[Settfa] *Changed Schedule* Tuesday 4 Mar, *Joseph McDonald*

[David Bradley-Williams]

Dear all,

Due to a mutual agreement between the speakers concerning availability, 
we have a change of speaker for tomorrow's seminar. Please see find the 
amended announcement immediately below.

With best wishes,
David

Tuesday, 4 March 2025 - 10:00 to 11:30
Place: IM, konírna

Speaker: Joseph McDonald, University of Alberta
Title: MacNeille completion, canonical completion, and duality for 
monadic ortholattices

Abstract
An ortholattice is a bounded lattice equipped with an order-inverting 
involutive complementation. A monadic ortholattice is an ortholattice 
equipped with a closure operator, known as a quantifier, whose closed 
elements form a sub-ortholattice. Monadic ortholattices generalize 
monadic Boolean algebras - the algebraic model of the classical 
predicate calculus in a single variable. Janowitz [3] first considered 
quantifiers on orthomodular lattices, and Harding [1] studied them, as 
well as cylindric ortholattices, for their connections to von Neumann 
algebras, in particular, to subfactors.

We show that the variety of monadic ortholattices is closed under 
MacNeille and canonical completions. In each case, the completion of A 
is obtained by forming an associated dual space X that is a monadic 
orthoframe. This is a set equipped with an orthogonality relation and an 
additional binary relation satisfying certain conditions. For the 
MacNeille completion, X is formed from the non-zero elements of A, and 
for the canonical completion, X is formed from the proper filters of A. 
In either case, the corresponding completion of A is then obtained as 
the complete ortholattice of bi-orthogonally closed subsets of X with an 
additional operation defined through the binary relation on X. With the 
introduction of a spectral topology on a monadic orthoframe along the 
lines of McDonald and Yamamoto [4], we obtain a dual equivalence between 
the category of monadic ortholattices and homomorphisms and the category 
of monadic upper Vietoris orthospaces and certain continuous frame 
morphisms. The duality presented here is obtained in ZF independently of 
the Axiom of Choice. This talk is based on joint work with John Harding 
and Miguel Peinado [2].

[1] Harding, J.: Quantum monadic algebras. Journal of Physics A: 
Mathematical and Theoretical. Vol 55 (2023)
[2] Harding, J., McDonald, J., Peinado, M.: Monadic ortholattices: 
completions and duality. Forthcoming in Algebra Universalis (2025)
[3] Janowitz, M.: Quantifiers and orthomodular lattices. Pacific Journal 
of Mathematics. Vol 13 (1963)
[4] McDonald, J., Yamamoto, K.: Choice-free duality for 
orthocomplemented lattices by means of spectral spaces. Algebra 
Universalis. Vol. 83 (2022)

For more information see the seminar web page at
https://www.math.cas.cz/index.php/events/seminar/6

On 2025-03-01 09:00, kubis at math.cas.cz wrote:
> Tuesday, 4 March 2025 - 10:00 to 11:30
> Place: IM, konírna
> 
>  Speaker: Wesley Fussner, ICS CAS
> Title: Amalgamation in Residuated Structures
> 
>  Abstract
> Residuated structures play an important role in several fields, with
> prominent examples coming from the algebra of binary relations,
> ordered groups, multiplicative ideal theory, as well as pure and
> applied logic. Due primarily to systematic connections between
> amalgamation and various logical interpolation properties, recent
> years have seen an extensive effort to understand amalgamation in
> residuated structures. This effort has proven extremely successful,
> and we now have a well-developed understanding of amalgamation in
> several of the most significant classes of residuated structures. This
> talk surveys some of landmark progress in this area, focusing on the
> last 5-10 years work and emphasizing the interplay between algebraic
> and logical techniques.
> 
>  For more information see the seminar web page at
> https://www.math.cas.cz/index.php/events/seminar/6
> 
>  Set Theory and Analysis mailing list
> settfa at math.cas.cz
> https://list.math.cas.cz/listinfo/settfa@math.cas.cz
> _______________________________________________
> Settfa mailing list
> Settfa at math.cas.cz
> https://list.math.cas.cz/listinfo/settfa

-- 
David Bradley-Williams PhD
Department of Abstract Analysis
Institute of Mathematics, Czech Academy of Sciences
Žitná 25
115 67 Praha 1
Czech Republic

Tel: +420-222 090 743

Url: http://davidbw.sdf.org/