Simplified axiomatic system of DRl-semigroups

Abstract

DRl-semigroups are a generalization of lattice ordered groups (l-groups) containing a.o. Boolean algebras, Brouwerian algebras and MV-algebras. Since their introduction in the 1960s their axiomatic system has been several times reduced.In this paper, we show that it can be simplified even further. Specifically, we show that the axiom ensuring compatibility of the semigroup operation + and the lattice operations is equivalent to a significantly weaker condition of monotonicity of + with respect to the implied order <=. Because the same equivalence holds for l-groups, we observe that the axiomatic systems of l-groups and DRl-semigroups are more aligned than originally thought.