Control categories provide a first-order framework for the denotational semantics of control structures in which programs are represented by algebraic expressions. The main innovation is the emphasis of an axiomatized case statement in terms of which program simplification has points in common with the manipulation of linear combinations in noncommutative ring theory. Assertion semantics is deduced and the Dijkstra composition rule is equivalent to determinism. Under development is a theory of canonical form based on algebraic reduction, matrix similarity invariants and equational and first-order characterization. The thesis is that lines of attack familiar to algebraists find new applications in theoretical computer science. Module-theoretic ideas applied to a simplified model express conditional statements in terms of guard action, leading to a canonical form expressed as a linear combination. A first-order axiomatization of iteration discovered with Arbib in 1978 emphasizes how iteration transforms and this leads to a proposed transformation theory analogous to techniques used to solve linear differential equations. The P.I. is a highly respected contributor to mathematical aspects of computer science. The proposed work is expected to make a useful contribution to computer science especially as it expands the understanding of the concept of control categories as introduced recently by the P.I. JUSTIFICATION
