The radical of a perfect residuated structure

In this paper we extend some properties of the radical of an MTL-algebra to the non-commutative case of a more general residuated structure, namely the image-algebra. For the particular case of pseudo-MTL algebras, some specific results are presented. We introduce the notion of a local additive measure on a perfect pseudo-MTL algebra and we prove that, with some additional conditions, every local additive measure can be extended to a Riečan state; a necessary and sufficient condition is given for such an extension to be a Bosbach state.