Abstract
We introduce a graph GE(L) of equivalence classes of zero divisors of a meet semilattice L with 0. The set of vertices of GE(L) are the equivalence classes of nonzero zero divisors of L and two vertices [x] and [y] are adjacent if and only if [x]∧[y]=[0]. It is proved that GE(L) is connected and either it contains a cycle of length 3 or GE(L)≅K2. It is known that two Boolean lattices L1 and L2 have isomorphic zero divisor graphs if and only if L1≅L2. This result is extended to the class of SSC meet semilattices. Finally, we show that Beck's Conjecture is true for GE(L) .
Highlights
The idea of a zero divisor graph was introduced by Beck in [1] to investigate the interplay between ring theoretic properties and graph theoretic properties
The graph of equivalence classes of zero divisors of commutative rings is well studied in Allen et al [13], Anderson and LaGrange [14], and Spiroff and Wickham [15]
We extend this concept to a meet semilattice L with 0
Summary
We introduce a graph GE(L) of equivalence classes of zero divisors of a meet semilattice L with 0. The set of vertices of GE(L) are the equivalence classes of nonzero zero divisors of L and two vertices [x] and [y] are adjacent if and only if [x] ∧ [y] = [0]. It is proved that GE(L) is connected and either it contains a cycle of length 3 or GE(L) ≅ K2. It is known that two Boolean lattices L1 and L2 have isomorphic zero divisor graphs if and only if L1 ≅ L2. This result is extended to the class of SSC meet semilattices. We show that Beck’s Conjecture is true for GE(L)
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