EXISTENCE OF STATES ON FUZZY STRUCTURES

In this paper we will investigate the existence of states on residuated lattices, pseudo-MTL algebras and pseudo-BL algebras. The main result of the paper consists of proving that any perfect pseudo-BL algebra admits a unique state. We also prove that there exist linear residuated lattices and linear pseudo-MTL algebras having no states. Similarly as in [4] we introduce the notions of Bosbach state and Riec̆an state on a residuated lattice. Obviously, all the results proved for residuated lattices are also valid in the case of pseudo-MTL algebras. Definition 1.1. A Bosbach state on a residuated lattice A is a function s: A → [0, 1] such that the following conditions hold for any 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. Example 1.2. ([5] Consider the set A = {0, a, b, c, 1} with 0 < a < b < c < 1 and the pseudo-MTL algebra A = (A,∧,∨,,→, , 0, 1) where the operations ,→, are given by the following tables: 0 a b c 1 0 0 0 0 0 0 a 0 a a a a b 0 a a b b c 0 a a c c 1 0 a b c 1 → 0 a b c 1 0 1 1 1 1 1 a 0 1 1 1 1 b 0 c 1 1 1 c 0 a b 1 1 1 0 a b c 1 0 a b c 1 0 1 1 1 1 1 a 0 1 1 1 1 b 0 b 1 1 1 c 0 b b 1 1 1 0 a b c