Generalized pseudo-EMV-effect algebras

EMV-algebras were recently introduced in Dvurečenskij and Zahiri (Fuzzy Sets Syst, 2019. https://doi.org/10.1016/j.fss.2019.02.013) as new structures generalizing both MV-algebras and Boolean rings. These algebras do not assume that they contain a top element. We present a non-commutative generalization of EMV-algebras, called pseudo-EMV-algebras. We show how from a pseudo-EMV-algebra we can derive a generalized pseudo-EMV-effect algebra and conversely, from a generalized effect algebra with a stronger type of the Riesz decomposition property we can derive a pseudo-EMV-algebra. We show that every generalized pseudo-EMV-effect algebra without top element can be embedded into a pseudo-MV-effect algebra with top element as a maximal and normal ideal of the pseudo-MV-effect algebra.