23 research outputs found
Noncommutative Frames Revisited
In this note, we correct an error in arXiv:1702.04949 by adding an additional
assumption of join completeness. We demonstrate with examples why this
assumption is necessary, and discuss how join completeness relates to other
properties of a skew lattice.Comment: 8 page
A non-commutative Priestley duality
We prove that the category of left-handed strongly distributive skew lattices
with zero and proper homomorphisms is dually equivalent to a category of
sheaves over local Priestley spaces. Our result thus provides a non-commutative
version of classical Priestley duality for distributive lattices and
generalizes the recent development of Stone duality for skew Boolean algebras.
From the point of view of skew lattices, Leech showed early on that any
strongly distributive skew lattice can be embedded in the skew lattice of
partial functions on some set with the operations being given by restriction
and so-called override. Our duality shows that there is a canonical choice for
this embedding.
Conversely, from the point of view of sheaves over Boolean spaces, our
results show that skew lattices correspond to Priestley orders on these spaces
and that skew lattice structures are naturally appropriate in any setting
involving sheaves over Priestley spaces.Comment: 20 page
Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup
We prove that the minimal cardinality of the semitransitive subsemigroup in
the singular part \IS_n\setminus \S_n of the symmetric inverse semigroup
\IS_n is , where is the greatest proper divisor of , and
classify all semitransitive subsemigroups of this minimal cardinality
Semitransitive and transitive subsemigroups of the inverse symmetric semigroups
We classify minimal transitive subsemigroups of the finitary inverse
symmetric semigroup modulo the classification of minimal transitive subgroups
of finite symmetric groups; and semitransitive subsemigroups of the finite
inverse symmetric semigroup of the minimal cardinality modulo the
classification of transitive subgroups of the minimal cardinality of finite
symmetric groups.Comment: 16 page
Duality for noncommutative frames
We characterize the left-handed noncommutative frames that arise from sheaves
on topological spaces. Further, we show that a general left-handed
noncommutative frame arises from a sheaf on the dissolution locale
associated to the commutative shadow of . Both constructions are made
precise in terms of dual equivalences of categories, similar to the duality
result for strongly distributive skew lattices in arXiv:1206.5848.Comment: 28 page