Minimal varieties of semirings with involution (Q1866835)

S. V. Polin characterized in 1980 all the minimal varieties of semirings. By a semiring with involution is meant a semiring \((S;+,\cdot)\) with one more unary operation \(^*\) satisfying the identities \((x+y)^*=x^*+y^*\), \((xy)^*=y^*x^*\), \((x^*)^*=x\). If \(^*\) is trivial, i.e. \(x^*=x\), then minimal varieties of semirings with this involution are those of Polin's list. The author characterizes all minimal varieties of semirings with non-trivial involution. These are four varieties generated by 3- or 4-element semirings and the family of varieties generated by the rings \(\mathbb{Z}_p\) (\(p\) prime) with zero-multiplication.