Complete congruence lattices of two related modular lattices

作者：Czedli Gabor^*

来源：ALGEBRA UNIVERSALIS, 2017, 78(3): 251-289.

DOI：10.1007/s00012-017-0457-9

• complete lattice congruence
• representation with complete congruence lattice
• modular lattice
• 2-distributive lattice

By a 1991 result of R. Freese, G. Gratzer, and E. T. Schmidt, every complete lattice A is isomorphic to the lattice Com(K) of complete congruences of a strongly atomic, 3-distributive, complete modular lattice K. In 2002, Gratzer and Schmidt improved 3-distributivity to 2-distributivity. Here, we represent morphisms between two complete lattices with complete lattice congruences in three ways. Namely, for let and be arbitrary complete lattices and let : be maps such that (i) is -preserving and 0-separating, (ii) is -preserving, and (iii) is -preserving. We prove that for there exist strongly atomic, 2-distributive, complete modular lattices and such that and, in addition, (i) is a principal ideal of and is represented by complete congruence extension, (ii) is a sublattice of and is represented by restriction, and (iii) is represented as the composite of a map naturally induced by a complete lattice homomorphism from to and the complete congruence generation in . Also, our approach yields a relatively easy construction that proves the above-mentioned 2002 result of Gratzer and Schmidt.

• 出版日期2017-11