A many-sorted algebraic theory of answer-zero and full-zero

This paper introduces a many-sorted algebraic theory distinguishing the zero value of a distinguished answer projection from the full-zero of an object. The theory is given by a signature with three sorts: answer values, carrier values, and full objects. The signature contains a full-object formation operation, two projections, and three zero constants of the corresponding sorts. The axioms fix the projections and define the full-zero as the pair of the answer-zero and carrier-zero. It is proved that, under nontriviality of the carrier sort, there exists a full object with zero answer projection that is not equal to the full-zero. Hence the zero value of the answer projection is not sufficient for equality of an object with the full-zero. The proof uses the projection axioms, the definition of the full-zero, and the existence of a nonzero carrier value. A finite nontrivial model is constructed, showing that the axioms are jointly satisfiable.