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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.

##### MSC:
 16Y60 Semirings 20M07 Varieties and pseudovarieties of semigroups 08B15 Lattices of varieties
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