Remarks on closed semimodules and quotient semimodules (Q1357259)

Let \((R,+,\cdot)\) be a semiring, \((M,+)\) a (left) \(R\)-semimodule and \((A,+)\) a subsemimodule of \((M,+)\). Then \((A,+)\) is called normal if for all \(a\in A\) and \(m\in M\) there is some \(a'\in A\) such that \(a+m=m+a'\), and \((A,+)\) is called closed if \(a+x\in A\) and \(y+a'\in A\) for some \(a,a'\in A\), \(x,y\in M\) imply \(x,y\in A\). Moreover, the relation \(\rho_A=\{(x,y)\in M\times M\mid x+a_1=y+a_2\) for some \(a_i\in A\}\) is called the normal relation defined by \((A,+)\). If \((A,+)\) is normal, then \(\rho_A\) is a congruence on \((M,+)\) and the quotient \(R\)-semimodule \(M/\rho_A\) is denoted by \(M/A\). The main result is: Let \(\mathcal A\) be the set of closed subsemimodules of \((M,+)\) containing the normal subsemimodule \((A,+)\) and \(\mathcal B\) the set of all closed subsemimodules of \(M/A\). Then \(f(H)=H/A\) for all \(H\in{\mathcal A}\) defines a bijection from \(\mathcal A\) onto \(\mathcal B\).