## The variety generated by an ai-semiring of order three.(English)Zbl 1489.16050

Semirings are algebras $$(S,+,\cdot)$$ such that the additive reduct is a commutative semigroup, the multiplicative reduct is a semigroup, and where multiplication distributes over addition. A semiring is an ai-semiring if it is addditively idempotent, i.e,. its additive reduct is a semilattice. The authors continue their investigation of varieties of semirings started in many earlier publications. In this paper they are interested in the finite basis problem for the varieties of ai-semirings generated by a unique three element semiring. After an introduction recalling some known results concerning the finite basis problem, they note that there are precisely six two element ai-semirings, and the variety generated by any one of them is finitely based. Up to isomorphism there are $$61$$ three element semirings. It is known from previous publications that each of certain $$45$$ of them generate a finitely based variety. The authors focus their attention on the remaining varieties. The main results of this paper shows that with a possible exception of one variety, all of them are finitely based. The equational base for each of these varieties is provided. Then it is conjectured that the one remaining variety is not finitely based.

### MSC:

 16Y60 Semirings 08B05 Equational logic, Mal’tsev conditions

### Keywords:

ai-semiring; identity; finitely based variety
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### References:

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