AI 中文总结
研究单演逆半群作为半群时的豪森性质，除单演自由逆半群外都具有该性质，且对于此特殊情况，判定两个有限生成子半群的交是否有限生成的问题可算法求解。
AI 中文摘要
若一个代数结构中任意两个有限生成子代数的交是有限生成的，则称该结构具有豪森性质。我们在单演逆半群的背景下探讨豪森性质。已知琼斯和特罗特（1989年）以及琼斯（2016年）的工作表明，每个单演逆半群作为逆半群（即相对于其逆子半群）具有豪森性质。本文中，我们将单演逆半群视为半群，即考虑其所有子半群。我们证明除了单演自由逆半群外，每个单演逆半群在这个更广泛的意义上都具有豪森性质。对于这个特殊情况，我们表明确定两个有限生成子半群的交是否有限生成的问题是算法可判定的。
英文摘要
An algebraic structure is said to have the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. We explore the Howson property in the context of monogenic inverse semigroups. It is known, due to work of Jones and Trotter (1989) and Jones (2016), that every monogenic inverse semigroup has the Howson property considered as an inverse semigroup, i.e. with respect to its inverse subsemigroups. In this paper, we consider monogenic inverse semigroups qua semigroups, i.e. we consider all their subsemigroups. We prove that every monogenic inverse semigroup possesses the Howson property in this broader sense, with the sole exception of the monogenic free inverse semigroup. For this exceptional case, we show that the problem of determining whether the intersection of two finitely generated subsemigroups is finitely generated is algorithmically decidable.
