ManyVal ’08 - Applications of Topological Dualities to Measure Theory in Algebraic Many- Valued Logic, May 19–21, 2008, University of Milan, Milan, Italy Coauthors: E. Vincekov´a IDEALS IN MV PAIRS S. PULMANNOVA The concept of an MV-algebra was introduced by Chang [4] as an algebraic basis for many-valued logic. It turned out that MV-algebras are a subclass of a more general class of effect algebras [7, 6]. Namely, MV-algebras are in one-to-one cor- respondence with lattice ordered effect algebras satisfying the Riesz decomposition property [2], the latter are called MV-effect algebras. In the study of congruences and quotients of effect algebras, a crucial role is played by so-called Riesz ideals [13, 8]. Namely, every Riesz ideal gives rise to a congruence, and if a congruence is generated by an ideal, then this ideal must be Riesz [8, 5], but there are congruences which are not induced by any ideal [1]. In effect algebras satisfying Riesz decomposition properties (and boolean algebras as well as MV-algebras belong to this class), every ideal is a Riesz ideal. In addition, every (effect algebra) ideal in MV-algebras is an MV-algebra ideal, and the corre- sponding congruence is an MV-algebra congruence, in particular, the quotient is an MV-algebra. Similar situation is in boolean algebras. On the other hand, not every effect algebra congruence in the latter structures is an MV-algebra (boolean alge- bra) congruence. It is well-known that every congruence on effect algebras preserves the Riesz decomposition properties, but not necessarily the lattice structure. An important relation between MV-algebras and boolean algebras is obtained taking into account that every MV-algebra admits a structure of a bounded distribu- tive lattice. Namely, let us now recall the concept of a boolean algebra R-generated by a bounded distributive lattice D. We say that D R-generates a boolean algebra B(D) iff it is its 0,1-sublattice and generates it as a boolean algebra. G. Jenˇ ca in his recent work [10] showed that when the lattice D is an MV-effect algebra, then there exists a surjective morphism of effect algebras ψ D : B(D) → D and B(D)/ ∼ ψ D is isomorphic to D ([10]). In [9], the question is solved, if we can express the morphism ψ D in terms of boolean algebras only, without using the structure of effect algebra. The answer in [9, Th. 4.1, Th. 3.9] says, that for every MV-effect algebra M , there exists a group G(M ) (subgroup of the automorphism group of B(M )) such that an equivalence relation on B(M ) associated with G(M ) equals ∼ ψ M and vice versa, under some special conditions on the group G, a pair (B,G) (BG-pair), produces an MV-effect algebra B/ ∼ G . The condition, or the special property inflicted on G, is that the BG-pair must be a so called MV-pair. Namely, a BG-pair (B,G) is called an MV-pair iff the following conditions are satisfied: (MVP1) For all a,b ∈ B,f ∈ G such that a ≤ b and f (a) ≤ b, there is h ∈ G such that h(a)= f (a) and h(b)= b. (MVP2) For all a,b ∈ B and x ∈ L(a,b), there exists m ∈ max(L(a,b)) with m ≥ x, where L(a,b)= {a∧f (b): f ∈ G} and max(L(a,b)) is the set of all maximal elements in L(a,b). 1