Ideals in MV pairs
by
S. Pulmannova
Mathematical Institute, Slovak Academy of Sciences, Bratoslava, Slovakia
Coauthors: E. Vinceková

The concept of an MV-algebra was introduced by Chang [] 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 [, ]. Namely, MV-algebras are in one-to-one correspondence with lattice ordered effect algebras satisfying the Riesz decomposition property [], 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 [, ]. Namely, every Riesz ideal gives rise to a congruence, and if a congruence is generated by an ideal, then this ideal must be Riesz [, ], but there are congruences which are not induced by any ideal []. 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. Jenca in his recent work [] showed that when the lattice D is an MV-effect algebra, then there exists a surjective morphism of effect algebras y_D: B(D)→ D and B(D)/ ~ _yD is isomorphic to D ([]). In [], the question is solved, if we can express the morphism y_D in terms of boolean algebras only, without using the structure of effect algebra. The answer in [, 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 ~ _yM 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).

Recently, it was proved by Jenca that, given an MV-pair (B, G), the quotient B/ ~ _G, where ~ _G is an equivalence relation naturally associated with G, is an MV-algebra, and conversely, to every MV-algebra there corresponds an MV-pair.

In this paper, we study relations between congruences of B and congruences of B/ ~ _G induced by a G-invariant ideal I of B. In addition we bring some relations between ideals in MV-algebras and in the corresponding R-generated boolean algebras. Our interest turns to the MV-pair property and the question is, if having an MV-pair (B, G), we get again an MV-pair (B/I, G') for a boolean algebra ideal I. If it was the case, then by the Theorem 3.9. in [], (B/I)/ ~ _G' is again an MV-effect algebra and another question may arise - is it the same structure as if we do the process in the reversed order, that is, get an MV-effect algebra B/ ~ _G and factorize it by an ideal I/ ~ _G? We answer these questions mainly affirmatively.We also study relations between ideals in MV-algebras and in the corresponding R-generated boolean algebras. We get, for MV-effect algebras M₁, M₂ and their R-generated boolean algebras, the following commuting diagram, where f is a surjective homomorphism of MV-algebras, f^* is a surjective homomorphism of boolean algebras that extends f and y₁, y₂ are effect algebra morphisms:

Finally, we study relations between states on MV-algebras, the corresponding R-generated Boolean algebras and MV-pairs.

References

[]

A. Avallone and P. Vitolo: Congruences and Ideals of Effect Algebras, Order 20 (2003), 67-77.

[]

A. Dvurecenskij, S. Pulmannová: New Trends in Quantum Structures. Kluwer Academic Publishers, Dordrecht, 2000.

[]

G. Grätzer: General Lattice Theory. Birkhäuser, Stuttgart, 1978.

[]

C. Chang: Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc. 89(1959), 74-80.

[]

G. Chevalier, S. Pulmannová: Some Ideal Lattices in Partial Abelian Monoids and Effect Algebras. Order 17 (2000), 75-92.

[]

F. Chovanec, F. Kôpka: D-lattices Inter. J. Theor. Phys. 34 (1995), 1297-1302.

[]

D.J. Foulis and M.K. Bennett:Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325-1346.

[]

S. Gudder and S. Pulmannová: Quotients of partial abelian monoids, Algebra univers. 47 (2002), 395-421.

[]

G. Jenca: A representation theorem for MV-algebras. Soft Computing, 11(6): 557-564 (2007).

[]

G. Jenca: Boolena algebras R-generated by MV-effect algebras. Fuzzy sets and systems 145 (2004), 279-285.

[]

F. Kôpka and F. Chovanec: D-posets, Math. Slovaca 44 (1994), 21-34.

[]

S. Koppelberg: Handbook of Boolean Algebras North Holland, Amsterdam, 1989.

[]

S. Pulmannová: Congruences in partial abelian monoids, Algebra univers.37 (1997), 119-140.

PDF

Date received: March 16, 2008