The clone generated by the median functions

Finite degree clones are undecidable

A clone of functions on a finite domain determines and is determined by its system of invariant relations (=predicates). When a clone is determined by a finite number of relations, we say that the clone is of finite degree. For each Minsky Machine $\mathcal{M}$ we associate a finitely generated clone $\mathcal{C}$ such that $\mathcal{C}$ has finite degree if and only if $\mathcal{M}$ halts, thus proving that deciding whether a given clone has finite degree is impossible.

关于在两个元素集上多排序关系克隆的格

On the lattice of multi-sorted relational clones on a two-element set

本文通过定义规范关系类和基本操作，提供了一个新的证明方法，展示了在两个元素域上多排序克隆的格结构及其有限生成性。

Closed function sets on groups of prime order

We give a full description of all sets of functions on the group $(\mathbb{ Z}_p, +)$ of prime order which are closed under the composition with the clone generated by $+$ from both sides. Thereby, we also get a description of all iterative algebras on $\mathbb{Z}_p$ which are closed under the composition with the clone generated by $+$ from both sides. As another consequence, there are infinitely many non finitely generated clones above $\operatorname{Clo}({\mathbb{ Z}_p \times \mathbb{ Z}_p, +})$ for $p > 2$.

Uniform Birkhoff

We show that pseudovarieties of finitely generated algebras, i.e., classes $C$ of finitely generated algebras closed under finite products, homomorphic images, and subalgebras, can be described via a uniform structure $U$ on the free algebra for $C$: the members of $C$ then are precisely those finitely generated algebras $A$ for which the natural mapping from the free algebra onto the term clone of $A$ is well-defined and uniformly continuous with respect to the uniformity $U$ and the uniformity of pointwise convergence on the term clone of $A$, respectively. Our result unifies earlier theorem…

On the clone of aggregation functions on bounded lattices

The main aim of this paper is to study aggregation functions on lattices via clone theory approach. Observing that the aggregation functions on lattices just correspond to $0,1$-monotone clones, as the main result we show that for any finite $n$-element lattice $L$ there is a set of at most $2n+2$ aggregation functions on $L$ from which the respective clone is generated. Namely, the set of generating aggregation functions consists only of at most $n$ unary functions, at most $n$ binary functions, and lattice operations $\wedge,\vee$, and all aggregation functions of $L$ are composed of them by…

Clones containing the Mal'cev operation of $\mathbb{Z}_{pq}$

We investigate finitary functions from $\mathbb{Z}_{pq}$ to $\mathbb{Z}_{pq}$ for two distinct prime numbers $p$ and $q$. We show that the lattice of all clones on the set $\mathbb{Z}_{pq}$ which contain the addition of $\mathbb{Z}_{pq}$ is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattice of all $(\mathbb{Z}_p,\mathbb{Z}_q)$-linearly closed clonoids to the $p+1$ power and the lattice of all $(\mathbb{Z}_q,\mathbb{Z}_p)$-linearly closed clonoids to the $q+1$ power. These lattices are studied in arXiv:1910.11…

源克隆为线性克隆、目标克隆为半格克隆或0-分离或1-分离克隆的布尔函数克隆类（clonoid）

Clonoids of Boolean functions with a linear source clone and a semilattice or 0- or 1-separating target clone

该研究推广Sparks定理，确定了特定布尔函数克隆配对下克隆类格的不可数性，结合已有结果完成了二元集上所有克隆配对的克隆类格基数的完整分类。

Finite generation of congruence preserving functions

We investigate when the clone of congruence preserving functions is finitely generated. We obtain a full description for all finite $p$-groups, and for all finite algebras with Mal'cev term and simple congruence lattice. The characterization for $p$-groups allows a generalization to a large class of expansions of groups.

阿贝尔无平方因子群的扩张

Expansions of abelian squarefree groups

该研究针对无平方因子数对应的模n整数集，证明了含加法运算的克隆格有限，给出其基数上界的构造方式，并证明这类克隆可由元数不超过各素因子最大值的函数集生成，推进了阿贝尔无平方因子群相关克隆结构的研究。