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Communicated by [editor] The "if-then-else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial or possibly general binary relations) that can be recombined using if-then-else. We show that this particular extension admits no finite complete axiomatization anddoi:10.1142/s0218196709005354 fatcat:5a2tfbfnnjgi5mzigymubllstm