Monadic Effect Algebras

Abstract

The main goal of this paper is to introduce and investigate the related theory on monadic effect algebras. First, we design the axiomatic system of existential quantifiers on effect algebras and then use it to give the definition of the universal quantifier and monadic effect algebras. Then, we introduce relatively complete subalgebra and prove that there exists a one-to-one correspondence between the set of all the existential quantifiers and the set of all the relatively complete subalgebras. Moreover, we characterize and give the generated formula of monadic ideals and prove that Riesz monadic ideals and Riesz monadic congruences can be mutually induced. Finally, we study the strong existential quantifier and characterize monadic simple and monadic subdirectly irreducible effect algebras.