A note on minors determined by clones of semilattices

On centralizers of finite lattices and semilattices

We study centralizer clones of finite lattices and semilattices. For semilattices, we give two characterizations of the centralizer and also derive formulas for the number of operations of a given essential arity in the centralizer. We also characterize operations in the centralizer clone of a distributive lattice, and we prove that the essential arity of operations in the centralizer is bounded for every finite (possibly nondistributive) lattice. Using these results, we present a simple derivation for the centralizers of clones of Boolean functions.

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。

源克隆和目标克隆为本质一元、线性、半格或0-分离、1-分离的布尔函数的克隆类（clonoid）

Clonoids of Boolean functions with essentially unary, linear, semilattice, or 0- or 1-separating source and target clones

本研究拓展Sparks定理，针对特定类型的克隆对(C₁,C₂)，确定了布尔函数的(C₁,C₂)-克隆类格的基数，分别处理了格不可数与有限的情形，并总结了该领域的已知结果。

三元素集上子极大克隆的minor等价分类

Submaximal clones over a three-element set up to minor-equivalence

本文研究三元素集上所有克隆在minor同态序下的偏序结构，证明了该偏序集恰好只有三个子极大元素，为理解CSP复杂度与minor恒等式之间的关系提供了关键结构信息。

自对偶运算的克隆格坍缩

The lattice of clones of self-dual operations collapsed

本文完整刻画三元集上自对偶运算克隆在保持子式映射序下的结构，证明其仅含可数多个等价类，并等价给出原始正可构造性序的结论。

关于三元素集上的马尔采夫克隆直至次等价

Mal'cev clones over a three-element set up to minor-equivalence

对三元素集上的马尔采夫克隆按次同态进行分类，这是三元素关系结构直至 pp 可构造性完全分类的又一步，还给出布拉托夫结果的替代证明，即三元素集上的马尔采夫克隆至多有 4 元关系基。