A construction of commutative nilsemigroups
A commutative nilsemigroup \(S\) is a commutative semigroup with zero 0 and for every element \(a\in S\) there is \(n\in\mathbb{Z}_+\) such that \(a^n=0\). In this paper the author constructs all nilsemigroups by means of the quotient semigroup of a free commutative semigroup \(F\) using the concept of ``coherent nests'' of equivalence relations on some set. This paper extends the author's earlier results [Commun. Algebra 19, No. 11, 3145-3172 (1991; Zbl 0755.20016)]. Let \(S\) be a semigroup, \(I\) an ideal of \(S\), and \(C\) a congruence on \(S\). If \(I\) is a \(C\)-class, then \(C\) is called \(I\)-congruence on \(S\). A nilsemigroup ideal \(I\) of \(S\) is an ideal of \(S\) such that \(S/I\) is a nilsemigroup. A coideal \(H\) of \(S\) is a subset \(H\) of \(S\) such that \(S\setminus H\) is an ideal of \(S\). In particular if \(H\) is a coideal of \(S\) and \(S\setminus H\) is a nilsemigroup ideal, then \(H\) is called a nilsemigroup coideal of \(S\). Let \(K\) denote the set of all maximal elements of \(H\). We assume that \(H\) is generated by its maximal elements. Given an \(I\)-congruence \(C\) and for every \(u\in H\) we define an equivalence relation \(C_u\) on \(K\) as follows: \[ kC_u\ell\text{ if and only if }k=\ell\text{ or }k,\ell\geq u\text{ and }k-uC\ell-u. \] The family \(N(C)=(C_u)_{u\in H}\) is called the nest of equivalence relations of \(C\). When \(H\) is generated by its maximal elements, every \(C_u\) is an equivalence relation on \(K\) and the following conditions are satisfied: for \(u,v\in H\), (N1) \(u\leq v\Rightarrow C_v\subseteq C_u\), (N2) \(C_u\subseteq D_u\) where \(aD_ub\) iff \(K:a=K:b\) in the sense of \textit{R. Croisot} [J. Math. Pure Appl., IX. Sér. 36, 373-417 (1957; Zbl 0079.25002)], (N3) if \(k,\ell,m,n\in K\) and \(k-\ell=m-n\), then for every \(u\in H\), \(kC_{p+u}\ell\iff mC_{q+u}n\) where \(p=k-k\land m\), \(q=m-k\land m\). A family \((C_u)_{u\in H}\) of equivalence relations on \(K\) satisfying (N1), (N2) and (N3) is called a coherent nest of equivalence relations on \(K\). For a free commutative semigroup \(F\) over \(X\), let \(\overline F=\prod_{x\in X}(\mathbb{N}\cup\{0\})\) be all families \(a=(a_x)_{x\in X}\) of nonnegative integers under addition \((a_x)_{x\in X}+(b_x)_{x\in X}=(a_x+b_x)_{x\in X}\). For \(H\), define \(\overline H\) by \(\overline H=\{\sup U\in\overline F\mid U\) is a directed subset of \(H\}\), define \(\overline K\) to be the set of maximal elements of \(\overline H\). Now let \(P\) be the nonnegative cone of \(\overline F\). The following is the most important main theorem of this paper. Let \(F\) be a free commutative semigroup and \(I\) a nilsemigroup ideal of \(F\), and let \((C_u)_{u\in\overline H}\) be a coherent nest of equivalence relations on \(\overline K\). Define \(C\) as follows: \(a,b\in P\), \(aCb\) if and only if either \(a,b\in I\) or \(a,b\in H\) and \(a+uC_ub+u\) for some \(u\in\overline H\). Then \(C\) is an \(I\)-congruence on \(P\) and its nest of equivalence relations is \((C_u)_{u\in\overline H}\). There is a one-to-one correspondence between \(I\)-congruences on \(P\) and coherent nests of equivalence relations on \(\overline K\).
scientific article; zbMATH DE number 1870464