Semigroup of generalized inverses of matrices Hanifa Zekraoui and Said Guedjiba Abstract. The paper is divided into two principal parts. In the first one, we give the set of generalized inverses of a matrix A a structure of a semigroup and study some algebraic properties like factorization and commutativity. We also define an equivalence relation in order to estab- lish an isomorphism between the quotient semigroup and the semigroup of projectors on R(A). In the second part, we study a relation between semigroups associated to equivalent matrices and establish a correspon- dence between the set of matrices and the set of associated semigroups. We also study some algebraic properties in this set like intersection and partial order. M.S.C. 2000: 15A09, 15A24, 20M15. Key words: Generalized inverse; equivalent matrices; projectors; partial order. 1 Introduction In the following, K represents the real or the complex field. Let A be an m × n matrix over K. The generalized inverse, or the {1}−inverse of A is an n×m matrix X over K, which satisfies the matrix equation AXA = A. If in addition X satisfies the equation XAX = X, then X is said to be a reflexive generalized inverse or {1, 2}−inverse of A. It is known that there are infinitely sets of {1}−inverses and {1, 2}−inverses of a matrix A. Many studies on generalized inverses and their applications have been done (see [1], [2], [5]). Some of them are algebraic; like the sum (see [4]) and the reverse order law of a product (see [7]). In some areas of binary matrices, the usage of the generalized inverse of a matrix might provide results as well, like the CP property for binary matrices (see [6]). In this paper we will study two principal parts. In the first one, we will give the set of {1}− inverses of a matrix A a structure of a semigroup and study some algebraic properties like factorization and commutativity. To study the property of factorization in these semigroups, we will use the Moore-Penrose inverse. We will also define an equivalence relation in order to establish an isomorphism between the quotient semigroup and the semigroup of projectors on R(A). The second part concerns the relation between semigroups associated to equivalent matrices and the Applied Sciences, Vol.12, 2010, pp. 146-152. c  Balkan Society of Geometers, Geometry Balkan Press 2010.