States on Residuated Skew Lattices

Abstract

In this paper, we introduce the notion of states and state operators on residuated skew lattices and investigate some related properties of them. The relationships between state operators and states on residuated skew lattices are discussed. We prove that every Bosbach state on a residuated skew lattice is a Riecan state and with an example we show that there is a Riecan state on a residuated skew lattice which is not a Bosbach state. Also, some conditions are given for a Riecan state on residuated skew lattice to be a Bosbach state. We present different types of state residuated skew lattices, like weak, strong, simple, local and state-morphism residuated skew lattices. Any weak state-morphism residuated skew lattice is a strong state residuated skew lattice and converse is true under an extra condition.