Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$ , we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $ , and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc.79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

中文翻译：

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具有相等间隙和缺陷的内射线性变换

让五是场上的无限维向量空间F然后让$I(V)$是所有单射偏线性变换的反半群五. 给定$\alpha \in I(V)$，我们表示域和范围$\阿尔法$经过${\mathop {\textrm {dom}}}\,\alpha $和${\mathop {\textrm {im}}}\,\alpha $, 我们称红衣主教$g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $和$d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $的“差距”和“缺陷”$\阿尔法$. 我们研究半群$A(V)$具有相等间隙和缺陷的所有单射偏线性变换，并刻画格林的关系和理想$A(V)$. 这类似于 Sanwong 和 Sullivan 的工作 ['具有相等间隙和缺陷的内射变换'，公牛。澳大利亚。数学。社会党。79(2009), 327–336] 对集合情况下类似定义的半群，但我们证明这些半群绝不是同构的。