Motivated by the pop-stack-sorting map on symmetric groups, Defant defined an operator $\mathsf{Pop}_M : M \to M$ for each complete meet-semilattice $M$ $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}).$$ This paper concerns dynamics of $\mathsf{Pop}_{\mathrm{Tam}_n}$, where $\mathrm{Tam}_n$ is $n$-th Tamari lattice. We say element $x\in \mathrm{Tam}_n$ $t$-$\mathsf{Pop}$-sortab...

The aim of this dissertation is to explore two different forms of intuitionistic quantum logic, one due to Isham, Butterfield and Döring, and the second due to Coecke. The two schemes have very different philosophical and physical motivations, and we explore how this leads to differences in the lattice extensions that they introduce. Isham, Butterfield and Döring suggested the so-called topos a...

We study abstract properties of intervals in the complete lattice of all meet-closed subsets ( -subsemilattices) of a -(meet-)semilattice S, where is an arbitrary cardinal number. Any interval of that kind is an extremally detachable closure system (that is, for each closed set A and each point x outside A, deleting x from the closed set generated by A and x leaves a closed set). Such closure s...

Abstract. For any finite unital commutative idempotent semigroup S, a unital semilattice, we show how to compute the amenability constant of its semigroup algebra l(S), which is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. Our theory applies to certain natural subalgebras of Fourier-Stieltjes alg...

In this paper the authors have studied the Glivenko congruence R in a 0-distributive nearlattice S defined by " () R b a ≡ if and only if 0 = ∧ x a is equivalent to 0 = ∧ x b for each S x ∈ ". They have shown that the quotient nearlattice R S is weakly complemented. Moreover, R S is distributive if and only if S is 0-distributive. They also proved that every Sectionally complemented nearlattice...