Effect algebras with state operator

Effect algebras have been introduced by Foulis and Bennett [2] (see also [3, 4] for equivalent definitions) for modeling unsharp measurements in quantum mechanical systems [5]. They are a generalization of many structures which arise in the axiomatization of quantum mechanics (Hilbert space effects [7]), noncommutative measure theory and probability (orthomodular lattices and posets, [6]), fuzzy measure theory and many-valued logic (MV-algebras [10, 9]). A state, as an analogue of a probability measure, is a basic notion in algebraic structures used in the quantum theories (see e.g., [8]), and properties of states have been deeply studied by many authors. In MV-algebras, states as averaging the truth value were first studied in [11]. In the last few years, the notion of a state has been studied by many experts in MV-algebras, e.g, [13, 12]. Another approach to the state theory on MV-algebras has been presented recently in [15]. Namely, a new unary operator was added to the MV-algebras structure as an internal state (or so-called state operator). MV-algebras with the added state operator are called state MValgebras. The idea is that an internal state has some properties reminiscent of states, but, while a state is a map from an MV-algebra into [0, 1], an internal state is an operator of the algebra. State MV-algebras generalize, for example, Hájek’s approach [14] to fuzzy logic with modality Pr (interpreted as probably with the following semantic interpretation: The probability of an event a is presented as the truth value of Pr(a). For a more detailed motivation of state MV-algebras and their relation to logic, see [15]. In [1], the notion of a state operator was extended from MV-algebras to the more general frame of effect algebras. A state operator is there defined as an additive, unital and idempotent operator on E. A state operator on E is called strong, if it satisfies the additional condition