Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Generalized Domination.

This thesis develops the theory of the everywhere domination relation between functions from one infinite cardinal to another. When the domain of the functions is the cardinal of the continuum and the range is the set of natural numbers, we may restrict our attention to nicely definable functions from R to N. When we consider a class of such functions which contains all Baire class one functions, it becomes possible to encode information into these functions which can be decoded from any dominator. Specifically, we show that there is a generalized Galois-Tukey connection from the appropriate domination relation to a classical ordering studied in recursion theory. The proof techniques are developed to prove new implications regarding the distributivity of complete Boolean algebras. Next, we investigate a more technical relation relevant to the study of Borel equivalence relations on R with countable equivalence classes. We show than an analogous generalized Galois-Tukey connection exists between this relation and another ordering studied in recursion theory.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113539/1/danhath_1.pd

Igre na Bulovim algebrama

The method of forcing is widely used in set theory to obtain various consistency proofs. Complete Boolean algebras play the main role in applications of forcing. Therefore it is useful to define games on Boolean algebras that characterize their properties important for the method. The most investigated game is Jech’s distributivity game, such that the first player has the winning strategy iff the algebra is not (ω, 2)-distributive. We define another game characterizing the collapsing of the continuum to ω, prove several sufficient conditions for the second player to have a winning strategy, and obtain a Boolean algebra on which the game is undetermined. Forsing je metod široko korišćen u teoriji skupova za dokaze konsistentnosti. Kompletne  Bulove algebre igraju glavnu ulogu u primenama forsinga. Stoga je korisno definisati igre na Bulovim algebrama koje karakterišu njihove osobine od značaja za taj metod. Najbolje proučena je Jehova igra, koja ima osobinu da prvi igrač ima pobedničku strategiju akko algebra nije (ω, 2)-distributivna. U tezi definišemo još jednu igru, koja karakteriše kolaps kontinuuma na ω, dokazujemo nekoliko dovoljnih uslova da bi drugi igraš imao pobedničku strategiju, i konstruišemo Bulovu algebru na kojoj je igra neodređena

Clones over Finite Sets and Minor Conditions

Achieving a classification of all clones of operations over a finite set is one of the goals at the heart of universal algebra. In 1921 Post provided a full description of the lattice of all clones over a two-element set. However, over the following years, it has been shown that a similar classification seems arduously reachable even if we only focus on clones over three-element sets: in 1959 Janov and Mučnik proved that there exists a continuum of clones over a k-element set for every k > 2. Subsequent research in universal algebra therefore focused on understanding particular aspects of clone lattices over finite domains. Remarkable results in this direction are the description of maximal and minimal clones. One might still hope to classify all operation clones on finite domains up to some equivalence relation so that equivalent clones share many of the properties that are of interest in universal algebra. In a recent turn of events, a weakening of the notion of clone homomorphism was introduced: a minor-preserving map from a clone C to D is a map which preserves arities and composition with projections. The minor-equivalence relation on clones over finite sets gained importance both in universal algebra and in computer science: minor-equivalent clones satisfy the same set identities of the form f(x_1,...,x_n) = g(y_1,...,y_m), also known as minor-identities. Moreover, it was proved that the complexity of the CSP of a finite structure A only depends on the set of minor-identities satisfied by the polymorphism clone of A. Throughout this dissertation we focus on the poset that arises by considering clones over a three-element set with the following order: we write C ≤_{m} D if there exist a minor-preserving map from C to D. It has been proved that ≤_{m} is a preorder; we call the poset arising from ≤_{m} the pp-constructability poset. We initiate a systematic study of the pp-constructability poset. To this end, we distinguish two cases that are qualitatively distinct: when considering clones over a finite set A, one can either set a boundary on the cardinality of A, or not. We denote by P_n the pp-constructability poset restricted to clones over a set A such that |A|=n and by P_{fin} we denote the whole pp-constructability poset, i.e., we only require A to be finite. First, we prove that P_{fin} is a semilattice and that it has no atoms. Moreover, we provide a complete description of P_2 and describe a significant part of P_3: we prove that P_3 has exactly three submaximal elements and present a full description of the ideal generated by one of these submaximal elements. As a byproduct, we prove that there are only countably many clones of self-dual operations over {0,1,2} up to minor-equivalence

An Algebraic Approach to Mso-Definability on Countable Linear Orderings

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues