Research Statement

Asher Kach

My research in effective algebra and computable structure theory lies at the interface of logic and algebra, centering on the effective content of algebraic objects and algebraic processes. The major motivating question is to determine how simple or how difficult it is to describe an algebraic object (e.g., an algebraic structure, an isomorphism between algebraic structures, an embedding between algebraic structures, etc.) or execute an algebraic process (e.g., factoring a polynomial into irreducible factors, finding the algebraic closure of a field, etc.). For many classes of algebraic structures, I am also interested in classical questions that have a logical aspect to them. In this statement, I discuss work and interests of mine pertaining to Euclidean domains and Boolean algebras. Though these results are not necessarily my strongest results, this work does best illustrate some of my current research interests. An annotated bibliography of all my work can be found online at http://www.math.uconn.edu/~kach/mathematics/mymathematics.html.

1 Background

Based on seminal work by Church, Kleene, and Turing, the notion of an algorithmic process was formalized in the mid-twentieth century, with the Turing Machine (essentially an idealized com- puter with infinite memory and unbounded time for computation) as a dominant computational paradigm. Fundamental to this formalization and the development of the fields of effective algebra and computable structure theory were the notions of computable relations, computable functions, and computable structures.

Definition. A function f : ω → ω (relation R ⊆ ω) is computable if there is a Turing Machine computing (accepting) it. Definition. A countably infinite structure (e.g., a linear order, a group, a field) is computable if its universe can be identified with the natural numbers ω = {0, 1, 2,... } in such a way that the functions and relations become computable operations on ω. Less formally, a function or relation is computable if it is intuitively computable, i.e., if it can be computed by a natural algorithm. Similarly, a structure is computable if it is intuitively computable.

Example. The group of integers Z under addition is a computable structure. By identifying the nonnegative integer n with the natural number 2n and the negative integer −n with the natural number 2n + 1, the integers Z are nicely identified with the natural numbers ω. Moreover, given natural numbers a and b, it is possible to (effectively) recover the integers za and zb they code, compute the integer za + zb, and determine the natural number coding the integer za + zb. A major aim of effective algebra and computable structure theory is to understand which alge- braic objects and algebraic processes are computable and, when they are not computable, how much extra computational power is necessary to describe or implement them.

2 Euclidean Domains

We recall a Euclidean domain is a commutative integral domain R (a commutative ring with no zero divisors) that can be endowed with a Euclidean ranking function φ, a function from the nonzero elements of R to the non-negative integers satisfying

(∀a 6= 0)(∀b 6= 0)(∃q)[a + bq = 0 or φ(a + bq) < φ(a)] . (†)

1 This says either b divides a or the remainder a+bq can be chosen to be “smaller” than the dividend a. From this definition, it would seem difficult to determine whether an arbitrary commutative integral domain is (is not) a Euclidean domain as one would need to check all possible ranking functions φ. An alternate characterization simplifies this task.

Definition (Samuel [7]). If R is a Euclidean domain, define sets Rn for n ∈ ω by recursion, with R0 := {d :(∃u)[ud = 1]} and Rn+1 := {d 6= 0 : (∀a)(∃q)[a + dq = 0 or a + dq ∈ Rn]}. Theorem (Samuel [7]). A commutative integral domain is a Euclidean domain if and only if R = ∪n∈ωRn. In this case, the function φR given by φR(x) = n where n is least so that x ∈ Rn is a ranking function for R and is pointwise least amongst all ranking functions for R. There is no reason to require the range of φ to be the non-negative integers. Provided the range is well-ordered, typical algorithmic processes (such as the division algorithm) would still always terminate. This observation motivates the following generalization. Definition (Samuel [7]). A commutative integral domain R is a transfinite Euclidean domain if there is a transfinite ranking function φ from the nonzero elements of R to the ordinals satisfying (†). Together with Rodney Downey, I showed a number of results pertaining to computable Euclidean domains. Though the question of whether there is a properly transfinite Euclidean domain remains open (i.e., whether there is a transfinite Euclidean domain that is not a Euclidean domain), we showed a positive answer in the effective setting. Theorem (Downey and Kach [3]). There is a computable Euclidean domain having no computable Euclidean function but having computable transfinite Euclidean functions. This shows that in the effective setting, extra power is gained from the ability to use larger well-ordered sets. On the other hand, this extra power is not always sufficient to yield a computable ranking function. Theorem (Downey and Kach [3]). There is a computable Euclidean domain having no computable transfinite Euclidean function. Indeed, there are computable Euclidean domains for which any trans- finite Euclidean function computes ∅0 (the Halting Problem). This is a strengthening of a result of Schrieber who showed there is a computable Euclidean 0 0 domain R in which φR has complexity ∅ (see [8]). It is natural to ask whether ∅ always computes a ranking function for a computable Euclidean domain. We showed that it does not. Theorem (Downey and Kach [3]). There is a computable Euclidean domain having no ∅0-computable Euclidean function. Indeed, there is a computable Euclidean domain R for which φR has complex- ity ∅00. Together with Paul Ellis and Reed Solomon, I am seeking to strengthen these results. Indeed, we are working on the following questions. Question. Is there, for each n ∈ ω, a computable Euclidean domain having no ∅(n)-computable ranking function φ? Question (Samuel [7]). Is there a properly transfinite Euclidean domain?

3 Boolean Algebras

We recall a Boolean algebra is a complemented distributed lattice. In comparison to most other classes of algebraic structures, the class of Boolean algebras is not very well understood in the effective setting for various reasons. For example, the following is a well-known open question asked by Downey and Jockusch (see [2]). We recall a structure is lown if it is A-computable for some lown set A, i.e., a set A satisfying A(n) = ∅(n).

2 Conjecture (The lown Conjecture). Is every lown Boolean algebra computable? Making use of Ketonen invariants (isomorphism invariants for countable Boolean algebras), I have shown a positive answer to this question for a natural class of Boolean algebras by isolating a necessary and sufficient condition for a depth zero and rank ω Boolean algebra to be computable. Roughly speaking, depth is a measure of the homogeneity of the Boolean algebra and rank is a measure of the size of the Boolean algebra. Theorem (Kach [4] and [5]). There is a natural two-to-one map between depth zero and rank ω Boolean algebras and nonempty sets S ⊆ ω.

0 ∅(2n+2) Definition. A set S ⊆ ω is Σ(2n+3) in the Feiner Hierarchy if there is an index e so that Φe (n)↓ if and only if n ∈ S. Theorem (Kach [4] and [5]). A depth zero and rank ω Boolean algebra is computable if and only if 0 the set it encodes is Σ(2n+3) in the Feiner Hierarchy.

Corollary (Kach [4] and [5]). If a depth zero and rank ω Boolean algebra is lown for some n ∈ ω, then it is computable. My intent is to continue this line of inquiry, emphasizing the isomorphism invariants by Ketonen in this line of study. Though a major goal is to settle Question 3, the general aim is to understand which Boolean algebras are computable. A particular step towards this aim that I seek to settle is the following. Question. Which depth ω and rank one Boolean algebras are computable? My interest in the effective properties of Boolean algebras generated a significant personal interest in the classical properties of countable Boolean algebras. For example, I am currently working with Steffen Lempp on resolving a question of Camerlo and Gao asking whether Vaught’s Conjecture holds for Boolean algebras in the infinitary language Lω1ω (see [1]). We conjecture the following.

Conjecture. If ϕ is a sentence in the infinitary language Lω1ω in the language of Boolean algebras extending the theory of Boolean algebras, then ϕ has continuum many models if it has uncountably many models. Three important classes of such infinitary sentences ϕ (related to the Ketonen invariants of the models of ϕ) have been isolated. For two of these classes, the conjecture has already been verified; for the remaining class, considerable progress has already been made. As the classical complexity of a Boolean algebra arises from its depth and the depth of its subalgebras, I am also very interested in better understanding the relationship between the structure of an algebra and the structure of its subalgebras. Together with Steffen Lempp, I am investigating to what extent the set of depths of subalgebras must be downward closed (see [6]). The following two theorems illustrate that the set of depths need not be downward closed, yet cannot be arbitrary either. Theorem (Kach and Lempp [6]). For each ordinal α, there is Boolean algebra of depth ωα for which every subalgebra has depth zero, depth one, or depth ωα. Theorem (Kach and Lempp [6]). Every Boolean algebra has a depth zero subalgebra. Every Boolean algebra of depth at least one has either a depth one subalgebra or a depth two subalgebra. The aim is to eventually characterize what sets can be realized as the set of depths of subalgebras. Question. What sets of countable ordinals can be realized as the set of depths of subalgebras of a countable Boolean algebra?

3 References

[1] Riccardo Camerlo and Su Gao. The completeness of the isomorphism relation for countable Boolean algebras. Trans. Amer. Math. Soc., 353(2):491–518 (electronic), 2001. [2] Rod Downey and Carl G. Jockusch. Every low Boolean algebra is isomorphic to a recursive one. Proc. Amer. Math. Soc., 122(3):871–880, 1994.

[3] Rodney G. Downey and Asher M. Kach. Euclidean functions of computable Euclidean domains. submitted. [4] Asher M. Kach. Characterizing the computable structures: Boolean algebras and linear orders. ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–The University of Wisconsin - Madison.

[5] Asher M. Kach. Depth zero Boolean algebras. Trans. Amer. Math. Soc., 362(8):4243–4265, 2010. [6] Asher M. Kach and Steffen Lempp. Downward closure of depth in countable boolean algebras. in preparation. [7] Pierre Samuel. About Euclidean rings. J. Algebra, 19:282–301, 1971.

[8] Leonard Schrieber. Recursive properties of Euclidean domains. Ann. Pure Appl. Logic, 29(1):59– 77, 1985.

4