半群上同余备格的完全分配性及Oehmke问题的若干注记

Complete Distributivity of the Lattice of Congruences on a Semigroup and Some Notes on Oehmke's Problems

本文研究半群上同余备格的完全分配性。证明了如下结果:设S是一个半群,C(S)是S上的同余备格,以下条件彼此等价:(1).C(S)是完全分配的;(2).S是特殊半群;(3).C(S)是由R(S)自由生成的;(4).对于Aα、β∈C(S),D(α∩β)=D(α)∪D(β).同时,对Oehmke提出的一些问题作了初步探讨。

Our purpose is to study complete distributivity of the lattice of congruences on a semigroup and prove the result: For a semigroup S the following are equivalent; (1)C(S) is completely distributive,(2)S is a special semigroup,(3)C(S) is freely generated by R(S),(4)For any α, β∩ C(S),D(α )= D(α) ∪D(β).Besides,we discuss some problems in put forward by Oehmke.