Page 209 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
Schur null preservers
Ying-Fen Lin, y.lin@qub.ac.uk
Queen’s University Belfast, United Kingdom
Coauthor: Donal O’Cofaigh
Schur multiplicative and Schur null preserving maps have been initially studied on matrix
spaces, these are preservers with respect to the Schur product defined on matrices. We ex-
tend those results on matrices over some function spaces. In this talk, I will first introduce
two different notions of Schur null preservers, then give some characterisation results on those
operators.
Linear functions preserving Green’s relations over fields
Artem Maksaev, tematematik@gmail.com
Lomonosov Moscow State University,
Moscow Center for Fundamental and Applied Mathematics, Russian Federation
The talk is based on the joined work with A. Guterman, M. Johnson, and M. Kambites [5].
Green’s relations are a number of equivalence relations and pre-orders which are defined
upon any semigroup. Introduced by J. Green [1] in 1951, they encapsulate the ideal structure
of the semigroup, and play a central role in almost every aspect of semigroup theory. The
following are the definitions of Green’s relations R, L, J , H, D.
Definition. Let M be a monoid. For a, b ∈ M, we say that:
(i) a R b if aM = bM;
(ii) a L b if Ma = Mb;
(iii) a J b if MaM = MbM;
(iv) a H b if a R b and a L b;
(v) a D b if there exists c ∈ M : a R c and c L b.
In particular, these are natural relations to define upon the set of n × n matrices over any
semiring, when viewed as a semigroup under matrix multiplication.
The investigation of linear transformations preserving natural functions, invariants and re-
lations on matrices has a long history, dating back to a result of Frobenius [1] describing maps
which preserve the determinant. During the past century a lot of effort has been devoted to the
development of this theory, particularly, for matrices over semirings.
In 2018, motivated by recent interest in the structure of the tropical semifield, A. Guter-
man, M. Johnson, and M. Kambites [3] characterized bijective linear maps which preserve (or
strongly preserve) each of Green’s relations on the space of n × n matrices over an anti-negative
semifield. The results of [3] are rather unusual, since they provided a classification for all semi-
fields except fields. Thus, the question arises, what the corresponding maps over fields are. The
aim of this talk is to answer it.
Let F be a field and Mn(F) be the monoid of n × n matrices over F. Based on a convenient
description of Green’s relations for Mn(F), we present a complete classification of linear maps
207
Schur null preservers
Ying-Fen Lin, y.lin@qub.ac.uk
Queen’s University Belfast, United Kingdom
Coauthor: Donal O’Cofaigh
Schur multiplicative and Schur null preserving maps have been initially studied on matrix
spaces, these are preservers with respect to the Schur product defined on matrices. We ex-
tend those results on matrices over some function spaces. In this talk, I will first introduce
two different notions of Schur null preservers, then give some characterisation results on those
operators.
Linear functions preserving Green’s relations over fields
Artem Maksaev, tematematik@gmail.com
Lomonosov Moscow State University,
Moscow Center for Fundamental and Applied Mathematics, Russian Federation
The talk is based on the joined work with A. Guterman, M. Johnson, and M. Kambites [5].
Green’s relations are a number of equivalence relations and pre-orders which are defined
upon any semigroup. Introduced by J. Green [1] in 1951, they encapsulate the ideal structure
of the semigroup, and play a central role in almost every aspect of semigroup theory. The
following are the definitions of Green’s relations R, L, J , H, D.
Definition. Let M be a monoid. For a, b ∈ M, we say that:
(i) a R b if aM = bM;
(ii) a L b if Ma = Mb;
(iii) a J b if MaM = MbM;
(iv) a H b if a R b and a L b;
(v) a D b if there exists c ∈ M : a R c and c L b.
In particular, these are natural relations to define upon the set of n × n matrices over any
semiring, when viewed as a semigroup under matrix multiplication.
The investigation of linear transformations preserving natural functions, invariants and re-
lations on matrices has a long history, dating back to a result of Frobenius [1] describing maps
which preserve the determinant. During the past century a lot of effort has been devoted to the
development of this theory, particularly, for matrices over semirings.
In 2018, motivated by recent interest in the structure of the tropical semifield, A. Guter-
man, M. Johnson, and M. Kambites [3] characterized bijective linear maps which preserve (or
strongly preserve) each of Green’s relations on the space of n × n matrices over an anti-negative
semifield. The results of [3] are rather unusual, since they provided a classification for all semi-
fields except fields. Thus, the question arises, what the corresponding maps over fields are. The
aim of this talk is to answer it.
Let F be a field and Mn(F) be the monoid of n × n matrices over F. Based on a convenient
description of Green’s relations for Mn(F), we present a complete classification of linear maps
207
