We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of ${\aleph }_k$-free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is ${\aleph }_k$-free if every subset of size $<{\aleph }_k$ is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a module G is...

By an $\in$-representation of a relation we mean its isomorphic embedding to $𝔼=\{\langle x,y\rangle ;x\in y\}$. Some theorems on such a representation are presented. Especially, we prove a version of the well-known theorem on isomorphic representation of extensional and well-founded relations in $𝔼$, which holds in Zermelo-Fraenkel set theory. This our version is in Zermelo-Fraenkel set theory false. A general theorem on a set-prolongation is proved; it enables us to solve the task of the representation in question.

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum, while ${\ell }_{\infty }$...

Some relatives of the Juhász Club Principle are introduced and studied in the presence of CH. In particular, it is shown that a slight strengthening of this principle implies the existence of a Suslin tree in the presence of CH.

We show that under ZFC, for every indecomposable ordinal α < ω₁, there exists a poset which is β-proper for every β < α but not α-proper. It is also shown that a poset is forcing equivalent to a poset satisfying Axiom A if and only if it is α-proper for every α < ω₁.

Assuming V = L, for every successor cardinal κ we construct a GCH and cardinal preserving forcing poset ℙ ∈ L such that in $L^{\mathbb{P}}$ the ideal of all non-stationary subsets of κ is Δ₁-definable over H(κ⁺).

The aim of this paper is to study a special type of fuzzy relations, the α-equivalences, as well as to consider the relation that connects these with the family of ε-partitions of the referential. Some classic equivalences between set, partitions and fuzzy relations are also studied.

The concept of a $$-closed subset was introduced in [1] for an algebraic structure $=(A,F,R)$ of type $$ and a set $$ of open formulas of the first order language $L\left(\right)$. The set $C_{(})$ of all $$-closed subsets of $$ forms a complete lattice whose properties were investigated in [1] and [2]. An algebraic structure $$ is called $$- hamiltonian, if every non-empty $$-closed subset of $$ is a class (block) of some congruence on $$; $$ is called $$- regular, if $=𝔽$ for every two $$, $𝔽\in$ whenever they have a congruence class $B\in C_{(})$ in common....

We investigate σ-entangled linear orders and narrowness of Boolean algebras. We show existence of σ-entangled linear orders in many cardinals, and we build Boolean algebras with neither large chains nor large pies. We study the behavior of these notions in ultraproducts.

In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets [18],...

We define ${\omega }^{\omega }$-directedness, investigate various properties to determine whether they have this property or not, and use our results to obtain easier proofs of theorems due to Laurence and Alster concerning the existence of a Michael space, i.eȧ Lindelöf space whose product with the irrationals is not Lindelöf.