ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

1. Logic

1991 MS Classification: Busneag Dumitru, University of Craiova, ROMANIA Valuations on Hilbert algebras A Hilbert algebra (Ha) is a triple (A, →,1), where A is a non set, → is a binary operation on A, 1∈ A is an Element such that the next axioms is verified for every x, y, z ∈ A :

(1) x → (y → x) = 1

(2) (x → (y → z)) → ((x → y) → (x → z)) = 1

(3) If x → y = y → x = 1, then x = y Examples 1. Classical propositional calculus, where → denotes the logical implication.

2. If (A, ≤) is a posed with 1, then (A, →,1) is a Ha, where for x, y ∈ A x → y = 1 iff x ≤ y and x → y = y if x ≤ y.

3. A Heyting algebra is a lattice A with 0 such that for every x, y ∈ A there exists an element x → y ∈ A with the property that for z ∈ A, x ∧ z ≤ y if z ≤ x → y . If (A, ∨, ∧, 0) is a Heyting algebra, then (A, →, 1) is a Ha, where 1 = a → a for an a ∈ A .

4. If (A, ∨, ∧, ⌝, 0, 1) is a Boolean lattice, then (A, →, 1) is a Ha, where for x, y ∈ A, x → y = (⌝x) ∨ y . The origin of Hilbert algebras is in Mathematical Logic . In 1923 Hilbert was the first who remarked the possibility of study a very interesting part of the classical calculus, taking as axioms only the Hilbert’ s ones concerningonly for logical implication (this field is known as ”positive implicative propositional calculus”) . If (A, →, 1) is a Ha, we define pseudo-valuation on A as rea-function v on A satisfing v(1) = 0 and v(x → y) ≥ v(y) - v(x) for any x, y ∈ A (v is called a valuation if x = 1 whenever v(x) = 0) . the aim of my communication is to prove that every pseudo-valuation (valuation) v includes a pseudo-metrics (metric) dv on A definite by dv(x, y) = v(x → y) + v(y → x) under wich → is uniformly continuos . Also, is proved many theorms of extension for pseudo-valuation .

REFERENCES 1. R. Balbes, Ph. Dwinger, Distributive lattices, University of Missouri Press, 1974.

2. D. Buşneag, Valuated Hilbert algebra, Analele Ştinţifice ale Universităţii ”Ovidius” Constanţa, vol. II, pp. 39-43, 1994 .

3. D. Buşneag, Hilbert algebras with valuations, Mathematica Japonica, 44, no. 2, pp. 285-289, 1996 .

4. D. Buşneag, Pseudo-metrics on Hibert algebras, Conference on Universal Algebra and Lattice Theory, Szeged, Hungary, July 15-19, 1996 .

5. A. Diego, Sur les Algebres de Hilbert, Collection de Logique Mathematique, Serie A, col. XXI, Gauthier-Vielars, Paris, 1966.

ICM 1998 Berlin · Short Communications and Posters