Local additive measures on perfect fuzzy structures
by
Lavinia Ciungu
State University of New York at Buffalo

In this paper we define and study the local states on perfect pseudo-MV algebras. The main result consists of proving that there is a one to one correspondence between the local states on strong perfect pseudo-MV algebras and the states on l-groups. As a generalization of these states, we introduce the notion of a local additive measure on a perfect pseudo-MTL algebra and we prove that, in some conditions, a local additive measure can be extended to a Riecan state. It is given a necessary and sufficient condition for a local additive measure on a perfect pseudo-MTL algebra to be a Bosbach state.

A pseudo-MV algebra (A, ⊕, ^-, ^~ , 0, 1) is called local if it has a unique maximal ideal. A local pseudo-MV algebra A is called perfect if for any x ∈ A, ord(x) < ∞ implies ord(x^-) = ∞.
The intersection of all maximal ideals of a pseudo-MV algebra is denoted by Rad(A) and it is called the radical of A.
A local pseudo-MV algebra A is perfect iff A = Rad(A)∪Rad(A)^*.
We denote by Id(a) the ideal generated by the element a ∈ A.
If A is a pseudo-MV algebra, we denote D(A)={x ∈ A | ord(x)=∞} and D(A)^*= {x ∈ A | x ≥ y^- for some y ∈ D(A)}.
We also have D(A)^*= {x ∈ A | x ≥ y ^~ for some y ∈ D(A)}.

Definition
If A is a perfect pseudo-MV algebra, then a local state on A is a function s : Rad(A) → R₊ satisfying the conditions:
(ls₁) s(0)=0;
(ls₂) s(x ⊕y) = s(x) + s(y) for all x, y ∈ Rad(A).
If a ∈ A such that Rad(A) = Id(a) then a local state s on A is normalized if s(a) = 1. A local state s is faithful if s(x) ≠ 0 for all x ∈ Rad(A),  x ≠ 0.

Definition
A perfect pseudo-MV algebra A is called strong perfect iff x^- = x ^~   for all  x ∈ A.

Proposition
Let A be a perfect pseudo-MV algebra and s a local state on A. Then, for all x, y ∈ Rad(A) the following hold:
(1) if x ≤ y then s(y) - s(x) = s(y *x^-) = s(x ^~ *y);
(2) s(x ∨y) + s(x ∧y) = s(x) + s(y);
(3) s(x ⊕y) + s(y *x) = s(x) + s(y).

Theorem
If A is a strong perfect pseudo-MV algebra and G an l-group such that G = D(A), then there is a one to one correspondence between the local states on A and the states on G. Under this correspondence, if Rad(A) = Id(a) and u = [a, 0] is a strong unit of G, the normalized local states on A are mapped onto normalized states on G.

If (A, ∧, ∨, *, →, ~ > , 0, 1) is a pseudo-MTL algebra, we will denote:
D(A) = {x ∈ A | ord(x) = ∞} and D(A)^* = {x ∈ A | ord(x) < ∞}.
The intersection of all maximal filters of a pseudo-MTL algebra A is called the radical of A and it is denoted by Rad(A). The pseudo-MTL algebra A is called local if it has a unique maximal filter and in this case Rad(A)=D(A). A is called perfect if it is good and for any x ∈ A, ord(x) < ∞ iff ord(x^-) = ∞ iff ord(x ^~ ) = ∞ (see [1]).
If A is a perfect pseudo-MTL algebra, then A=Rad(A) ∪Rad(A)^*.
We define a binary operation ⊕ on a pseudo-MTL algebra A by
x⊕y:=(y ^~ *x ^~ )^- for all x, y ∈ A.
If A is a good pseudo-MTL algebra we say that two elements x, y ∈ A are orthogonals, denoted x   ⊥  y, if x^{- ~} ≤ y ^~ .

Lemma
Let A be a perfect pseudo-MTL algebra.
(1) if x, y ∈ Rad(A)^*, then x and y are orthogonals;
(2) if x, y ∈ Rad(A), then x and y are not orthogonals.

Let A be a pseudo-MTL algebra and X ⊆ A\{0}. An element x ∈ A is called X-zero divisor if there is y₁, y₂ ∈ X sucht that x*y₁=y₂*x=0. If 0 is the only Rad(A)-zero divisor of A, then A is called relative free of zero elements.

Definition
A Bosbach state on a pseudo-MTL algebra A is a function s:A→ [0, 1] such that the following conditions hold for all x, y ∈ A:
(bs₁) s(x)+s(x→ y)=s(y)+s(y→ x);
(bs₂) s(x)+s(x ~ > y)=s(y)+s(y ~ > x);
(bs₃) s(0)=0 and s(1)=1.

If x and y are two orthogonal elements of a pseudo-MTL algebra A, then we define a partial operation "+" on A by x+y:=x ⊕y.

Definition
Let A be a good pseudo-MTL algebra. A Riecan state or additive measure on A is a function s:A→ [0, 1] such that the following conditions hold for all x, y ∈ A :
(rs₁) if x  ⊥ y, then s(x+y)=s(x)+s(y);
(rs₂) s(1)=1.

It was proved in [2] that every Bosbach state on a pseudo-MTL algebra A is a Riecan state, but the converse is not true. Moreover, we proved in [3] that every perfect pseudo-MTL algebra admits at least a Bosbach state.
According to the previous Lemma, for all x, y ∈ Rad(A)^* we have x  ⊥ y, so the operation + is defined for all elements of Rad(A)^*.

Definition
If A is a perfect pseudo-MTL algebra, then a local additive measure on A is a function s: Rad(A)^* → [0, 1] satisfying the conditions:
(ls₁) s(x + y) = s(x) + s(y) for all x, y ∈ Rad(A)^*;
(ls₂) s(0)=0.

Examples
Let A be a perfect pseudo-MTL algebra. Then:
(1) The function s: Rad(A)^* → [0, 1], s(x)=0 for all x ∈ Rad(A)^* is a local additive measure on A;
(2) If S is a Riecan state on A, then s=S/Rad(A)^* is a local additive measure on A.

According to the previous Lemma it follows that the function s is well defined, i. e. x⊕y ∈ Rad(A)^* for all x, y ∈ Rad(A)^*.

Proposition
If s is a local additive measure on the perfect pseudo-MTL algebra A, then the following hold for all x, y ∈ Rad(A)^*:
(1) s(x^{- ~} )=s(x);
(2) s(x)+s(y^--)=s((y^-*x ^~ )^-) and  s(x)+s(y ^{~ ~} )=s((y ^~ *x^-) ^~ );
(3) s(x^--)+s((x ^~ *x^-) ^~ )=s(x ^{~ ~} )+s((x^-*x ^~ )^-);
(4) s(x) ≤ s((x^- *x ^~ )^-) and s(x) ≤ s((x ^~ *x^-) ^~ ).

If s is a local additive measure on the perfect pseudo-MTL algebra A, then we define the function s^*: Rad(A)→ [0, 1] by s^*(x)=1-s(x^-⊕x ^~ ) for all x ∈ Rad(A).

Proposition
If s is a local additive measure on the perfect pseudo-MTL algebra A, then the following hold for all x, y ∈ Rad(A):
(1) s^*(1)=1;
(2) s^*(x^{- ~} )=s^*(x);
(3) s^*(x⊕y)=1-[s(y^-*x^-) + s(y ^~ *x ^~ )];
(4) 1+s^*(x) ≤ s^*(x^--) + s^*(x ^{~ ~} );
(5) s^*(x⊕y)=s^*(x)+s^*(y)  iff  s(y^-*x^-)=s(y ^~ *x ^~ )=0;
(6) min{s(x^-), s(x ^~ )} ≤ 1/2.

Theorem (Extension theorem)
Let A be a perfect pseudo-MTL algebra relative free of zero divisors. Then every local additive measure on A can be extended to a Riecan state on A.

Theorem
Let A be a perfect pseudo-MTL algebra relative free of zero divisors. The extension of a local additive measure s on A is a Bosbach state on A if and only if s(x)=0 for all x ∈ Rad(A)^*.

References
[1] L. C. Ciungu, Some classes of pseudo-MTL algebras, Bull. Math. Soc. Sci. Math. Roumanie 3(2007), 223-247.
[2] L. C. Ciungu, Bosbach and Riecan states on residuated lattices, Journal of Applied Functional Analysis, to appear.
[3] L. C. Ciungu, On the existence of states on fuzzy structures, Southeast Asian Bulletin of Mathematics, to appear.

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Date received: January 17, 2008