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Thunderbird
The Algebra and Logic seminar is a research and learning seminar organized by the algebra and logic research group of the Department of Mathematics at the University of Colorado at Boulder. The scope of the seminar includes all topics with links to algebra, logic, or their applications, like general algebra, logic, model theory, category theory, set theory, set-theoretic topology, or theoretical computer science. If you have questions regarding this seminar, please contact Keith Kearnes or Peter Mayr.
In order to study a certain 3-dimensional analogue of the Mandelbrot set, I needed a notion of holomorphy for the corresponding 3-dimensional real algebra. While this motivation (to generalize the basic elements of complex dynamics) is analytic, the focus of this talk will be the connection between holomorphic functions over a real algebra and first-order properties of the algebra. It turns out that, under the assumption of a single (left or right) divisible element, the holomorphic functions over an arbitrary real algebra form a vector space. Properties such as the Product Rule and Chain Rule are equivalent to purely algebraic properties of the underlying real algebra. When our real algebra has a multiplicative identity element, the Product Rule is precisely equivalent to being both commutative and associative. I will present proofs of some of these statements, which are quite succinct once the proper groundwork has been done.
An (abstract) minion M is a functor from the category of finite sets to the category of sets. So, for every finite set X, there is a set M(X) (the 'X-ary' part of M), and for every function f: X -> Y there is a function M(f) : M(X) -> M(Y) such that M preserves the identity and composition. Every variety V naturally gives rise to a minion F_V, where F_V(X) is the free algebra generated by X and F_V(f) : F_V(X) -> F_V(Y) is the unique homomorphism extending f: X -> Y. In this talk I would like to discuss a nice (multilinear) description of this minion for the variety of n-step nilpotent groups. This description is based on concepts from combinatorics on words (subword counting) and a result by Fox, Lyndon and Chen from 1958, and motivated by the study of uniform algorithms for constraint satisfaction problems (CSPs).
Distalty is a model-theoretic tameness-notion, which generalizes o-minimality and strengthens NIP, containing important unstable theories such as DLO, real closed fields and Presburger arithmetic. Its consequences include the strong Erdős–Hajnal property. One interesting problem is that of preserving tameness when adding new predicates or functions. I will discuss some results around this problem from existing literature and some from my master’s thesis and from ongoing joint work with Alessio Mansutti and Michael Benedikt, all concerning distal expansions of Presburger arithmetic and RCF by functions and by additional relations on distinguished subsets. These include distality of Semenov arithmetic—the natural numbers with addition, order and the function n ↦→ 2n —and the corresponding expansions by the Fibonacci sequence, the Factorial function and other fast-growing functions.
Datalog fragments have been used a lot to study constraint satisfaction problems (CSPs) that are solvable in polynomial time. Well known fragments are linear Datalog, symmetric linear Datalog, and arc monadic Datalog, better known as arc consistency. We introduce n-almost symmetric Datalog and study n-almost symmetric linear arc monadic Datalog. We characterize the finite relational structures whose CSP is solved by this Datalog fragment as those that can be primitive positively constructed from the transitive tournament on n + 2 vertices. We also give characterizations in terms of a certain homomorphism duality (which we call n-fixed unfolded caterpillar duality) and in universal-algebraic terms (the existence of k-absorptive operations and of operations forming an elevator chain of length n + 1).
Almost symmetric linear arc monadic datalog and transitive tournaments
Automorphism groups of structures in classical and continuous logic provide a rich source of examples of Polish groups and model-theoretic methods are central in their study. The setting in which there is a perfect correspondence (equivalence of categories) between model theory and groups is that of aleph_0-categorical structures and Roelcke precompact Polish groups. I will survey what is known about this and then discuss new work which extends the correspondence to a broader one between locally Roelcke precompact groups (studied by Rosendal) and locally aleph_0-categorical structures, which we define. This is joint work with Itai Ben Yaacov.
R. A. Dean asked whether a free lattice can contain an infinite strictly ascending chain of sublattices, each isomorphic to the free lattice FL(3) on three generators. Ralph Freese recorded the question in his appendix on free lattices in Grätzer's General Lattice Theory. In joint work with Freese, we show that the answer is no. The proof turns such a chain into a sequence of self-embeddings of FL(3). Using the fact that a lattice embedding never lowers an element's complexity, we show that no three self-embeddings of FL(3) can be stacked coherently. Freeness is essential: an ascending chain of 3-generated sublattices of a free lattice need not stabilize. We found the proof, and formally verified it in Agda, with help from AI agents working under my direction. I will describe that process, what agents did, what we did, and what the formal verification certifies.
Dean's problem on ascending chains in free lattices
Holomorphy over real algebras