Distributive Closed Inverse Subsemigroup Lattices

Cheong, Kyeong Hee

doi:10.1007/pl00005934

Cited by 1 publication

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) In contrast with the the previous proposition, the convex closure of an aperiodic, completely semisimple, faintly archimedean inverse semigroup need retain none of those properties. (2) In contrast with Proposition 4.7, the order ideal generated by an aperiodic, completely semisimple pseudo-archimedean inverse semigroup need retain none of those properties.…”

Section: Properties Preserved By Convex Closurementioning

confidence: 90%

“…We remark that in [2] there are found necessary and sufficient conditions in order that each of the above retractions should extend to the whole lattice Co (S).…”

Section: Proposition 52 Let U Be An Order Ideal Of An Inverse Semigmentioning

confidence: 99%

“…By semisimplicity, for each a ∈ N S , aa −1 ||a −1 a and so according to (2) implies that φ preserves L and R. As noted above, a −1 φ = (aφ) −1 .…”

Section: Co -And L-isomorphismsmentioning

confidence: 99%

See 2 more Smart Citations

Inverse semigroups determined by their lattices of convex inverse subsemigroups II

Cheong¹,

Jones

2003

Algebra Universalis

Self Cite

View full text Add to dashboard Cite

In Part I of this paper we showed that the study of Co -isomorphisms of inverse semigroups, that is, isomorphisms between the lattices of convex inverse subsemigroups of two such semigroups, can be reduced to consideration of those that induce an isomorphism between the respective semilattices of idempotents. We go on here to prove two somewhat complementary theorems. We show that the inverse semigroups Co -isomorphic to the bicyclic semigroup are precisely the well-known simple, aperiodic, inverse ω-semigroups B d . And we show that for completely semisimple inverse semigroups (those that contain no bicyclic subsemigroup), Co -isomorphisms that induce an isomorphism between the semilattices of idempotents of the respective inverse semigroups are entirely equivalent to L-isomorphisms with the same property. (An L-isomorphism is an isomorphism between the lattices of all inverse subsemigroups.) Combining this result, known results on L-isomorphisms and the main theorem of Part I yields a complete determination of Co -isomorphisms for broad classes of semigroups. For some slightly narrower classes it is known that every Co -isomorphism of necessity induces an isomorphism on the semilattice of idempotents, yielding theorems on their Co -determinability.

show abstract

Section: Properties Preserved By Convex Closurementioning

confidence: 90%

“…We remark that in [2] there are found necessary and sufficient conditions in order that each of the above retractions should extend to the whole lattice Co (S).…”

Section: Proposition 52 Let U Be An Order Ideal Of An Inverse Semigmentioning

confidence: 99%