Subatomicity in rank-2 lattice monoids (Q6597270)

Let \(M\) be a cancellative and commutative monoid (written additively), and \(\mathcal{U}(M)\) (\(\mathcal{A}(M)\)) be the set of its invertible elements (of its atoms, respectively). An element \(b\in M\backslash \mathcal{U}(M)\) is called \textit{atomic} (respectively, \textit{almost atomic}, \textit{quasiatomic}) provided that \(b \in \langle \mathcal{A}(M)\rangle\) (respectively, \(b \in gp(\langle \mathcal{A}(M)\rangle)\), \(b \in \langle \mathcal{A}(M)\rangle - M : = \{a - m \colon a \in \langle \mathcal{A}(M)\rangle\) and \(m \in M\}\); here \(gp(\langle \mathcal{A}(M)\rangle)\) denotes the difference group of \(\mathcal{A}(M)\)). The monoid \(M\) is called 1) \textit{atomic} (\textit{almost atomic}, \textit{quasiatomic}) if every element in \(M\backslash \mathcal{U}(M)\) is atomic (respectively, almost atomic, quasiatomic); 2) \textit{nearly atomic} if there exists \(b \in M\) such that \(b+m\) is atomic for all \(m\in M\); 3) a \textit{lattice monoid} if it is a submonoid of a finite rank free abelian group; 4) a \textit{Furstenberg monoid} provided that every element of \(M\backslash \mathcal{U}(M)\) is divisible by an atom. If \(M\) is a submonoid of \(\mathbb{Z}^2\) then it is called \textit{rationally supported} if there exists a line \(L\) in \(\mathbb{Z}^2\) with rational slope through the origin such that all elements of \(M\) belong either to the upper or lower subspace determined by \(L\). Let \(M\) be a rationally supported lattice monoid of \(\mathbb{Z}^2\). It is proved that: 1) if \(\vert\mathcal{A}(M)\vert=\infty\), then \(M\) is atomic; 2) If \(M\) is nearly atomic, then \(M\) is atomic; 3) \(M\) is a Furstenberg monoid. It is also proved that a monoid \(M\) is quasiatomic if and only if every nonzero prime ideal of \(M\) contains an atom. The article contains a large number of examples.