zbMATH — the first resource for mathematics

Varieties of idempotent semirings with commutative addition. (English) Zbl 1084.16039
Let \(\mathbf B\) be the variety of all bands, \(\mathbf{RB}\) the subvariety of all regular bands, and \({\mathbf S}\ell\) the subvariety of all semilattices. A semiring \((S,+,\cdot)\) is called idempotent, if \((S,+)\) and \((S,\cdot)\) belong to \(\mathbf B\). For any subvariety \(\mathbf V\) of \(\mathbf B\), denote by \({\mathbf V}^+\) the variety of all idempotent semirings \((S,+,\cdot)\) such that \((S,+)\in{\mathbf V}\), and likewise for \({\mathbf V}^\cdot\).
In the paper the lattice \({\mathbf L}({\mathbf S}\ell^+)\) of subvarieties of \({\mathbf S}\ell^+\) is investigated. For any \(F\in{\mathbf B}\), a semiring \(\overline P_f(F)\in{\mathbf S}\ell^+\) is constructed which yields a free semiring \(\overline P_f(F{\mathbf B}_X)\in{\mathbf S}\ell^+\) on the set \(X\) if \(F{\mathbf B}_X\) denotes the free band on \(X\).
As a first step, \((S,\cdot)\in\mathbf{RB}\) is shown for every semiring \((S,+,\cdot)\in{\mathbf S}\ell^+\), i.e. \({\mathbf S}\ell^+=\mathbf{RB}^\cdot\cap{\mathbf S}\ell^+\). A further investigation of \(\overline P_f(F)\in{\mathbf S}\ell^+\) for \(F\in\mathbf{RB}\) yields that the mapping \({\mathbf L}(\mathbf{RB})\to{\mathbf L}({\mathbf S}\ell^+)\) given by \({\mathbf V}\mapsto{\mathbf V}^\cdot\cap{\mathbf S}\ell^+\) for any subvariety \(\mathbf V\) of \(\mathbf{RB}\) is an injective \(\cap\)-homomorphism. Therefore there are exactly 13 subvarieties of \({\mathbf S}\ell^+\) in the image of this \(\cap\)-homomorphism. As the main result it is shown that these 13 subvarieties generate a sublattice of \({\mathbf L}({\mathbf S}\ell^+)\) consisting of 20 subvarieties. Moreover, models of free semirings in these 20 varieties are given.

16Y60 Semirings
20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties
Full Text: DOI