This work presents a model-theoretic approach to the study of first-order theories of classes of BL-chains. Among other facts, we present several classes of BL-algebras, generating the whole variety of BL-algebras, whose first-order theory has quantifier elimination. Model-completeness and decision problems are also investigated. Then we investigate classes of BL-algebras having (or not having) the amalgamation property or the joint embedding property and we relate the above properties to the existence of ultrahomogeneous models.

We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to ω, answering a question by López-Abad. We show that k-colored Milliken-Taylor ultrafilters have at least k+1 near coherence classes of ultrafilters in its projection to ω. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model.

Shelah has shown that $\aleph_1$-categoricity for Abstract Elementary Classes (AECs) is not absolute in the following sense: There is an example $K$ of an AEC (which is actually axiomatizable in the logic $L(Q)$) such that if $2^{\aleph_0}<2^{\aleph_1}$ (the weak CH holds) then $K$ has the maximum possible number of models of size $\aleph_1$, whereas if Martin's Axiom at $\aleph_1$ (denoted by MA) holds then $K$ is $\aleph_1$-categorical. In this note we extract the properties from Shelah's example which make both parts work resulting in our definitions of condition A and condition B, and then we show that for any AEC satisfying these two conditions, neither of these implications can be reversed.

Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. Previously, the author proved that this theory is supersimple. In supersimple theories there is a notion of rank defined in analogy with Lascar U-rank for superstable theories. It is also possible to define a notion of dimension for types in DCFA based on transcendence degree of realization of the types. In this paper we compute the rank of a model of DCFA, give some properties regarding rank and dimension, and give an example of a definable set with finite rank but infinite dimension. Finally we prove that for the case of definable subgroup of the additive group being finite-dimensional and having finite rank are equivalent.

A number of results have been obtained concerning Borel structures starting with Silver and Friedman followed by Harrington, Shelah, Marker, and Louveau. Friedman also initiated the model theory of Borel (in fact totally Borel) structures. By this we mean the study of the class of Borel models of a given first-order theory. The subject was further investigated by Steinhorn. The present work is meant to go further in this direction. It is based on the assumption that the study of the class of, say, countable models of a theory reduces to analyzing a single $\omega_1$-saturated model. The question then arises as to when such a model can be totally Borel. We present here a partial answer to this problem when the theory under investigation is superstable.

We employ the notions of "sequential function" and "interrogation" (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using Longley's preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is that every realizability topos is a geometric quotient of a realizability topos on a total combinatory algebra.