Each finite algebra $\mathbf A$ induces a lattice~$\mathbf L_{\mathbf A}$ viathe quasi-order~$\to$ on the finite members of the variety generatedby~$\mathbf A$, where $\mathbf B \to \mathbf C$ if there exists a homomorphismfrom $\mathbf B$ to~$\mathbf C$. In this paper, we introduce the question:`Which lattices arise as the homomorphism lattice $\mathbf L_{\mathbf A}$induced by a finite algebra $\mathbf A$?' Our main result is that each finitedistributive lattice arises as~$\mathbf L_{\mathbf Q}$, for some quasi-primalalgebra~$\mathbf Q$. We also obtain representations of some other classes oflattices as homomorphism lattices, including all finite partition lattices, allfinite subspace lattices and all lattices of the form $\mathbf L\oplus \mathbf1$, where $\mathbf L$ is an interval in the subgroup lattice of a finite group.
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机译：每个有限代数$ \ mathbf A $通过〜$ \ mathbf A $生成的有限元中的拟阶数〜$ \ to $诱发一个阵元〜$ \ mathbf L _ {\ mathbf A} $ B \到\ mathbf C $，如果存在从$ \ mathbf B $到〜$ \ mathbf C $的同态。在本文中，我们引入一个问题：“由有限代数$ \ mathbf A $引起的同构晶格$ \ mathbf L _ {\ mathbf A} $会出现哪些晶格？”我们的主要结果是，对于一些拟原始代数〜$ \ mathbf Q $，每个有限分布格都以〜$ \ mathbf L _ {\ mathbf Q} $的形式出现。我们还获得了一些其他类的晶格作为同态格的表示形式，包括所有有限分区格，所有子空间格和所有形式为$ \ mathbf L \ oplus \ mathbf1 $的格，其中$ \ mathbf L $是子组中的区间有限群的晶格。
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