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The variety $${\mathcal Z}$$ Z of commutative additively and multiplicatively idempotent semirings is studied. We prove that $${\mathcal Z}$$ Z is generated by a single subdirectly irreducible three-element semiring and it has a canonical form for its terms. Hence, $${\mathcal Z}$$ Z is locally finite despite the fact that it is residually large. The word problem in $${\mathcal Z}$$ Z is solvable.
We prove that the variety $${{\mathscr {V}}}$$ V of commutative multiplicatively idempotent semirings satisfying $$x+y+xyz\approx x+y$$ x + y + x y z ≈ x + y is generated by a single three-element semiring. Moreover, we describe a normal form system for terms in $${{\mathscr {V}}}$$ V and we show that the word problem in $${{\mathscr {V}}}$$ V is solvable. Although...
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