Proof. By Corollary 2 of 1.1, Z[(0: Z,)]2Z[(0: S)], and since S-(0:S) = {0}, S c 2 [ ( 0 : S)]. By 1.2, Z[(0: Z J^Z j and the result follows. Corollary 2. // Z t and Z 2 are bofh Z-closed subsets of V, then Z x = Z 2 i/, and onfy if, (0:Z 1 ) = (0:Z 2 ). This follows since, if (0: Z t ) = (0: Z 2 ), then by 1.2, Z x = Z[(0: Z x )] = Z[(0: Z 2 )] = Z 2 . Corollary 2 puts zero sets and their annihilators in one-one correspondence. The next proposition shows this correspondence is lattice

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... . Proposition 1.3. Let Z x and Z 2 be Z-closed subsets of an N-group V • Z x c Z 2 if, and only if, (0: Z x )^(0: Z 2 ). Proof. Clearly if Z t <=Z 2 , then (0: Z x )^(0: Z 2 ). If (0: Z x )^(0: Z 2 ), then Z[(0: Z i ) ] s Z[(0: Z 2 )], By Corollary 2 of 1.1. From 1.2, Z[(0: Z t )] = Z t for i = 1, 2, and the proposition holds. Let AT be a near-ring, V an N-group, A a subset of V and a an element of N. As in the case of functions we denote the sets {va: v e A} and {v e V: va e A} by Aa and Aa" 1 , respectively. Proposition 1.4. 7/ V is an N-group, then Z x a~l = Z[a(0: Z^], /or any Z-closed subset Z x o/ V and a in the zero-symmetric part of N.