Related papers: On sigma-entangled linear orders

We construct Boolean Algebras answering questions of Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with ``few automorphisms'',…

We introduce the idea of a weakly entangled linear order, and show that it is consistent for a Suslin line to be weakly entangled. We generalize the notion of entangled linear orders to $\omega_1$-trees, and prove that an $\omega_1$-tree is…

This paper grew as a continuation of [Sh462] but in the present form it can serve as a motivation for it as well. We deal with the same notions, and use just one simple lemma from there. Originally entangledness was introduced in order to…

We deal with some of problems posed by Monk and related to cardinal invariant of ultraproducts of Boolean algebras. We also introduce and investigate some new cardinal invariants.

We investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain "doubles" of linear groups, and obtain new examples of finitely generated…

We construct, in ZFC, a sequence of Boolean algebras for which the product of Lengths is strictly smaller than the Length of the product algebra.

The superamalgamation property is a strong form of the amalgamation property which applies to ordered structures; it has found many applications in algebraic logic. We show that superamalgamation has some interest also from the pure…

We prove the consistency with ZFC of ``the length of an ultraproduct of Boolean algebras is smaller than the ultraproduct of the lengths''. Similarly for some other cardinal invariants of Boolean algebras.

We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.

We prove that if $\leq$ is an analytic partial order then either $\leq$ can be extended to a (boldface) $\Delta^1_2$ linear order similar to an antichain in $2^{<\omega_1}$ ordered lexicographically or a certain Borel partial order $\leq_0$…

We prove that for any superatomic Boolean Algebra of cardinality >beth_omega there is an automorphism moving uncountably many atoms. Similarly for larger cardinals. Any of those results are essentially best possible.

We try to build, provably in ZFC, for a first order T a model in which any isomorphism between two Boolean algebras is definable. The problem, compared to [Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do not…

We study operator algebras associated to integral domains. In particular, with respect to a set of natural identities we look at the possible nonselfadjoint operator algebras which encode the ring structure of an integral domain. We show…

We construct a model of the Open Graph Axiom (OGA) in which there is a 2-entangled Suslin line $S$. Consequently, in this model, there is a 2-entangled uncountable linear order, but no such order is separable. This resolves a problem posed…

We continue our studies on semilattice ordered algebras. This time we accept constants in the type of algebras. We investigate identities satisfied by such algebras and describe the free objects in varieties of semilattice ordered algebras…

The class of closed graphs by a linear ordering on their sets of vertices is investigated. A recent characterization of such a class of graphs is analyzed by using tools from the proper interval graph theory.

We deal with several pcf problems; we characterize another version of exponentiation: number of kappa-branches in a tree with lambda nodes, deal with existence of independent sets in stable theories, possible cardinality of ultraproduct,…

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $\kappa,\lambda$ are infinite cardinals such that $\kappa^{+++} \leq \lambda$, $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}= \kappa^+$, and…

We prove a Theorem about the relationship between the Depth of the ultraproduct of Boolean algebras, divided by an ultrafilter, and the products of the depths of each component. This answers (partly) an open problem of Monk.

Entangled linear orders were first introduced by Abraham and Shelah. Todor\v{c}evi\'c showed that these linear orders exist under $\mathsf{CH}$. We prove the following results: (1) If $\mathsf{CH}$ holds, then, for every $n > 0$, there is…