In this expository article we use topological ideas, notably compactness, to establish certain basic properties of orderable groups. Many of the properties we shall discuss are well-known, but I believe some of the proofs are new. These will be used, in turn, to prove some orderability results, including the left-orderability of the group of PL homeomorphisms of a surface with boundary, which are fixed on at least one boundary component.
Let be a -ary lattice term. A -pointed lattice , will be called a -lattice (or a test lattice if is not specified), if is generated by and, in addition, for any -ary lattice term satisfying , in , the lattice identity holds in all lattices. In an elementary visual way, we construct a finite -lattice for each . If is a canonical lattice term,...
In the paper an additive closure operator on an abelian unital -group is introduced and one studies the mutual relation of such operators and of additive closure ones on the -algebra .
Let be a Hilbert algebra. The monoid of all unary operations on generated by operations , which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of . This semilattice is isomorphic to the semilattice of finitely generated filters of , it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of . Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of , and the...