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1.
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
2.
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
3.
It is well known that the free group on a non-empty set can be totally ordered and, further, that each compatible latttice ordering on a free group is a total ordering. On the other hand, Saitô has shown that no non-trivial free inverse semigroup can be totally ordered. In this note we show, however, that every free inverse monoid admits compatible lattice orderings which are closely related to the total orderings on free groups.These orderings are natural in the sense that the imposed partial ordering on the idempotents coincides with the natural partial ordering. For this to happen in a lattice ordered inverse semigroup, the idempotents must form a distributive lattice. The method of construction of the lattice orderings on free inverse monoids can be applied to show that naturally lattice ordered inverse semigroups with a given distributive lattice E of idempotents can have arbitrary Green's relation structure. Analogous results hold for naturally -semilatticed inverse semigroups. In this case, there is no restriction on the semilattice E of idempotents.We also show that every compatible lattice ordering on the free monogenic inverse monoid is of the type considered here. This permits us to prove that there are precisely eight distinct compatible lattice orderings on this semigroup. They belong to two families, each of which contains four members, of conjuguate lattice orderings. 相似文献
4.
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
consider the smallest regular semigroups that contain precisely one of each of
these four types. We show that, to within isomorphism and dualisomorphism,
there are six such semigroups and characterise them as quotient semigroups of
certain regular Rees matrix semigroups. 相似文献
5.
A. H. Clifford 《Semigroup Forum》1972,5(1):137-144
Call a semigroup S left unipotent if eachℒ-class of S contains exactly one idempotent. A structure theorem for bisimple left unipotent semigroups is given which reduces
to that of N. R. Reilly [8] for bisimple inverse semigroups. A structure theorem, alternative to one given by R. J. Warne
[13], is given for the case when the band ES of idempotents of S is an ω-chain of right zero semigroups, and two applications of it are made.
This research was partially supported by a grant from the National Science Foundation. 相似文献
6.
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
7.
8.
A semigroup is tight if each of its congruences is uniquely determined by each
of the congruence classes. Bisimple inverse semigroups are tight, and tight
semigroups are either simple or congruence-free with zero. Although congruence-free
semigroups are tight, they are not necessarily bisimple. We construct
tight inverse semigroups and tight inverse monoids that are neither bisimple
nor congruence-free. 相似文献
9.
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.
Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described
(as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that
contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we
call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that TX contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that TX contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7. 相似文献
10.
Mária B. Szendrei 《Semigroup Forum》1980,20(1):1-10
In the present paper we deal with two problems concerning orthodox semigroups. M. Yamada raised the questions in [6] whether
there exists an orthodox semigroup T with band of idempotents E and greatest inverse semigroup homomorphic image S for every
band E and inverse semigroup S which have the property that
is isomorphic to the semilattice of idempotents of S, and if T exists then whether it is always unique up to isomorphism.
T. E. Hall [1] has published counter-examples in connection with both questions and, moreover, he has given a necessary and
sufficient condition for existence. Now we prove a more effective necessary and sufficient condition for existence and deal
with uniqueness, too. On the other hand, D. B. McAlister's theorem in [4] saying that every inverse semigroup is an idempotent
separating homomorphic image of a proper inverse semigroup is generalized for orthodox semigroups. The proofs of these results
are based on a theorem concerning a special type of pullback diagrams. In verifying this theorem we make use of the results
in [5] which we draw up in Section 1. The main theorems are stated in Section 2. For the undefined notions and notations the
reader is referred to [2]. 相似文献
11.
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
12.
Peter R. Jones 《Semigroup Forum》2006,73(3):330-344
A lattice isomorphism between inverse semigroups S and T is an isomorphism between their lattices of inverse subsemigroups.
When S is combinatorial, it has long been known that a bijection is induced between S and T. Various authors have introduced
successively weaker "archimedean" hypotheses under which this bijection is necessarily an isomorphism, naturally inducing
the original lattice isomorphism. Since lattice-isomorphic groups need not have the same cardinality, extending these techniques
to the non-combinatorial case requires some means of tying the subgroups to the rest of the semigroup. Ershova showed that
if S has no nontrivial isolated subgroups (subgroups that form an entire D-class) then again a bijection exists between S
and T. Recently, this technique has been successfully exploited, by Goberstein for fundamental inverse semigroups and by
the author for completely semisimple inverse semigroups, under two different finiteness hypotheses. In this paper, we derive
further properties of Ershova's bijection(s) and formulate a "quasi-connected" hypothesis that enables us to derive both Goberstein's
and the author's earlier results as corollaries. 相似文献
13.
14.
15.
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
16.
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
17.
18.
19.
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