用单一域公理刻画有限生成域

Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods.

The elementary closure of the class Nr_nCA_m for m\geq n+1 is not finitely axiomatizable, futhermore for any finite k\geq 1, there is A\in Nr_ωCA_{\omeg+k}that is not SNr_ωCA_{ω+k+1}

We show that for 1<n<m, the class Nr_nCA_m known to be non-elementary is pseudo elementary. When n and m are finite we use a two sorted theory, when n is finite and m infinite we use a three sorted one, and finally when both are infinite we use a four sorted defining theory. Our non finite axiomatizability result, follows from the fact that for 2<n<m, and any r\in ωthere exists a finite (Monk like) algebra C(m,n,r), such that C(m,n,r)\in Nr_nCA_m C(m,n,r)\notin SNr_nCA_{m+1}, and any non trivial ultraproduct on r of such algebras in in ElNr_nCA_m. Finally we use such algebras, to show that for…

对周期小的l-预群变种进行公理化

Axiomatizing small varieties of periodic l-pregroups

本文研究了周期小的l-预群变种的公理化，揭示了有限连接的格结构，并characterize了有限子直接不可约代数的性质。

Characteristic formulas over intermediate logics

We expand the notion of characteristic formula to infinite finitely presentable subdirectly irreducible algebras. We prove that there is a continuum of varieties of Heyting algebras containing infinite finitely presentable subdirectly irreducible algebras. Moreover, we prove that there is a continuum of intermediate logics that can be axiomatized by characteristic formulas of infinite algebras while they are not axiomatizable by standard Jankov formulas. We give the examples of intermediate logics that are not axiomatizable by characteristic formulas of infinite algebras. Also, using the Goede…

Locally Finite Quasivarieties of MV-algebras

In this paper we show that every locally finite quasivariety of MV-algebras is finitely generated and finitely based. To see this result we study critical MV-algebras. We also give axiomatizations of some of these quasivarieties.

合成代数几何的基础

A Foundation for Synthetic Algebraic Geometry

本文基于Kock和Blechschmidt的工作，在Zariski topos内部建立代数几何基础，使用同伦类型论结合三个公理作为其内部语言，提出“Zariski局部选择”公理并通过立方模型证明其合理性，贡献了高阶类型在上同调推理中的应用。

Finite axiomatizability for profinite groups

A group is $\textit{finitely axiomatizable}$ (FA) in a class $\mathcal{C}$ if it can be determined up to isomorphism within $\mathcal{C}$ by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups. Reasons why certain groups cannot be FA are also discussed.

Defining R and G(R)

We show that for Chevalley groups G(R) of rank at least 2 over a ring R the root subgroups are essentially (nearly always) the double centralizers of corresponding root elements. In very many cases this implies that R and G(R) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields. For such groups it then follows that the group G(R) is finitely axiomatizable in the appropriate class of groups provided R is finitely axiomatizable in the corresponding class of rings.

关于自由伪补分配格的全称理论

On the universal theory of the free pseudocomplemented distributive lattice

本文证明了自由伪补分配格的全称理论是可判定的，给出了递归公理化，并描述了可嵌入自由代数的有限生成伪补分配格。

带加法与左乘 $ω$ 的序数的等式理论

Equational theory of ordinals with addition and left multiplication by $ω$

本文证明了序数结构 $\langle ω^ω, +, ω\cdot \rangle$ 的等式理论有限可公理化，并提供了线性时间判定算法。

Axioms and Decidability for Type Isomorphism in the Presence of Sums

We consider the problem of characterizing isomorphisms of types, or, equivalently, constructive cardinality of sets, in the simultaneous presence of disjoint unions, Cartesian products, and exponentials. Mostly relying on results about polynomials with exponentiation that have not been used in our context, we derive: that the usual finite axiomatization known as High-School Identities (HSI) is complete for a significant subclass of types; that it is decidable for that subclass when two types are isomorphic; that, for the whole of the set of types, a recursive extension of the axioms of HSI exi…

非初等模态逻辑，假设 P ≠ NP

Elementary modal logics and circuit complexity

本文证明有限公理化初等模态逻辑的有效性问题属于P，并利用图论复杂性结果，在假设P≠NP下构造无限多个非初等的K4-稳定逻辑，条件性否定回答开放问题。

Characterizing categoricity in several classes of modules

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. $Theorem.$ Assume $R$ is an associative ring with unity. 1. The class of locally pure-injective $R$-modules is $λ$-categorical in $all$ $λ> |R|+\aleph_0$ if and only if $R \cong M_n(D)$ for $D$ a division ring and $n \geq 1$. 2. The class of flat $R$-modules is $λ$-categorical in $all$ $λ> |R| + \aleph_0$ if and only if $R \cong M_n(k)$ for $k$ a local ring such that its maximal ideal is left $T$-nilpotent and $n \geq 1$. 3. Assume $R$ is a commutative ring.…

A Dependent Bi-Coloured Field

We have considered a Fraisse class of finitely generated ordered real fields with a colour predicate. A predimension map is defined on finite sets and the Fraisse limit of the class is axiomatized by a theory $T$, which is proved to be dependent. The theory is proved to be non-distal with $dp-rank=\aleph_0$.