We investigate the lattice of clones that are generated by a set of functions that are induced on a finite field \({\mathbb {F}}\) by monomials. We study the atoms and coatoms of this lattice and investigate whether this lattice contains infinite ascending chains, or infinite descending chains, or infinite antichains.We give a connection between the lattice of these clones and semi-affine algebras. Furthermore, we show that the sublattice of idempotent clones of this lattice is finite and every idempotent monomial clone is principal.

中文翻译：

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有限域上的单项式克隆格

我们研究了由单项式在有限域\({\mathbb {F}}\)上诱导的一组函数生成的克隆格。我们研究了这个晶格的原子和涂层，并研究了这个晶格是否包含无限上升链，或无限下降链，或无限反链。我们给出了这些克隆的晶格与半仿射代数之间的联系。此外，我们证明了该格的幂等克隆的亚格是有限的，并且每个幂等单项式克隆都是主要的。