One of the longstanding problems in universal algebra is the question ofwhich finite lattices are isomorphic to the congruence lattices of finitealgebras. This question can be phrased as which finite lattices can berepresented as lattices of equivalence relations on finite sets closed undercertain first order formulas. We generalize this question to a differentcollection of first-order formulas, giving examples to demonstrate that our newquestion is distinct. We then prove that every lattice $\m M_n$ can berepresented in this new way. [This is an extended version of a paper submittedto \emph{Algebra Universalis}.]
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机译：通用代数中长期存在的问题之一是哪个有限晶格与有限代数的同余格同构。这个问题可以用短语来表示，哪些有限晶格可以表示为在不确定的一阶公式封闭的有限集上的等价关系的晶格。我们将此问题归纳为不同的一阶公式集合，并举例说明我们的新问题是独特的。然后，我们证明每个晶格$ \ m M_n $都可以用这种新方式表示。 [这是提交给\ emph {Algebra Universalis}的论文的扩展版本。]
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