In the 1920s Heyting attempted at axiomatizing constructive geometry. Recently, von Plato used different concepts to axiomatize it. He used 14 axioms to formulate constructive apartness geometry, seven of which have occurrences of negation. In this paper we show with the help of ANDP, a theorem prover based on natural deduction, that four new axioms without negation, shorter and more intuitive, can replace seven of von Plato's 14 ones. Thus we obtained a near negation-free new system consisting of 11 axioms.

We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices.