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INJECTIVE LINEAR TRANSFORMATIONS WITH EQUAL GAP AND DEFECT

Part of: Semigroups Basic linear algebra

Published online by Cambridge University Press:  18 May 2021

C. MENDES ARAÚJO Open the ORCID record for C. MENDES ARAÚJO [Opens in a new window]  and
S. MENDES-GONÇALVES Open the ORCID record for S. MENDES-GONÇALVES [Opens in a new window]
C. MENDES ARAÚJO
Affiliation:
CMAT–Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal e-mail: clmendes@math.uminho.pt
S. MENDES-GONÇALVES*
Affiliation:
CMAT–Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal
*
e-mail: smendes@math.uminho.pt
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Abstract

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.

Keywords

idealsinjective linear transformationsGreen’s relationsquasiresiduals

MSC classification

Primary: 20M20: Semigroups of transformations, etc.
Secondary: 15A04: Linear transformations, semilinear transformations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 106 - 116
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The research of the authors was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020.

References

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