On \(\phi\)-absorbing primary elements in lattice modules (Q333765)

Summary: Let \(L\) be a \(C\)-lattice and let \(M\) be a lattice module over \(L\). Let \(\phi : M \to M\) be a function. A proper element \(P \in M\) is said to be \(\phi\)-absorbing primary if, for \(x_1, x_2, \ldots, x_n \in L\) and \(N \in M\), \(x_1 x_2 \cdots x_n N \leq P\) and \(x_1 x_2 \cdots x_n N \nleq \phi(P)\) together imply \(x_1 x_2 \cdots x_n \leq(P : 1_M)\) or \(x_1 x_2 \cdots x_{i - 1} x_{i + 1} \cdots x_n N \leq \root M \of P\), for some \(i \in \{1,2, \ldots, n \}\). We study some basic properties of \(\phi\)-absorbing primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as \(\phi\)-absorbing elements and \(\phi\)-absorbing primary elements are unified.