Extension of combinatory logic to a theory of combinatory representation

AbstractIn theoretical computer science and mathematics the models of combinatory logic are of significance in various ways. In particular within the discipline algebra, varieties are represented in the graph models of Engeler using some set of equations. In this paper, such representations of algebraic classes are presented entirely within the theory of combinatory logic. Under formal conditions this is achieved in a uniform manner by equations composed only of a few basic combinators. Some conditions are first-order logic formulae and yield the semi-universal combinatory models that provide for varieties using equations. The other, second-order logic axioms, are added to the former conditions yielding the universal combinatory models. In those models, varieties may also be presented as solutions of the equations. These results are extended to universal and algorithmic classes