A characterization of semirings which are subdirect products of a distributive lattice and a ring
A semiring \(S\) is an algebraic structure \((S,+,\cdot)\) consisting of a non-empty set \(S\) together with two binary operations \(+\) and \(\cdot\) on \(S\) such that \((S,+)\) and \((S,\cdot)\) are semigroups connected by ring-like distributivity. The additive identity (if it exists) of a semiring is called zero and is denoted by \(0\). An additively commutative semiring \(S\) with zero satisfying \(0\cdot x=x\cdot 0=0\) for all \(x\in S\) is called a hemiring. A halfring is an additively cancellative hemiring. A subsemiring \(T\) of a semiring \(S\) is called a distributive sublattice of \(S\) iff \(T\) is a (distributive) lattice with respect to the operations defined in \(S\). A \(k\)-ideal \(K\) of a semiring \(S\) is an ideal of \(S\) (\(KS,SK\subseteq K\)) such that if \(a\in K\), \(x\in S\), and if either \(a+x\in K\) or \(x+a\in K\), then \(x\in K\). A semigroup \(S\) is called \(E\)-inversive iff for every \(a\in S\), there exists \(x\in S\) such that \(ax\) is an idempotent of \(S\). An additively commutative semiring \(S\) is called \(E\)-inversive iff its additive reduct \((S,+)\) is an \(E\)-inverse semigroup. A semiring \(S\) is called an inverse semiring iff \((S,+)\) is an inverse semigroup. An additively commutative semiring \(S\) is called a Clifford semiring iff \(S\) is an inverse semiring such that the set of all additive idempotents is a distributive sublattice of \(S\) as well as a \(k\)-ideal of \(S\). A subdirect product \(H\) of a semiring \(S\) and a ring \(R\) is called inverse iff \((e,r)\in H\) implies \((e,-r)\in H\) for every additively idempotent \(e\) of \(S\) and \(r\in R\). A Boolean semiring is a semiring in which each element is already multiplicatively idempotent. In this paper the author defines a strong distributive lattice of semirings and he proves that a semiring \(S\) is a subdirect product of a distributive lattice and a ring if \(S\) is an \(E\)-inversive strong distributive lattice of halfrings. Further a Clifford semiring which is, in fact, an inversive subdirect product of a distributive lattice and a ring, is characterized as a strong distributive lattice of rings. Finally, a semiring \(S\) is a subdirect product of a distributive lattice and a Boolean ring iff \(S\) is a commutative Boolean semiring satisfying \(a+2ab=a\) for all \(a,b\in S\). A semiring \(S\) is a direct product of a distributive lattice and a Boolean ring iff \(S\) is a subdirect product and for any \(a,b\in S\), there exists \(c\in S\) such that \(a+c=2(a+c)\) and \(2b=2c\).
strong distributive lattices of semirings
scientific article; zbMATH DE number 1330655