Classes of Algebras and closure operations

关于半环的$z$-理想与$z$-闭包运算，I

On $z$-ideals and $z$-closure operations of semirings, I

本文研究半环的$z$-理想，引入并分析了$z$-素、$z$-半素等几类特殊理想及其性质，利用$z$-闭包算子证明了它们与相应理想类的等价性。

Closure operations, Continuous valuations on monoids and Spectral spaces

We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations on monoids. We prove that the collection of continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space.

Galois connection for sets of operations closed under permutation, cylindrification and composition

We consider sets of operations on a set A that are closed under permutation of variables, addition of dummy variables and composition. We describe these closed sets in terms of a Galois connection between operations and systems of pointed multisets, and we also describe the closed sets of the dual objects by means of necessary and sufficient closure conditions. Moreover, we show that the corresponding closure systems are uncountable for every A with at least two elements.

On Galois Connections between External Operations and Relational Constraints: Arity Restrictions and Operator Decompositions

We study the basic Galois connection induced by the "satisfaction" relation between external operations $A^n\rightarrow B$ defined on a set $A$ and valued in a possibly different set $B$ on the one hand, and ordered pairs $(R,S)$ of relations $R\subseteq A^m$ and $S\subseteq B^m$, called relational constraints, on the other hand. We decompose the closure maps associated with this Galois connection, in terms of closure operators corresponding to simple closure conditions describing the corresponding Galois closed sets of functions and constraints. We consider further Galois correspondences by r…

关于半环中减法闭包算子在理想运算上的分配性

On the distributivity of the subtractive closure operator over ideal operations in hemirings

研究半环中减法闭包算子在理想运算上的分配性，引入库拉托夫斯基半环和核半环，给出反例与正面结果，如指出$\mathbb{N}_0$等是核半环，$T = [0,1]$是库拉托夫斯基半环等，还纠正相关断言。

Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings

Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the \emph{prime $k$-spectrum} of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide…