Related papers: The finite basis problem for additively idempotent…
We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific $4$-element additively idempotent semirings, $S_{(4,545)}$ and $S_{(4,634)}$, whose additive…
We establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. Applying this condition, we prove that the six-element additively idempotent semiring $SR_6$ has no finite basis for its identity.…
We show that the additively idempotent semiring $S_7^0$ has no finite basis for its equational theory. This answers an open problem posed by Jackson et al. (J. Algebra 611 (2022), 211--245).
We study the finite basis problem for $4$-element additively idempotent semirings whose additive reducts have two minimal elements and one coatom. Up to isomorphism, there are $112$ such algebras. We show that $106$ of them are finitely…
We study the finite basis problem for $4$-element additively idempotent semirings whose additive reducts are quasi-antichains. Up to isomorphism, there are $93$ such algebras. We show that with the exception of the semiring $S_{(4, 435)}$,…
We study the finite basis problem for 4-element additively idempotent semirings whose additive reducts are semilattices of height 1. Up to isomorphism, there are 58 such algebras. We show that 49 of them are finitely based and the remaining…
We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a $4$-element…
We study the finite basis problem for additively idempotent semirings satisfying the identity $xy \approx xz$. Let $\mathbf{R}$ denote the variety of all such semirings. Yue et al. (2025, Algebra Universalis, DOI:10.1007/s00012-025-00908-5)…
We present some general results implying nonfinite axiomatisability of many additively idempotent semirings with finitely based semigroup reducts. The smallest is a $3$-element commutative example, which we show also has \texttt{NP}-hard…
We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial…
The present paper is a continuation of \cite{jrz} and is devoted to the study of limit varieties of additively idempotent semirings. A limit variety is a nonfinitely based variety whose proper subvarieties are all finitely based. We present…
This paper establishes the existence of a finitely based finite semiring whose variety contains a continuum of subvarieties; such a variety is said to be of type \(2^{\aleph_0}\). Using the homomorphism theory of Kneser graphs, we prove…
We provide a complete classification of matrix semirings $\mathbf{M}_n(S)$ over two-element additively idempotent semirings $S$ with respect to the finite basis property.Our main theorem shows that for every integer $n \geq 2$,the semiring…
Since for the classification of finite (congruence-)simple semirings it remains to classify the additively idempotent semirings, we progress on the characterization of finite simple additively idempotent semirings as semirings of…
We prove that the interval $[\mathsf{V}(S_7),\mathsf{V}(B_2^1)]$ in the lattice of additively idempotent semiring (ai-semiring) varieties has the cardinality of the continuum,where $S_7$ is the smallest nonfinitely based ai-semiring (a…
The finite basis property is often connected with the finite rank property, which it entails. Many examples have been produced of finite rank varieties which are not finitely based. In this note, we establish a result on nilpotent…
The 5-element Brandt semigroup $B_2$ admits the structure of a naturally semilattice-ordered inverse semigroup, thus becoming an additively idempotent semiring with the operation of taking greatest lower bounds as the semiring addition. For…
For every semilattice $\mathcal{S}=(S,+)$, the set $\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the 3-element chain has no…
Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively…
In this paper, we focus on the variety $\mathbf{NF}_3$ generated by all flat semirings with $3$-nilpotent multiplicative reducts. By introducing graph semirings, we characterize all subdirectly irreducible members of $\mathbf{NF}_3$. We…