Lattice-ordered groups, as well as -algebras (pseudo -algebras), are both particular cases of dually residuated lattice-ordered monoids (-monoids for short). In the paper we study ideals of lower-bounded -monoids including -algebras. Especially, we deal with the connections between ideals of a -monoid and ideals of the lattice reduct of .
The article presents the definitions of action and of its omission, formulated in some logical theories of action. The concern of the analysis is to point at manifold possibilities of precising the intuitive sense of discussed concepts depending on the theory. The utility of a particular definition may be evaluated when the domain of applicability of the theory is taken into account ; application often becomes justification for the choice of a given definition.
In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product , where 2 is 2 = 0,1 with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X = ω,ℝ. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.