Coauthors: E. Vinceková IDEALS IN MV PAIRS

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 correspondence 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 corresponding 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 algebra) 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 distributive 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. Jenča 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 ψD : B(D)→ D and B(D)/∼ψD is isomorphic to D ([10]). In [9], the question is solved, if we can express the morphism ψD 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 ∼ψM 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: