Mathematics

We study the first-order axiomatisability of finite semiring interpretations or, equivalently, the question whether elementary equivalence and isomorphism coincide for valuations of atomic facts over a finite universe into a commutative semiring. Contrary to the classical case of Boolean semantics, where every finite structure can obviously be axiomatised up to isomorphism by a first-order sentence, the situation in semiring semantics is rather different, and strongly depends on the underlying semiring. We prove that for a number of important semirings, including min-max semirings, and the semirings of positive Boolean expressions, there exist finite semiring interpretations that are elementarily equivalent but not isomorphic. The same is true for the polynomial semirings that are universal for the classes of absorptive, idempotent, and fully idempotent semirings, respectively. On the other side, we prove that for other, practically relevant, semirings such as the Viterby semiring, the tropical semiring, the natural semiring and the universal polynomial semiring N[X], all finite semiring interpretations are first-order axiomatisable (and thus elementary equivalence implies isomorphism), although some of the axiomatisations that we exhibit use an infinite set of axioms.

We introduce the notion of (Γ,E) -determinacy for Γ a pointclass and E an equivalence relation on a Polish space X . A case of particular interest is the case when E= E G is the (left) shift-action of G on S G where S=2={0,1} or S=ω . We show that for all shift actions by countable groups G , and any "reasonable" pointclass Γ , that (Γ, E G ) -determinacy implies Γ -determinacy. We also prove a corresponding result when E is a subshift of finite type of the shift map on 2 Z .

Based on the notion of thin sets introduced recently by T.~Banakh, Sz.~Głąb, E.~Jabłońska and J.~Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of admissible messages (i.e. subsets of the Cantor cube Z ω 2 ). Using Banach-Mazur games we prove that all protocols detecting errors are Baire spaces and generic (in particular maximal) ones are not neither Borel nor meager. We also show that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit. This result is related to so-called xor-sets defined by D.~Niwiński and E.~Kopczyński in 2014.

This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom. The third and final part of the paper examines the consistency strength of choiceless large cardinals, including a proof that assuming DC, the existence of an elementary embedding from V λ+3 to V λ+3 implies the consistency of ZFC + I 0 . By a recent result of Schlutzenberg, an elementary embedding from V λ+2 to V λ+2 does not suffice.

We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.

Assume ZF (without the Axiom of Choice). Let j: V ε → V δ be a non-trivial ∈ -cofinal Σ 1 -elementary embedding, where ε,δ are limit ordinals. We prove some restrictions on the constructibility of j from V δ , mostly focusing on the case ε=δ . In particular, if ε=δ and j∈L( V δ ) then δ has cofinality ω . However, assuming ZFC+I 3 , with the appropriate ε=δ , one can force to get such j∈L( V V[G] δ ) . Assuming Dependent Choice and that δ has cofinality ω (but not assuming V=L( V δ ) ), and j: V δ → V δ is Σ 1 -elementary, we show that there are "perfectly many" such j , with none being "isolated". Assuming a proper class of weak Lowenheim-Skolem cardinals, we also give a first-order characterization of critical points of embeddings j:V→M with M transitive. The main results rely on a development of extenders under ZF (which is most useful given such wLS cardinals).

Hrushovski's suggestion, given in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics , 2012], to capture the structure of the 1-analysable covers of a theory T using simplicial groupoids definable in T is realized here. The ideas of Haykazyan and Moosa, found in ["Functoriality and uniformity in Hrushovski's groupoid-cover correspondence," Annals of Pure and Applied Logic , 2018] are used, and extended, to define an equivalence of categories. Finally, a couple of examples are studied with these new tools.

We introduce extensions by rules of the extensional level of the Minimalist Foundation which turn out to be equivalent to constructive and classical axiomatic set theories.

We state conditions for which a definable local homomorphism between two locally definable groups G , G ??can be uniquely extended when G is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Thm. 9.1] (see Corollary 2.2). We also prove that Theorem 10.2 in [3] also holds for any definably connected definably compact semialgebraic group G not necessarily abelian over a sufficiently saturated real closed field R ; namely, that the o-minimal universal covering group G ? of G is an open locally definable subgroup of H (R) 0 ? for some R -algebraic group H (Thm. 3.3). Finally, for an abelian definably connected semialgebraic group G over R , we describe G ? as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative R -algebraic groups (Theorem 3.4)

The optimality of the Erdős-Rado theorem for pairs is witnessed by the colouring Δ κ :[ 2 κ ] 2 →κ recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which Δ κ is an \emph{extremal} such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of Δ -regressive and almost Δ -regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether Δ κ has the minimal cardinality of any \emph{maximal} triangle-free or odd-cycle-free colouring into κ . We resolve the question positively for odd-cycle-free colourings.