\(k\)-furcus semigroups. (Q1950953)

Let \(S\) be an atomic monoid (every element can be expressed up to units as a finite product of irreducible elements). Then \(S\) is \(k\)-forcus if whenever an element can be expressed as a product of at least \(k\) non-unit elements, then it can be factored as the product of exactly \(k\) irreducible elements (also known as atoms). The authors prove that if \(S\) is a \(k\)-forcus monoid, then it does not contain strong atoms, its elasticity and tame degree are infinite, and the catenary degree and critical length are between \(3\) and \(k+1\). They also prove that for every \(x\in S\), \(k\) times the elasticity of \(x\) is in \(\mathbb N\cup\{\infty\}\). For \(k\in\{2,3\}\) the Delta set of \(S\) is just \(\{1\}\), and for \(k>3\), \(1\) is in the Delta set of \(S\), which is contained in \(\{1,\dots,k-2\}\). In the third section, the notion of quasi \(k\)-forcus monoid is introduced: every non-unit has a factorization of length at most \(k\). These monoids do not contain strong atoms, their elasticity is also not bounded, the catenary degree is less than \(k+1\), and the critical length is greater than or equal to 3 and less than or equal to \(k+1\). The Delta set contains 1 and it is contained in \(\{1,\dots,k-3\}\). The authors also offer bounds for the sets of lengths of these semigroups and \(k\)-forcus semigroups. The last section is devoted to divisor graphs. For an element \(x\) in a monoid the graph \(G(x)\) is the graph with vertices \(a\) atoms such that \(a\mid x\), and edges \(ab\) such that \(ab\mid x\). Loops are allowed, and labeled with \(n\), if \(a^n\mid x\) but \(a^{n+1}\) does not divide \(x\). The authors prove that every nonunit with a factorization of length greater than \(k\) have nonconnected associated graph. Observe that connectedness does not depend on loops, and precisely these graphs without loops correspond to the graphs introduced by \textit{J. C. Rosales} [in Int. J. Algebra Comput. 6, No. 4, 441-455 (1996; Zbl 0863.20026)] (and later generalized to several kinds of monoids). Rosales proved that every nonconnected graph yields relations for a minimal presentation of the monoid. Thus the authors are ensuring that every \(x\) with a factorization with length greater than \(k\) yields at least a relation in the minimal presentation of the monoid. The paper contains several examples and figures that make the reading appealing.