The Strength of the Gr\"atzer-Schmidt Theorem
Arch. Math. Log.(2014)
摘要
The Gr\"atzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. A lattice is algebraic if it is complete and generated by its compact elements. We show that the set of indices of computable lattices that are complete is $\Pi^1_1$-complete; the set of indices of computable lattices that are algebraic is $\Pi^1_1$-complete; and that there is a computable lattice $L$ such that the set of compact elements of $L$ is $\Pi^1_1$-complete. As a corollary, there is a computable algebraic lattice that is not computably isomorphic to any computable congruence lattice.
更多查看译文
关键词
computability theory,lattice theory
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络