Arithmetic of the units of number fields gives a lot of information about number fields. According to Dirichlet's unit theorem, the structure of units of the orders in number fields. In this talk, we are interested in exact expression of the units. Moreover, we will share the conjectures for units. We will talk about our results in this direction if time permits.

Universal Coalgebra is a general theory of state based systems, encompassing all sorts of automata, transition systems, Kripke structures, etc., be they deterministic, non-deterministic, probabilistic, second order, or else. This diversity of examples is covered by choosing as signature F of the mentioned coalgebras an appropriate Set-Functor, rather than just a sequence of arities, like in Universal Algebra. Preservation properties of that signature functor F determine much of the structure of the corresponding classes of coalgebras. Of particular interest, it turns out, is the question, whether F preserves preimages or (weak) pullbacks. This provides only the backdrop and the motivation for us to study a class of functors arising from classical Universal Algebra as "free-algebra-functors" where FV(X) is taken as the base set of the free algebra over X in a variety V. We show amongst other things, that FV(-) preserves preimages if and only if V can be defined by essentially balanced (also called regular) equations. Regarding preservation of kernel pairs, the picture is still fragmented. For instance, if V is a Malcev variety then FV weakly preserves kernel pairs, and if V is n-permutable and FV weakly preserves kernel pairs, then V is Malcev. In joint work with R. Freese we study the case when V is a variety L of lattices, with further order preserving operations permitted. It turns out that FL(-) always weakly preserves kernel pairs. In purely lattice theoretical terms, this shows that in any lattice variety L any balanced equation p(u1,...,um) = q(v1,...,vn), where {u1,...,um} = {v1,...,vn} must have an obvious reason, namely that both p and q are substitution instances of a common "ancestor"-term s, from which the said equation arises by further identification of variables, resulting in syntactically identical terms. For arbitrary nontrivial idempotent varieties V of algebras we additionally show, using Olshak's theorem, that (strong) preservation of pullbacks is impossible. Free algebra functors are special instances of copower functors which we can define in any concrete and cocomplete category. For instance, if M is a (commutative) monoid, the functor M[X] arising as X-fold sum of M, preserves preimages iff M is positive, and weakly preserves pullbacks iff M is (refinable) equidivisible.

Lucid reasons behind balanced identities

Dec. 12, 2023 3:30pm (MATH 3…

Topology

Justin Barhite

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Traces arise in many different places in math: traces of matrices, characters of group representations, and even the Euler characteristic of a CW complex! There are very general notions of trace, expressed in the language of category theory, that capture these examples of traces and whose properties imply familiar results like the Lefschetz fixed point theorem and the induction formula for characters. The formalism of traces doesn't tell the whole story though; there are some constructions that feel trace-like in certain ways but also have a distinct flavor, and what's really needed to explain them from this category-theoretic perspective is a dual notion of "cotrace." I will talk about some of these things that I have been working to understand by developing a theory of bicategorical cotraces.