We consider the internal rate of return (IRR) decision rule in capital budgeting problems with fuzzy cash flows. The possibility distribution of the IRR at any r ≥ 0, is defined to be the degree of possibility that the (fuzzy) net present value of the project with discount factor r equals to zero. Generalizing our earlier results on fuzzy capital budegeting problems [Car99] we show that the possibility distribution of the {IRR} is a highly nonlinear function which is getting more and more unbalanced...

The aim of this paper is to define and study cardinal (direct) and ordinal operations of addition, multiplication, and exponentiation for $n$-ary relational systems. $n$-ary ordered sets are defined as special $n$-ary relational systems by means of properties that seem to suitably generalize reflexivity, antisymmetry, and transitivity from the case of $n=2$ or 3. The class of $n$-ary ordered sets is then closed under the cardinal and ordinal operations.

We calculate the cardinal characteristics of the $\sigma$-ideal $\mathscr{H}𝒩\left(G\right)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $cov\left(\mathscr{H}𝒩\right(G\left)\right)\le 𝔟\le max\{𝔡,non(𝒩\left)\right\}\le non\left(\mathscr{H}𝒩\right(G\left)\right)\le cof\left(\mathscr{H}𝒩\right(G\left)\right)>min\{𝔡,non(𝒩\left)\right\}$. If $G={\prod }_{n\ge 0}{G}_n$ is the product of abelian locally compact groups $G_n$, then $add\left(\mathscr{H}𝒩\right(G\left)\right)=add\left(𝒩\right)$, $cov\left(\mathscr{H}𝒩\right(G\left)\right)=min\{𝔟,cov(𝒩\left)\right\}$, $non\left(\mathscr{H}𝒩\right(G\left)\right)=max\{𝔡,non(𝒩\left)\right\}$ and $cof\left(\mathscr{H}𝒩\right(G\left)\right)\ge cof\left(𝒩\right)$, where $𝒩$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $cof\left(\mathscr{H}𝒩\left(G\right)\right)>2^{{\aleph }_0}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of...

We study cardinal coefficients related to combinatorial properties of partitions of $\omega$ with respect to the order of almost containedness.

We deal with some problems posed by Monk [Mo 1], [Mo 3] and related to cardinal invariants of ultraproducts of Boolean algebras. We also introduce and investigate several new cardinal invariants.

We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most ${\left(2^{\aleph ₀}\right)}^V$ levels of size ω. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of regular and of 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda }\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\right(\beta ):\beta <\alpha ]$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda ₀i>\lambda ₁>...>{\lambda }_{n-1}$ and ordinals $\alpha ₀,...,{\alpha }_{n-1}$ such that $\alpha =\alpha ₀+\cdots +{\alpha }_{n-1}$ and $f=f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i{\in }_{{\lambda }_i}\left({\alpha }_i\right)$. Under GCH we prove that if α < ω₂ then (i) ${}_{\omega }\left(\alpha \right)=s{\in }^{\alpha }\omega ,\omega ₁:s\left(0\right)=\omega$; (ii) if λ > cf(λ) = ω, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega ₁-closedin\alpha$; (iii) if cf(λ) = ω₁, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha$; (iv) if cf(λ) > ω₁, ${}_{\lambda }\left(\alpha \right){=}^{\alpha }\lambda$. This yields a complete characterization of the classes (α) for all α < ω₂,...

The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine homsets, monomorphisms, epimorpshisms and isomorphisms [7] within a free-object category [1] and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property and the Pullback Lemma is formalized [15]. In the last part of the article we formalize the pullback of functors [14] and it...

We continue the work of [2] and prove that for λ successor, a λ-categorical theory T in $L_{\kappa *,\omega }$ is μ-categorical for every μ ≤ λ which is above the $\left(2^{LS\left(T\right)}\right)⁺$-beth cardinal.