Geometriae Dedicata 65: 355–366, 1997. 355 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Isomorphisms of Linear Semigroups Dedicated to Prof. Dr. H. M¨ aurer, on the occasion of his 60th birthday MARTIN SCHWACHH ¨ OFER and MARKUS STROPPEL Fachbereich Mathematik der TH, Schloßgartenstr. 7, D-64289 Darmstadt, Germany (Received: 4 September 1995) Abstract. A linear semigroup is a subsemigroup of the semigroup of all endomorphisms of a vector space over a (not necessarily commutative) field. In this note it is shown that every isomorphism of linear semigroups that contain all rank-one-operators is induced by a semilinear bijection of the corresponding vector spaces, unless these vector spaces have dimension 1. Mathematics Subject Classifications (1991): 20M15, 20M20. Key words: semigroups, rank-one-operators, isomorphisms, anti-isomorphisms, projective spaces. 1. Introduction This paper evolved from a part of the first author’s doctoral dissertation [9]. The authors gratefully acknowledge stimulating discussions with H. M¨ aurer, and also with K. H. Hofmann and T. Grundh¨ ofer. Throughout, let and be (not necessarily commutative) fields and let , be right vector spaces over and , respectively. The dual space of linear forms from to is then a left vector space over . We study linear semigroups, that is, subsemigroups of End ( ), the semigroup of all -linear maps from to . Of particular importance in this paper will be the rank-one-operators of ; i.e., linear maps : such that dim 1. We will denote the set of rank-one-operators of by M ( ); the subsemigroup generated by M ( ) is M ( ): M ( ) 0 . A subsemigroup of End ( ) is called a wide linear semigroup if it contains M ( ) (and, therefore, also M ( )). This notion does not depend on the given embedding in End ( ), see Lemma 2.6. Our Main Theorem 6.2 asserts that every isomorphism of wide linear semigroups is induced by a semilinear bijection between the corresponding vector spaces. In a recent paper [5], S. H. Hochwald considers the semigroup End ( ) for a vector space of finite dimension over , and the subsemigroup End ( ) (these are wide linear semigroups). He proves that every spectrum-preserving endomorphism of these semigroups is induced by a linear bijection of . In a