Strong congruence in atomistic lattices

Main Article Content

H. S. Ramananda

Abstract

In this paper, for an atomistic lattice L satisfying the ascending and descending chain conditions, it is proved that• the congruence lattice Con L is isomorphic to the sublattice of the set of all standard elements of L;• every congruence relation of L is representable.Further, the notion of a strong congruence relation in lattices is introduced, some examples of strong congruence relations are given and it is proved that in a sectionally complemented lattice satisfying the ascending and descending chain conditions, every congruence relation is strong.

Article Details

How to Cite
Ramananda, H. (2016). Strong congruence in atomistic lattices. Acta Mathematica Universitatis Comenianae, 85(2), 337-346. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/254/394
Section
Articles

References

1. Dwight Duffus, Claude Laflamme, Maurice Pouzet and Robort Woodrow: Convex sublattices of lattice and a fixed point property, Eprint arXiv:1108.5210 (2011), 1-29.

2. G. Gr¨atzer: General Lattice Theory, Birkh¨auser Verlag, Second Edition, 1998.

3. S. Lavanya and S. Parameshwara Bhatta: A new approach to the lattice of convex sublattices of a lattice, Algebra Univers., 35 (1996), 63-71.

4. S. Lavanya: Some new ordering on the set of all convex sublattices and the set of all sublattices of a lattice, Mangalore University, Ph.D thesis, 1994.

5. S. Lavanya and S. Parameshwara Bhatta: Finite distributive lattices which are isomorphic to products of chains, Algebra Univers., 37 (1997), 448-457.

6. L. Libkin: Direct decompositions of atomistic algebraic lattices, Algebra Univers. 33 (1995), 127-135.

7. Michael Siff and Thomas Reps: Identifying module via concept analysis,IEEE Transactions on Software Engineering, 25 (1999), 749-766

8. S. Parameshwara Bhatta and H. S Ramananda: On ideals and congruence relation in trellises, Acta Math. Univ. Comenianae, 2(2010) 209-216.

9. S. Radeleczki: Some structure theorems for algebraic atomistic lattices, Acta Math. Hungar., 86 (2000), 1-15.

10. G. Richter: On the structure of lattices in which every element is a join of join-irreducible elements, Periodica Mathematica Hungerica, 13 (1982), 47-69.

11. B. Seselja and A. Tepavcevic: A note on atomistic weak congruence lattices, Discrete Mathematics, 308 (2008), 2054-2057.