Page 96 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 96
NCOMMUTATIVE STRUCTURES WITHIN ORDER STRUCTURES, SEMIGROUPS
AND UNIVERSAL ALGEBRA (MS-67)
Duality for noncommutative frames
Jens Hemelaer, jens.hemelaer@uantwerpen.be
University of Antwerp, Belgium
Coauthors: Karin Cvetko-Vah, Lieven Le Bruyn
Noncommutative frames were introduced by Karin Cvetko-Vah as a noncommutative general-
ization of frames, similar to how skew lattices generalize lattices. The concept of a noncommu-
tative frame was motivated by Lieven Le Bruyn’s construction of a noncommutative topology
on the points of the Arithmetic Site of Connes and Consani. In this talk, we will extend the
duality between locales and frames to the noncommutative world. Our approach is inspired by
an earlier paper of Bauer, Cvetko-Vah, Gehrke, van Gool and Kudryavtseva, that introduced a
noncommutative Priestley duality.
The talk is based on joint work with Karin Cvetko-Vah and Lieven Le Bruyn.
Quasibands and nonassociative, noncommutative lattices
Michael Kinyon, mkkinyon@gmail.com
University of Denver, United States
Nonassociative idempotent magmas arise naturally in various settings such as the faces of a
building or chains in modular lattices. In this talk I will describe a variety of magmas we call
quasibands, which arise as (sub)reducts of bands (idempotent semigroups): in a band (B, ·),
define a new operation ◦ by x ◦ y = xyx. This is analogous to how quandles arise as subreducts
of groups under the conjugation operations. The quasiband operation ◦ is sometimes used as
a notational shorthand (especially in the theory of noncommutative lattices) or to characterize
band properties. For instance, (B, ◦) is associative, hence a left regular band, if and only if
(B, ·) is a regular band.
The variety of quasibands is defined by 4 identities. A main result is that this is precisely the
variety of ◦-subreducts of bands. In addition, I will talk about the natural preorder and natural
partial order on a quasiband, the center of a quasiband, the relationship between free quasibands
and free bands, and some enumeration of quasibands for low orders.
In the case of a noncommutative lattice (B, ∧, ∨), the corresponding “double quasiband”
(B, , ) can be viewed as a nonassociative, noncommutative lattice. I will discuss the rela-
tionship between the two structures for some of the more commonly studied classes of noncom-
mutative lattices (quasilattices, paralattices, skew lattices, etc.)
This is joint work with Tomaž Pisanski (Ljubljana). My own work is partially supported by
Simons Foundation Collaboration Grant 359872.
Covering skew lattices
Jurij Kovicˇ, jurij.kovic@siol.net
UP FAMNIT, Koper, Slovenia, and IMFM, Ljubljana, Slovenia
We introduce the concept of covering skew lattices, using the analogy with the covering graphs,
and we explain some new discovered phenomena in different forms.
94
AND UNIVERSAL ALGEBRA (MS-67)
Duality for noncommutative frames
Jens Hemelaer, jens.hemelaer@uantwerpen.be
University of Antwerp, Belgium
Coauthors: Karin Cvetko-Vah, Lieven Le Bruyn
Noncommutative frames were introduced by Karin Cvetko-Vah as a noncommutative general-
ization of frames, similar to how skew lattices generalize lattices. The concept of a noncommu-
tative frame was motivated by Lieven Le Bruyn’s construction of a noncommutative topology
on the points of the Arithmetic Site of Connes and Consani. In this talk, we will extend the
duality between locales and frames to the noncommutative world. Our approach is inspired by
an earlier paper of Bauer, Cvetko-Vah, Gehrke, van Gool and Kudryavtseva, that introduced a
noncommutative Priestley duality.
The talk is based on joint work with Karin Cvetko-Vah and Lieven Le Bruyn.
Quasibands and nonassociative, noncommutative lattices
Michael Kinyon, mkkinyon@gmail.com
University of Denver, United States
Nonassociative idempotent magmas arise naturally in various settings such as the faces of a
building or chains in modular lattices. In this talk I will describe a variety of magmas we call
quasibands, which arise as (sub)reducts of bands (idempotent semigroups): in a band (B, ·),
define a new operation ◦ by x ◦ y = xyx. This is analogous to how quandles arise as subreducts
of groups under the conjugation operations. The quasiband operation ◦ is sometimes used as
a notational shorthand (especially in the theory of noncommutative lattices) or to characterize
band properties. For instance, (B, ◦) is associative, hence a left regular band, if and only if
(B, ·) is a regular band.
The variety of quasibands is defined by 4 identities. A main result is that this is precisely the
variety of ◦-subreducts of bands. In addition, I will talk about the natural preorder and natural
partial order on a quasiband, the center of a quasiband, the relationship between free quasibands
and free bands, and some enumeration of quasibands for low orders.
In the case of a noncommutative lattice (B, ∧, ∨), the corresponding “double quasiband”
(B, , ) can be viewed as a nonassociative, noncommutative lattice. I will discuss the rela-
tionship between the two structures for some of the more commonly studied classes of noncom-
mutative lattices (quasilattices, paralattices, skew lattices, etc.)
This is joint work with Tomaž Pisanski (Ljubljana). My own work is partially supported by
Simons Foundation Collaboration Grant 359872.
Covering skew lattices
Jurij Kovicˇ, jurij.kovic@siol.net
UP FAMNIT, Koper, Slovenia, and IMFM, Ljubljana, Slovenia
We introduce the concept of covering skew lattices, using the analogy with the covering graphs,
and we explain some new discovered phenomena in different forms.
94
