After a survey of some known lattice results, we determine the greatest idempotent (resp. compact) solution, when it exists, of a finite square rational equation assigned over a linear lattice. Similar considerations are presented for composite relational equations.

We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that: For example, the lattice M2ω is a sublattice of the loc...

Let X be a locally finite tree and let G = Aut(X). Then G is naturally a locally compact group. A discrete subgroup Γ ≤ G is called an X-lattice, or a tree lattice if Γ has finite covolume in G. The lattice Γ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X-lattices (Γ′,Γ) with Γ ≤ Γ′, starting from ‘edgeindexed graphs’ (A′, i′) a...

We prove that the lattice of varieties contains almost no compact elements.

An operator T : E → X between a Banach lattice E and a Banach space X is called b-weakly compact if T (B) is relatively weakly compact for each b-bounded set B in E. We characterize b-weakly compact operators among o-weakly compact operators. We show summing operators are b-weakly compact and discuss relation between Dunford–Pettis and b-weakly compact operators. We give necessary conditions fo...

We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that.

Complete eigenvalue spectra of the staggered Dirac operator in quenched 4d compact QED are studied on 83 × 4 and 83 × 6 lattices. We investigate the behavior of the nearest-neighbor spacing distribution P (s) as a measure of the fluctuation properties of the eigenvalues in the strong coupling and the Coulomb phase. In both phases we find agreement with the Wigner surmise of the unitary ensemble...