9 papers
Strong nonfinite basability of the three-element flat semiring
Mengya Yue, Xiaolei Shao
We prove that the three-element flat semiring is strongly nonfinitely based within the class of additively idempotent semirings. More generally, let be the subva…
Matrix varieties over : dimension-three stability and continuum-sized subvariety intervals
Jun Jiao, Xiaolei Shao
We study matrix semirings over the three-element flat additively idempotent semiring with elements one, a, and infinity, where the square of a is infinity. We determine the associa…
The finite basis problem for triangular Boolean matrix semirings and incidence semirings
Jun Jiao, Xiaolei Shao
An explicit finite identity basis is given for the eight-element semiring of upper triangular Boolean \(2\times2\) matrices in the signature \((+,\cdot)\). The basis consists of a…
Finite bases and joins for semirings defined by the divisibility order
Xiaolei Shao, Zidong Gao
We study additively idempotent semirings obtained from commutative words by equipping their subwords with the divisibility order. Every finite semiring associated with a power of o…
Nonfinitely based intervals of power semiring varieties
Xiaolei Shao, Zidong Gao
We study intervals in the lattice of additively idempotent semiring varieties associated with power semirings of finite groups. We prove that every variety lying above the nonempty…
The unique limit subvariety of the max-plus variety and finite basis properties
Xiaolei Shao, Mengya Yue
We prove that the variety generated by the max-plus algebra on the nonnegative integers has a unique limit subvariety, namely the variety generated by the six-element semiring cons…
The variety generated by all additively idempotent semirings of order four
Mengya Yue, Xiaolei Shao
We prove that the variety generated by all four-element additively idempotent semirings has no finite basis for its identities.
A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety
Xiaolei Shao, Miaomiao Ren, Zidong Gao
We present an explicit infinite equational basis for the six-element additively idempotent semiring and prove that is nonfinitely based. We also give a complete descr…
The finite basis problem for the power semirings of finite groups
Zidong Gao, Miaomiao Ren, Xiaolei Shao +1
For any group , the set of all nonempty subsets of forms an additively idempotent semiring under set-theoretic union and elementwise multiplication, called the power semirin…