of a regular semigroup is well-known. Here we investigate naturally ordered regular semigroups
that have an inverse transversal. Such semigroups are necessarily locally inverse and the inverse transversal is a quasi-ideal.
After considering various general properties that relate the imposed order to the natural order, we highlight the situation
in which the inverse transversal is a monoid. The regularity of Green’s relations is also characterised. Finally, we determine
the structure of a naturally ordered regular semigroup with an inverse monoid transversal. 相似文献
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Elton Pasku 《Semigroup Forum》2011,83(1):75-88
We prove that the category of Clifford semigroups and prehomomorphisms CSP\mathcal{CSP} is isomorphic to a certain subcategory of the category of diagrams over groups. Under this isomorphism, Clifford semigroups
are identified with certain functors. As an application of the isomorphism theorem, we show that the category with objects
commutative inverse semigroups having the same semilattice of idempotents and with morphisms, the inverse semigroup homomorphisms
that fix the semilattice, imbeds into a category of right modules over a certain ring. Also we find a very close relationship
between the cohomology groups of a commutative inverse monoid and the cohomology groups of the colimit group of the functor
giving the monoid. 相似文献
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Every inverse semigroup possesses a natural partial order and therefore convexity with respect to this order is of interest.
We study the extent to which an inverse semigroup is determined by its lattice of convex inverse subsemigroups; that is, if
the lattices of two inverse semigroups are isomorphic, how are the semigroups related? We solve this problem completely for
semilattices and for inverse semigroups in general reduce it to the case where the lattice isomorphism induces an isomorphism
between the semilattices of idempotents of the semigroups. For many inverse semigroups, such as the monogenic ones, this case
is the only one that can occur. In Part II, a study of the reduced case enables us to prove that many inverse semigroups,
such as the free ones, are strictly determined by their lattices of convex inverse subsemigroups, and to show that the answer
obtained here for semilattices can be extended to a broad class of inverse semigroups, including all finite, aperiodic ones.
Received September 24, 2002; accepted in final form December 15, 2002. 相似文献
6.
In this paper, we consider the lattice Subf S of full subsemigroups of an inverse semigroup S. Our first main theorem states
that for any inverse semigroup S, Subf S is a subdirect product of the lattices of full subsemigroups of its principal factors,
so that Subf S is distributive [meet semidistributive, join semidistributive, modular, semimodular] if and only if the lattice
of full subsemigroups of each principal factor is. To examine such inverse semigroups, therefore, we need essentially only
consider those which are 0-simple. For a 0-simple inverse semigroup S (not a group with zero), we show that in fact each of
modularity, meet semidistributivity and join semidistributivity of Subf S is equivalent to distributivity of S, that is, S
is the combinatorial Brandt semigroup
with exactly two nonzero idempotents and two nonidempotents. About semimodularity, however, we concentrate only on the completely
0-simple case, that is, Brandt semigroups. For a Brandt semigroup S (not a group with zero), semimodularity of Subf S is equivalent
to distributivity of Subf S. Finally, we characterize an inverse semigroup S for which Subf S is a chain. 相似文献
7.
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
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Emil Daniel Schwab 《代数通讯》2013,41(5):1779-1789
We study the incidence algebra of the reduced standard division category of a combinatorial bisimple inverse monoid [with (E(S), ≤) locally finite], and we describe semigroups of poset type (i.e., a combinatorial inverse semigroup for which the corresponding Möbius category is a poset) as being combinatorial strict inverse semigroups. Up to isomorphism, the only Möbius-division categories are the reduced standard division categories of combinatorial inverse monoids. 相似文献
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研究范畴与半群通过幂等元双序建立的一种自然联系.对每个有幂等元的半群S,其幂等元生成的左、右主理想之集通过双序ω~e,ω~r自然确定两个有子对象、有像且每个包含都右可裂的范畴L(S),R(S),其中态射的性质与S中元素的富足性、正则性有自然对应.利用这个联系,我们定义了"平衡(富足、正规)范畴"概念.对任一平衡(富足、正规)范畴■,我们构造其"锥半群"■,证明■左富足(富足、正则),且每个平衡(富足、正规)范畴■都与某左富足(富足、正则)半群S的左主理想范畴L(S)(作为有子对象的范畴)同构. 相似文献
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Orthodox semigroups whose idempotents satisfy a certain identity 总被引:2,自引:0,他引:2
Miyuki Yamada 《Semigroup Forum》1973,6(1):113-128
An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy
[xyx=yx]. Bisimple left [right] inverse semigroups have been studied by Venkatesan [6]. In this paper, we clarify the structure
of general left [right] inverse semigroups. Further, we also investigate the structure of orthodox semigroups whose idempotents
satisfy the identity xyxzx=xyzx. In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies
xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup. 相似文献
12.
It is proved that any pseudovariety of finite semigroups generated by inverse semigroups, the subgroups of which lie in some
proper pseudovariety of groups, does not contain all aperiodic semigroups with commuting idempotents. In contrast we show
that every finite semigroup with commuting idempotents divides a semigroup of partial bijections that shares the same subgroups.
Finally, we answer in the negative a question of Almeida as to whether a result of Stiffler characterizing the semidirect
product of the pseudovarieties ofR-trivial semigroups and groups applies to any proper pseudovariety of groups. 相似文献
13.
Lattice orders on the semigroup ring of a positive rooted monoid are constructed, and it is shown how to make the monoid ring into a lattice-ordered ring with squares positive in various ways. It is proved that under certain conditions these are all of the lattice orders that make the monoid ring into a lattice-ordered ring. In particular, all of the partial orders on the polynomial ring A[x] in one positive variable are determined for which the ring is not totally ordered but is a lattice-ordered ring with the property that the square of every element is positive. In the last section some basic properties of d-elements are considered, and they are used to characterize lattice-ordered division rings that are quadratic extensions of totally ordered division rings. 相似文献
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Directed graphs have long been used to gain an understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Frucht’s Theorem. We investigate two inverse semigroups defined over undirected graphs constructed from the notions of subgraph and vertex set induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these inverse semigroups. We prove a characterization theorem that states that every graph has a unique associated inverse semigroup up to isomorphism allowing for an algebraic restatement of the Edge Reconstruction Conjecture. 相似文献
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《代数通讯》2013,41(8):2929-2948
Abstract A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed. 相似文献
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Amal AlAli 《代数通讯》2017,45(11):4667-4678
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Using group congruences, we obtain necessary and sufficient conditions for an ordered E-inversive semigroup to be a Dubreil-Jacotin semigroup. We also determine when such a semigroup is naturally ordered. In particular,
when the subset of regular elements is a subsemigroup it contains a multiplicative inverse transversal. 相似文献
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R. Exel 《Semigroup Forum》2009,79(1):159-182
By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse
semigroup
in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of
Booleanness present in the semilattice of idempotents of
. After observing that the Vagner–Preston representation is not tight, we exhibit a canonical tight representation for any
inverse semigroup with zero, called the regular tight representation. We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative.
The lack of faithfulness is however completely understood as long as we restrict to continuous inverse semigroups, a class generalizing the E
*-unitaries.
Partially supported by CNPq. 相似文献
20.
T. S. BLYTH M. H. ALMEIDA SANTOS 《数学学报(英文版)》2006,22(6):1705-1714
An ordered pair (e, f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. We have shown previously that there are four distinct types of skew pairs of idempotents. Here we investigate the ubiquity of such skew pairs in full transformation semigroups. 相似文献