arXiv:2611.14551v1 Announce Type: new Abstract: We first show that the nonfinitely based six-element ai-semiring $SR_6$ has finitely based full matrix semirings for every $n\ge2$, with a four-identity basis relative to the ai-semiring axioms. To the best of our knowledge, this is the first known example of this phenomenon. We then construct a countably infinite flat ai-semiring $P$ whose scalar and two-by-two matrix varieties are nonfinitely based, whereas its matrix varieties in sizes $n\ge3$ admit a five-identity relative basis. For both examples, the intervals from the scalar variety to each matrix variety of size at least two have cardinality $2^{\aleph_0}$. Finally, we establish transfer results for stronger forms of nonfinite basability and a finite-basis dichotomy for matrix semirings of size at least two over scalars with a genuine zero generating a locally finite variety: finite basability in the multiplicatively nilpotent case, and inherent nonfinite basability otherwise.
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