生成交换代数上模理论的模型完备化

有限生成交换幺半群有界对一作用的有限Borel渐近维数

Finite Borel asymptotic dimension of bounded-to-one commutative monoid actions

本文证明有限生成交换幺半群的有界对一作用具有有限渐近维数，并给出显式上界，回答了Shinko等人的问题并推广其定理。

Profiniteness and representability of spectra of Heyting algebras

We prove that there exist profinite Heyting algebras that are not isomorphic to the profinite completion of any Heyting algebra. This resolves an open problem from 2009. More generally, we characterize those varieties of Heyting algebras in which profinite algebras are isomorphic to profinite completions. It turns out that there exists largest such. We give different characterizations of this variety and show that it is finitely axiomatizable and locally finite. From this it follows that it is decidable whether in a finitely axiomatizable variety of Heyting algebras all profinite members are p…

From non-commutative diagrams to anti-elementary classes

Anti-elementarity is a strong way of ensuring that a class of structures , in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form L $\infty$$λ$. We prove that many naturally defined classes are anti-elementary, including the following: $\bullet$ the class of all lattices of finitely generated convex {\ell}-subgroups of members of any class of {\ell}-groups containing all Archimedean {\ell}-groups; $\bullet$ the class of all semilattices of finitely generated {\ell}-ideals of members of any nontrivial quasivariety of {\ell…

The Diophantine problem in finitely generated commutative rings

We study systems of polynomial equations in infinite finitely generated commutative associative rings with an identity element. For each such ring $R$ we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that the Diophantine problem (decidability of systems of polynomial equations) in $O$ is reducible to the same problem in $R$. If, in particular, $R$ has positive characteristic or, more generally, if $R$ has infinite rank, then we further obtain a…

The Hahn embedding theorem for a class of residuated semigroups

Hahn's embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn's theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial sublex product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess o…

交换幺半群上的预序空间

The space of preorders on a commutative monoid

研究有限生成交换幺半群上与幺半群结构兼容的全预序，用分治技术给出构造性描述，证明其通用理论可判定，还得到多项式环上自由模单项式序新描述。

Model theory and the DME: a survey

Recent work using the model theory of differentially closed fields to answer questions having to do with the Dixmier-Moeglin equivalence for (noncommutatve) finitely generated noetherian algebras, and for (commutative) finitely generated Poisson algebras, is here surveyed, with an emphasis on the model-theoretic and differential-algebraic-geometric antecedents.

Studying the Diophantine problem in finitely generated rings and algebras via bilinear maps

We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that the Diophantine problem (decidability of systems of polynomial equations) in $O$ is Karp-reducible to the same problem in $R$. In several cases we further obtain an interpretation by systems of equations of the ring $\mathbb{F}_p[t]$ in …

Hilbert spaces and ${C}^\ast$-algebras are not finitely concrete

We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an analogous result for the category of commutative unital $C^\ast$-algebras with $\ast$-homomorphisms. This implies, in particular, that this category is not axiomatizable by a first-order theory, a strengthening of a conjecture of Bankston.