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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.

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)
Full Text: DOI
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