Generalized \(\mathcal N\)-semigroups (Q2709046)

All semigroups in this paper are commutative so the operation is denoted by \(+\). The authors define a generalized semigroup to be a cancellative semigroup \(S\) which is not a group and which contains at least one Archimedean element, where \(x\in S\) is called Archimedean if (\(\exists k\in\mathbb{N}\setminus\{0\}\), and \(\exists z\in S\)) \(kx=y+z\). Let \((G,+)\) be a group, \((\mathbb{N},+)\) be the monoid of the nonnegative integers and assume \(I\colon G\times G\to\mathbb{N}\) satisfies (1) \(\forall g_1,g_2\in G\), \(I(g_1,g_2)=I(g_2,g_1)\), (2) \(\forall g_1,g_2,g_3\in G\), \(I(g_1,g_2)+I(g_1+g_2,g_3)=I(g_2,g_3)+I(g_1,g_2+g_3)\). Define the operation \(+_I\) on \(\mathbb{N}\times G\) as follows: \((a_1,g_1)+_I(a_2,g_2)=(a_1+a_2+I(g_1,g_2),g_1+g_2)\) then \((\mathbb{N}\times G,+_I)\) is a generalized semigroup. Conversely a generalized semigroup \((S,+)\) is isomorphic to \((\mathbb{N}\times S/R_m,+_I)\) where \(m\) is an Archimedean element and \(R_m\) is defined by \(xR_my\Leftrightarrow (\exists k,k'\in\mathbb{N})\), \(x+km=y+k'm\). The authors assume \(I(0,0)=0\) or 1. If \(I(0,0)=0\), then \((S,+_I)\) is a monoid but if \(I(0,0)=1\), then \((S,+_I)\) has no idempotent element. They introduce \(\mathcal N\)-monoids, namely, an \(\mathcal N\)-monoid is a cancellative monoid which is not a group and which contains at least one Archimedean element. Since, generally, a cancellative semigroup can be embedded into a group, every generalized \(\mathcal N\)-semigroup can be embedded into an \(\mathcal N\)-monoid. According to the authors, we need only study \(\mathcal N\)-monoids. The remaining part of this paper is devoted to the following: reduced monoids, torsionfree monoids, reduced valuation monoids. A monoid is called reduced if its only unit is its identity element. A submonoid \(S\) of a group \(H\) is called a valuation monoid of \(H\) if for all \(h\in H\), \(\{h,-h\}\cap S\neq\emptyset\). A monoid \((\mathbb{N}\times G,+_I)\) is a reduced valuation monoid of \((\mathbb{Z}\times G,+_I)\) if and only if \(I(g,-g)=1\) for all \(g\in G\setminus\{0\}\). If \((\mathbb{N}\times G,+_I)\) is an \(\mathcal N\)-monoid with \(I(g,-g)=1\) for all \(g\in G\), then \((\mathbb{N}\times G,+_I)\) is a reduced valuation monoid of \((\mathbb{Z}\times G,+_I)\) with at least one Archimedean element.NEWLINENEWLINENEWLINEReviewer's comment: It is a nice theme for the authors to consider that \(I\) satisfies only \((T1)\), \((T2)\) from the reviewer's five conditions, and it is also a good idea that they introduce \(\mathcal N\)-monoids. The reviewer, however, mentions that the generalized \(\mathcal N\)-semigroup without idempotent is an \(\overline{\mathcal N}\)-semigroup, namely a sub-Archimedean CCIF-semigroup in the paper of the reviewer, \textit{H. B. Hamilton} and \textit{T. E. Nordahl} [Pac. J. Math. 61, 441-456 (1975; Zbl 0358.20073)]. Of course the starting point in this paper is different from the reviewer's paper of 1975.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00028].