For a partial lattice L the so-called two-point extension is defined in order to extend L to a lattice. We are motivated by the fact that the one-point extension broadly used for partial algebras does not work in this case, i.e. the one-point extension of a partial lattice need not be a lattice. We describe these two-point extensions and prove several properties of them. We introduce the concept of a congruence on a partial lattice and show its relationship to the notion of a homomorphism and its connections with congruences on the corresponding two-point extension. In particular we prove that the quotient L/E of a partial lattice L by a congruence E on L is again a partial lattice and that the two-point extension of L/E is isomorphic to the quotient lattice of the two-point extension L* of L by the congruence on L* generated by E. Several illustrative examples are enclosed.
Congruences of fork extensions of slim semimodular lattices
对于一个细的、平面的、半模格$L$和覆盖正方形~ $S$，G.~ Czédli和E. T.~ Schmidt引入了分叉扩张$L[S]$，它也是一个纤细的平面半模格。我们研究了$L$的同余何时扩张到$L[S]$。引入了$L[S]$的并不可约同余$\boldsymbolγ(S)$.我们确定它何时是新的，在这个意义上，它不是由$L$的并不可约同余生成的。当它是新的时，我们非常详细地描述同余$\boldsymbolγ(S)$。主要结果如下：\emph{In the order of join-irreducible congruences of a slim, planar, semimodular lattice $L$, the congruence $\boldsymbolγ(S)$ has \emph{at most two covers.}}
Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations
Grätzer和Schmidt从“抽象代数的同余格的刻画”（Acta Sci. Math.（Szeged）24（1963），34-59）适合于为每个有限分配格是可表示的这一众所周知的事实提供另一种证明，这被看作是有限格表示问题的一种特殊情况。这个证明的建设汇集了伯克霍夫的表示定理有限分配格，强调布尔格时，代表有限格，和透视偏序集的不平等的基础上。也许有可能推广这种方法中使用的技术。除了上述的表示定理外，本文的两个定理只使用了初等工具。特别是不依赖于群论的概念或技巧（参见Péter Pál Pálfy和Pavel Puddyak），也不依赖于众所周知的方法，用于证明某些有限格是可表示的（参见William J. DeMeo），例如闭包方法。
