This paper applies the theory of semi-coherent structures (also known as monotonic boolean functions) to the problem of linking records across databases. We use the parameter estimates from a best fitting generalized linear model to derive the equivalent semi-coherent structure function, which we then call our best fitting record linkage rule. In this paper, we first describe the application area of record linkage, followed by a description of the Fellegi-Sunter model of record linkage. We then describe how the record linkage problem can be encompassed in partial order theory, and use the theory of semi-coherent structures to develop a record linkage approach that generalizes the Fellegi-Sunter model. We develop a method for estimating an inequality constrained generalized linear model and, using the parameter estimates, determine the semi-coherent structure that best fits the training data. This best-fitting semi-coherent structure is our final estimated decision rule. We illustrate the approach with a factorial experiment with simulated data.