论文标题
在有限晶格和半层次上的方程式系统的解决方案集
Solution sets of systems of equations over finite lattices and semilattices
论文作者
论文摘要
字段上均匀线性方程的系统集合的解决方案集被描述为子空间，即在线性组合下封闭的集合。我们的目标是通过类似的闭合条件，通过任意有限代数来表征方程系统的解决方案集。我们表明，在给定代数的术语操作的克隆的中央器下，解决方案集始终是关闭的；此外，centralizer是唯一可以表征解决方案集的克隆。如果每个中央封闭的集合是有限代数上方程系统的所有解决方案的集合，那么我们说代数具有属性（SDC）。我们的主要结果是描述了有限的晶格和具有属性（SDC）的半层次：我们证明有限的晶格具有属性（SDC），并且仅当它是布尔晶格，并且有限的半静力属性具有属性（SDC）（SDC），并且仅当它是分布式时。
Solution sets of systems of homogeneous linear equations over fields are characterized as being subspaces, i.e., sets that are closed under linear combinations. Our goal is to characterize solution sets of systems of equations over arbitrary finite algebras by a similar closure condition. We show that solution sets are always closed under the centralizer of the clone of term operations of the given algebra; moreover, the centralizer is the only clone that could characterize solution sets. If every centralizer-closed set is the set of all solutions of a system of equations over a finite algebra, then we say that the algebra has Property (SDC). Our main result is the description of finite lattices and semilattices with Property (SDC): we prove that a finite lattice has Property (SDC) if and only if it is a Boolean lattice, and a finite semilattice has Property (SDC) if and only if it is distributive.
