AI 中文总结
本文研究箍形代数上的余核，给出格利文科型定理，特别关注项可定义余核，明确描述任意恰当瓦伊斯伯格箍形代数簇上定义乘法余核的项，还表明相关问题在基本箍形代数或BL - 代数簇上与在某些瓦伊斯伯格箍形代数簇上等价，并给出例子。
AI 中文摘要
在偏序幺半群\(\mathbf{A}\)上，余核\(\delta\)是一个内部算子，满足对所有\(a,b\in A\)，\(\delta(a)\cdot\delta(b)\leq\delta(a\cdot b)\)且\(\delta(a)\cdot\delta(1)=\delta(a)\)。若对所有\(a,b\in A\)，\(\delta(a\cdot b)=\delta(a)\cdot\delta(b)\)成立，则余核是乘法的。本文聚焦于箍形代数（它推广了MV - 代数和BL - 代数等代数类）上余核的研究。我们给出了余核的格利文科型定理，特别关注项可定义余核。主要结果是明确描述了在任意恰当的瓦伊斯伯格箍形代数簇的每个结构上定义乘法余核的所有项。还表明在基本箍形代数或BL - 代数簇上寻找定义（乘法）余核的项的问题，等同于在某些瓦伊斯伯格箍形代数簇或某些对的簇上寻找此类项，并给出了非平凡有趣的例子。
英文摘要
A conucleus $δ$ on a partially ordered monoid $\mathbf{A}$ is an interior operator that satisfies $δ(a) \cdot δ(b) \leq δ(a \cdot b)$ and $δ(a) \cdot δ(1) = δ(a)$ for all $a,b \in A$. A conucleus is multiplicative if the equality $δ(a \cdot b) = δ(a) \cdot δ(b)$ holds for all $a,b \in A$. In this article we focus on the study of conuclei on hoops, structures which generalize well-known classes of algebras, such as the class of MV-algebras and BL-algebras. Among several results, we provide a Glivenko-type theorem for conuclei. Special emphasis is given to term definable conuclei. The main result of this article is an explicit description of all terms that define a multiplicative conucleus on every structure of an arbitrary proper variety of Wajsberg hoops. We also show that the problem of finding terms that define (multiplicative) conuclei on a variety of basic hoops or BL-algebras is equivalent to finding such terms on some variety or some pair of varieties of Wajsberg hoops. We provide nontrivial interesting examples.
