Congruence Lattices of a Finite Isoform Lattices
K. Thirugnanasambandam, S. Mahendra kumar · IOSR Journal of Mathematics · 2016
DEFINITION: 1.1.1Let L be a lattice.Let be a congruence of L. Then is said to be isoform, if any two congruence classes of are isomorphic as lattices. DEFINITION: 1.1.2A lattice L is said to be isoform if all congruences of L are isoform. DEFINITION: 1.1.3A lattice L is said to be regular, if whenever two congruences share a congruence class, then the congruences are the same. NOTE: 1.1.4An isoform lattice is always regular. NOTATION: 1.1.5For a lattice L, we denote by L and i L the smallest and the largest congruence on L, respectively.C n will denote the n element chain.B n will denote the Boolean algebra with 2 n elements.For a bounded lattice A with bounds 0 and 1, A -will denote the lattice A-{0,1} EXAMPLE: 1.1.6Consider the Boolean algebra B 2 , with 4 elements.Its congruence lattice is also B 2 .It has four congruences, namely, the null congruence , the all congruence i and two non-trivial congruences 1 and 2 . 1 has two congruence classes { {0,a}, {1,b} } and 2 has two congruence classes { {0,b} } , {a,1} }. Congruence Lattices Of A Finite Isoform Lattices