The variety of commutative additively and multiplicatively idempotent semirings
The semirings in the title are algebras \((S,+,\cdot)\) with two semilattice structures \((S,+)\) and \((S,\cdot)\), where multiplication distributes over addition. They are quite well known under the name of meet-distributive bisemilattices, or bisemilattices with one distributive law. Several results from a long list of references are related to the results of the paper. Let \(V\) be the variety of meet-distributive bisemilattices. The (5-element) lattice of all subvarieties of \(V\) was fully described in [\textit{R. McKenzie} and the reviewer, Proc. Klagenfurt Conf. 1978, 213--218 (1979; Zbl 0419.06003)]. Among other things, this paper contains ``(a) a normal form for terms'' and a remark (a direct consequence of the main result) that ``(b) the variety \(V\) is generated by any one subdirectly irreducible algebra which is neither a lattice nor a semilattice.'' Free algebras in \(V\) were fully described in [the reviewer, Demonstratio Math. 13, 565--572 (1980, Zbl 0048.08006)]. All subdirectly irreducible \(V\)-algebras with at least one semilattice reduct a chain were characterised in [the reviewer, Algebra Universalis 10, 36--47, (1980, Zbl 0434.06005)]. A representation theorem for meet-distributive bisemilattices was given in [the reviewer and \textit{J. D. H. Smith}, J. Algebra 70, 78--88 (1981; Zbl 0457.06002)]. In the paper under review, the authors consider the same algebras with additional constants \(1\) as a neutral element for the multiplication, and \(0\) as both a zero of the multiplication and a neutral element for the addition. They describe some subdirectly irreducible algebras with both reducts being chains, and derive (a) and (b) (for those subdirectly irreducible algebras).
scientific article; zbMATH DE number 6893043
scientific article