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The lattice of idempotent distributive semiring varieties. (English) Zbl 0947.16036
A semiring $$(S,+,\cdot)$$ is an algebra with two binary operations $$+$$ and $$\cdot$$ such that the reducts $$(S,+)$$ and $$(S,\cdot)$$ are semigroups and in which the two distributive laws $$x(y+z)=xy+xz$$, $$(y+z)x=yx+zx$$ are satisfied. A band is a semigroup in which every element is an idempotent. An idempotent semiring is a semiring where both the additive reduct and multiplicative reduct are bands. A distributive semiring is a semiring which satisfies the dual two distributive laws $$x+yz=(x+y)(x+z)$$, $$yz+x=(y+x)(z+x)$$.
In the present paper, the variety $$ID$$ of all idempotent distributive semirings is investigated. A solution is given for the word problem for free idempotent distributive semirings. Using this solution the lattice $$L(ID)$$ of subvarieties of $$ID$$ is determined. It turns out $$L(ID)$$ is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the lattice $$L(B)$$ of all band varieties. In the semigroup theory, it is well-known that $$L(B)$$ is countable infinite and distributive. Therefore $$L(ID)$$ is countable infinite and distributive. Using the known semigroup fact about finite bases of any band variety, it is proved that every subvariety of $$ID$$ is finitely based.

##### MSC:
 16Y60 Semirings 08B15 Lattices of varieties 20M07 Varieties and pseudovarieties of semigroups 16R10 $$T$$-ideals, identities, varieties of associative rings and algebras 08A50 Word problems (aspects of algebraic structures)
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