有限分配格的同余表示中的并不可约数
On the number of join-irreducibles in a congruence representation of a finite distributive lattice
摘要：
对于有限格L，设EL表示L上的并依赖关系的自反闭包和传递闭包，定义在L的所有并不可约元的集合J（L）上。我们刻画的关系的形式EL，如下：定理。设E是有限集P上的拟序，则下列条件等价：（i）存在有限格L使得（J（L），EL）同构于拟序集（P，E）. (ii)对于有限格L，设je（L）=| J（L）|-|J（Con L）|其中Con L是L的同余格.众所周知，不等式je（L）$\ge$ 0成立。对于有限的分配格D，我们定义并过剩函数：JE（D）= min（je（L））|D是D的一部分）。给出了有限分配格D的并过剩函数的计算公式。该公式意味着JE（D）$\le$（2/3）|J（D）|，对于任意有限分配格D，常数2/3是最佳可能的.这个公式的一个特例给出了有限下界格的同余格的一个刻划。
英文摘要：
For a finite lattice L, let EL denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form EL, as follows: Theorem. Let E be a quasi-ordering on a finite set P. Then the following conditions are equivalent: (i) There exists a finite lattice L such that (J(L),EL) is isomorphic to the quasi-ordered set (P,E). (ii) There are not exactly two elements x in P such that p E x, for any p in P. For a finite lattice L, let je(L) = |J(L)|-|J(Con L)|, where Con L is the congruence lattice of L. It is well-known that the inequality je(L) $\ge$ 0 holds. For a finite distributive lattice D, let us define the join-excess function: JE(D) = min(je(L) | Con L isomorphic to D). We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that JE(D) $\le$ (2/3)| J(D)|, for any finite distributive lattice D; the constant 2/3 is best possible. A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.
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