We prove that a finite atomistic lattice can be represented as a lattice of quasivarieties if and only if it is isomorphic to the lattice of all subsemilattices of a finite semilattice. This settles a conjecture that appeared in the context of [11].

Given a locale $L$ and a join semilattice $J$ with bottom element $0_J$, a new concept $(\sigma ,J)$ called $L$-slice is defined,where $\sigma$ is as an action of the locale $L$ on the join semilattice $J$. The $L$-slice $(\sigma ,J)$ adopts topological properties of the locale $L$ through the action $\sigma$. It is shown that for each $a\in L$, ${\sigma }_a$ is an interior operator on $(\sigma ,J)$.The collection $M=\{{\sigma }_a;a\in L\}$ is a Priestly space and a subslice of $L$-$Hom(J,J)$. If the locale $L$ is spatial we establish an isomorphism between the $L$-slices $(\sigma ,L)$ and $(\delta ,M)$. We have shown that the fixed set of ${\sigma }_a$,...

We use a set theoretic approach to consensus by viewing an object as a set of smaller pieces called “bricks”. A consensus function is neutral if there exists a family D of sets such that a brick s is in the output of a profile if and only if the set of positions with objects that contain s belongs to D. We give sufficient set theoretic conditions for D to be a lattice filter and, in the case of a finite lattice, these conditions turn out to be necessary. Ourfinal result, which involves a finite...