Boolean algebras R-generated by MV-effect algebras

We prove that every MV-effect algebra M is, as an effect algebra, a homomorphic image of its R-generated Boolean algebra. We characterize central elements of M in terms of the constructed homomorphism. 1. Definitions and basic relationships An effect algebra is a partial algebra (E;⊕, 0, 1) with a binary partial operation ⊕ and two nullary operations 0, 1 satisfying the following conditions. (E1) If a⊕ b is defined, then b⊕ a is defined and a⊕ b = b⊕ a. (E2) If a ⊕ b and (a ⊕ b) ⊕ c are defined, then b ⊕ c and a ⊕ (b ⊕ c) are defined and (a⊕ b)⊕ c = a⊕ (b⊕ c). (E3) For every a ∈ E there is a unique a′ ∈ E such that a⊕ a′ = 1. (E4) If a⊕ 1 exists, then a = 0 Effect algebras were introduced by Foulis and Bennett in their paper [5]. Independently, Kôpka and Chovanec introduced an essentially equivalent structure called D-poset (see [10]). Another equivalent structure, called weak orthoalgebras was introduced by Giuntini and Greuling in [6]. We refer to [4] for more information on effect algebras and similar algebraic structures. For brevity, we denote an effect algebra (E;⊕, 0, 1) by E. In an effect algebra E, we write a ≤ b iff there is c ∈ E such that a⊕ c = b. It is easy to check that every effect algebra is cancellative, thus ≤ is a partial order on E. In this partial order, 0 is the least and 1 is the greatest element of E. Moreover, it is possible to introduce a new partial operation ; b a is defined iff a ≤ b and then a⊕ (b a) = b. It can be proved that a⊕ b is defined iff a ≤ b′ iff b ≤ a′. Therefore, it is usual to denote the domain of ⊕ by ⊥. If a ⊥ b, we say that a and b are orthogonal. Let E1, E2 be effect algebras. A mapping φ : E1 7→ E2 is called a homomorphism iff φ(1) = 1 and a ⊥ b implies that φ(a) ⊥ φ(b) and then φ(a⊕ b) = φ(a)⊕ φ(b). A homomorphism φ is an isomorphism iff φ is bijective and φ−1 is a homomorphism. Note that even if both E1 and E2 are lattice ordered, a homomorphism need not to preserve joins and meets. An MV-algebra (c.f. [2], [12]) is a (2, 1, 0)-type algebra (M ; ,¬, 0), such that satisfying the identities (x y) z = x (y z), x z = y x, x 0 = x, ¬¬x = x, x ¬0 = ¬0 and x ¬(x ¬y) = y ¬(y ¬x). 1991 Mathematics Subject Classification. Primary 06C15; Secondary 03G12,81P10.