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A new axiomatisation for domain and codomain on semirings and Kleene algebras is proposed. It is simpler, more general and more flexible than a predecessor, and it is particularly suitable for program analysis and construction via automated deduction. Different algebras of domain elements for distributive lattices, (co-)Heyting algebras and Boolean algebras arise by adapting this axiomatisation. Modal operators over all these domain algebras can then easily be defined. The calculus of the previous axiomatisation arises as a special case. An application in terms of a fully automated proof of a modal correspondence result for Löb’s formula is also presented.