The Ordered Rectangular Band of Reflexive Generalized Inverses of a Matrix

This article studies the algebraic structure of the set of all reflexive generalized inverses of a real matrix using a sandwich-type binary operation and examines the compatibility of certain matrix order relations with this operation. The concept of generalized inverses arises when dealing with singular or rectangular matrices, where a standard inverse does not exist. The study shows that equipped with a sandwich operation,the entire set of reflexive generalized inverses of a matrix forms the structure of a particular type of semigroup, known as a rectangular band. Further, several algebraic properties of this semigroup are investigated in detail. In particular, the compatibility of certain well-known matrix order relations, namely Sussman's order and Mitsch's order, with the sandwich operation is examined. It is shown that these order relations are preserved under the defined operation, which enables the semigroup of reflexive generalized inverses to be viewed naturally as an ordered matrix semigroup. The results obtained in this study contribute to a deeper understanding of the relationship between generalized inverse theory, semigroup structures, and matrix partial orders, thereby providing a useful framework for further research on algebraic and order-theoretic properties of generalized inverses and their applications in diverse fields.