发表机构
Chongqing University of Technology（重庆理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限幂零半群的完全幂半环的有限等式基，通过有限关系结构的对偶性给出判据，并建立直积判据与反例。
AI 中文摘要
我们研究有限半群（包括空集）的完全幂半环在无常量的加法与乘法签名下的有限等式基。对于非平凡的有限幂零半群，我们关联一个有限关系结构，记录非零的有序乘积。我们证明完全幂半环是有限基的当且仅当该结构具有有限对偶性，并将此条件与其一阶可定义性及其核的平方的可拆卸性联系起来。定量界将障碍大小与恒等基中所需变量数目联系起来。在幂零指数为 $d$ 的交换情形中，该判据简化为存在一个具有非零 $(d-1)$ 次幂的元素。我们还为具有二元群理想的半群建立了一个非有限基障碍，并证明了一个有限提升定理。这些结果产生了一个直积判据和一个五元素反例，表明恒等纤维条件并非充分。论证使用了等式逻辑、有限关系对偶性和显式代数构造。
英文摘要
We investigate finite equational bases for full power semirings of finite semigroups, including the empty set, in the constant-free signature with addition and multiplication. For a nontrivial finite nilpotent semigroup, we associate a finite relational structure recording the ordered products that are nonzero. We prove that the full power semiring is finitely based if and only if this structure has finite duality, and relate this condition to first-order definability and dismantling of the square of its core. Quantitative bounds connect obstruction size with the number of variables required in an identity basis. In the commutative case of nilpotency index $d$, the criterion reduces to the existence of an element with nonzero $(d-1)$st power. We also establish a nonfinite-basis obstruction for semigroups with a two-element group ideal and prove a finite lifting theorem. These results yield a direct-product criterion and a five-element counterexample to sufficiency of the identity-fibre condition. The arguments use equational logic, finite relational duality, and explicit algebraic constructions.
