{"ID":23595663,"CreatedAt":"2026-09-18T05:42:42.291333095Z","UpdatedAt":"2026-09-18T05:42:42.291333095Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19957","arxiv_id":"2609.19957","title":"Explicit equational bases for the power semirings of $S_7$","abstract":"For every semigroup $S$, the set $\\mathcal{P}(S)$ of all subsets of $S$ and the set $\\mathcal{P}^{+}(S)$ of all nonempty subsets of $S$ form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of $S$, respectively. We investigate the finite basis problem for the full and nonempty power semirings $\\mathcal{P}(S_7)$ and $\\mathcal{P}^{+}(S_7)$ of the multiplicative reduct of $S_7$, where $S_7$ is the unique nonfinitely based three-element additively idempotent semiring. We provide explicit infinite equational bases for both and prove that they are nonfinitely based. For $\\mathcal{P}^{+}(S_7)$, we establish a new sufficient condition for an additively idempotent semiring to be nonfinitely based and apply it to obtain the required result. Moreover, we show that the interval $[\\mathsf{V}(\\mathcal{P}^{+}(S_7)), \\mathsf{V}(\\mathcal{P}(S_7))]$ in the lattice of additively idempotent semiring varieties has the cardinality of the continuum.","short_abstract":"For every semigroup $S$, the set $\\mathcal{P}(S)$ of all subsets of $S$ and the set $\\mathcal{P}^{+}(S)$ of all nonempty subsets of $S$ form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of $S$, respectively. We investigate the fin...","url_abs":"https://arxiv.org/abs/2609.19957","url_pdf":"https://arxiv.org/pdf/2609.19957v1","authors":"[\"Mengya Yue\",\"Miaomiao Ren\",\"Zidong Gao\"]","published":"2026-09-17T09:28:59Z","proceeding":"math.GR","tasks":"[\"math.GR\"]","methods":"[]","has_code":false}