States on finite monoidal t-norm based algebras

The aim of this paper is to study the existences of Bosbach states and Rie?an states on finite monoidal t-norm based algebras (MTL-algebras for short). We give some examples to show that there exist MTL-algebras having no Bosbach states and Rie?an states. The conditions under which MTL-algebras have Bosbach states and Rie?an states are investigated, respectively. We prove that Rie?an states on MTL-algebras are reduced to states on IMTL-algebras. Furthermore, the necessary and sufficient conditions for finite linearly ordered locally finite MTL-algebras and peculiar MTL-algebras having Bosbach states and Rie?an states are obtained, respectively. In addition, the notions of pseudo-quasi-equivalent and a subalgebra under pseudo-quasi-equivalent are proposed and some of their properties are investigated.