A divisibility relation on ultrafilters is defined as follows: F (vertical bar) over tilde G if and only if every set in F upward closed for divisibility also belongs to G. After describing the first omega levels of this quasiorder, in this paper we generalize the process of determining the basic divisors of an ultrafilter. First we describe these basic divisors, obtained as (equivalence classes of) powers of prime ultrafilters. Using methods of nonstandard analysis we define the pattern of an ultrafilter: the collection of its basic divisors as well as the multiplicity of each of them. All such patterns have a certain closure property in an appropriate topology. We isolate the family of sets belonging to every ultrafilter with a given pattern. We show that every pattern with the closure property is realized by an ultrafilter. Finally, we apply patterns to obtain an equivalent condition for an ultrafilter to be self-divisible.
Positive region plays a fundamental role in rough set-based attribute reduction. We study positive regions of decision systems and of binary relations in rough set theory within the framework of reverse mathematics and computability theory. First, we propose the notion of infinite decision systems and prove that the existence of positive regions of decision systems is equivalent to arithmetic comprehension over the weak base theory RCA0. We also show that the complexity of positive regions of computable decision systems lies exactly in π02 of the arithmetic hierarchy. Next, we study positive regions of equivalence relations and binary relations. We show that the existence of each of the two positive regions is equivalent to arithmetic comprehension over RCA0; however, the exact complexity of positive regions of computable equivalence relations lies in π01 and the exact complexity of positive regions of computable binary relations lies in ∑02 of the arithmetic hierarchy.
The selection method is one of the methods to prove that various modal logics have the finite model property. For a given formula that is satisfiable in some model, we select a finite tree-like submodel, while preserving the satisfiability of the observed formula. In this paper, we adapt the selection method for the inquisitive modal logic InqML. We first define a tree-like model in the inquisitive setting and show that each satisfiable formula is satisfiable in a tree-like model. Then, using the notions of n-bisimulation and characteristic formulas, we show that InqML has the finite tree model property, i.e., each satisfiable formula is satisfiable in a finite tree-like model. Furthermore, we show analogous results for the inquisitive modal logic InqML⇒, and as a consequence we obtain the decidability of InqML⇒.
A divisibility relation on ultrafilters is defined as follows: F | G if and only if every set in F upward closed for divisibility also belongs to G. After describing the first w levels of this quasiorder, in this paper we generalize the process of determining the basic divisors of an ultrafilter. First we describe these basic divisors, obtained as (equivalence classes of) powers of prime ultrafilters. Using methods of nonstandard analysis we define the pattern of an ultrafilter: the collection of its basic divisors as well as the multiplicity of each of them. All such patterns have a certain closure property in an appropriate topology. We isolate the family of sets belonging to every ultrafilter with a given pattern. We show that every pattern with the closure property is realized by an ultrafilter. Finally, we apply patterns to obtain an equivalent condition for an ultrafilter to be self-divisible.
This paper studies effective aspects of semiperfect rings from the standpoint of reverse mathematics. Based on first-order Jacobson radicals of rings, we define a ring R with the Jacobson radical Jac(R) to be semiperfect if the quotient ring R/Jac(R) is semisimple, and idempotents of the quotient ring can be lifted to R. Using elementary matrix operations in linear algebra, we show that RCA0 proves a characterization of semiperfect rings in terms of idempotents of rings. Semiperfect rings are generalizations of semisimple rings and local rings, and semiperfect rings R with R/Jac(R) simple are isomorphic to matrix rings over local rings. Based on the effective characterization of semiperfect rings via idempotents, we prove the structure theorem of semiperfect rings R with R/Jac(R) simple in RCA0. Left perfect rings or right perfect rings are always semiperfect. Finally, we provide a proof for the structure theorem of one-sided perfect rings R with R/Jac(R) simple in WKL0.
