For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or antichains. This allows us to prove some consequences of the Boolean Prime Ideal theorem using arguments in the style of those that use Zorn’s Lemma, or Martin’s Axiom.

we study different completeness definitions for two categories of lattice-valued cauchy spaces and the relations between these definitions. we also show the equivalence of a so-called completion axiom and the existence of a completion.

We present a new method for proving completeness of ordered chaining calculi for resolution-based rst-order reasoning with transitive relations. Previously, chaining rules were postulated and a posteriori ver-iied in semantic completeness proofs. We synthesize the rules from the transitivity axiom. Completeness of our calculi then follows from cor-rectness of their construction. Our method yiel...

We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems a...

In this paper, a new class of generalized separation axioms (briefly, g-Tg-separation axioms) whose elements are called g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, g-Tg,N-axioms is defined in terms sets g-Tg-sets) topological spaces Tg-spaces) and the properties characterizations Tg-space endowed with each such discussed. The study shows that g-Tg,F-axiom implies g-Tg,K-axiom, g-Tg,H-axiom g-Tg,F-axiom, g-...

We deal with a wide range of colors in our daily life. They are such ubiquitous phenomena that is hard and next to impossible to imagine even a single entity (be it an object, place, living creature, etc) devoid of them. They are like death and tax which nobody can dispense with. This omnipresence of colors around us has also made its way through abstract and less tangible entities via the inte...