State-morphism pseudo-effect algebras

The notion of a state-morphism on pseudoeffect algebras is introduced, and pseudo-effect algebras with distinguished state-morphisms are studied under the name state-morphism pseudo-effect algebras (SMPEAs). It is shown that every SMPEA admits a representation as a (total) state-morphism algebra, and some results from the general theory of state-morphism algebras (that is, algebras endowed with a distinguished idempotent endomorphism called a state-morphism), recently developed by Botur and Dvurečenskij, can be applied. In particular, it is shown that under suitable conditions, a SMPEA can be embedded into a so-called diagonal one, realized by a direct product of the SMPEA with itself endowed with a suitable natural statemorphism.