Convergent and fundamental sequences are studied in a half linearly cyclically ordered group G with the abelian increasing part. The main result is the construction of the Cantor extension of G.

This paper is a continuation of a previous author’s article; the result is now extended to the case when the lattice under consideration need not have the least element.

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 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,...