A ``lattice theoretic'' proof of the independence of the automorphism group, the congruence lattice, and the subalgebra lattice of an infinitary algebra (Q805644)

The paper gives a very simple proof of the following theorem: Let \(L_ c\) and \(L_ s\) be complete algebraic lattices with more than one element, and let G be a group. Then there exists an infinitary algebra A such that the congruence lattice of A is isomorphic to \(L_ c\), the subalgebra lattice of A is isomorphic to \(L_ s\), and the automorphism group of A is isomorphic to G. The author constructs the algebra A starting from a complete lattice K which has the complete congruence relation lattice isomorphic to \(L_ c\); this is possible by a previous result of G. Grätzer and H. Lasker. The proof presented here is simpler than the previous one contained in Appendix 7 of the second edition of Grätzer's book: Universal algebra (1979; Zbl 0412.08001). However, the older proof concerns a more general case, in which \(L_ c\) and \(L_ s\) are \({\mathfrak m}\)-complete algebraic lattices with \({\mathfrak m}\) an infinite regular cardinal.