Dudek, Wiesław Aleksander

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Institute of Mathematics AS CR, v. v. i.

Acta Universitatis Palackianae Olomucensis. Facultas RerumNaturalium. MathematicaWiesław Aleksander DudekOn ideals in Hilbert algebrasActa Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Vol. 38 (1999), No.1, 31--34Persistent URL: http://dml.cz/dmlcz/120398Terms of use:© Palacký University Olomouc, Faculty of Science, 1999Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czZDt/ Acta Univ. Palacki. Olomuc, Fac. rer. nat. ^ Mathematica 38 (1999) 31-34 On Ideals in Hilbert Algebras WIESLAW A. DUDEK Institute of Mathematics, Technical University, Wybrzeie Wyspianskiego 27, 50-370 Wroclaw, Poland e-mail: dudek@im.pwr.wroc.pl (Received September 30, 1998) Abstract Connections between deductive systems of Hilbert algebras and ideals introduced by I. Chajda and R. Hala§ are described. Key words: Hilbert algebra, congruence, deductive system, ideal. 1991 Mathematics Subject Classification: 06F35, 03G25 The concept of Hilbert algebra was introduced in early 50-ties by L. Henkin and T . Skolem for some investigations of implication in intuicionistic and other non-classical logics. In 60-ties, these algebras were studied especially by A. Horn and A. Diego from algebraic point of view. A. Diego proved (cf. [4]) that Hilbert algebras form a variety which is locally finite. Hilbert algebras were treated by D. Busneag (cf. [1], [2]) and Y. B. Jun (cf. [5]) and some of their filters forming deductive systems were recognized. I. Chajda and R. Halas introduced in [3] the concept of ideal in Hilbert algebra and described connections between such ideals and congruences. In this note we describe connections between such ideals and deductive systems. Since there exist various modifications of the definition of Hilbert algebra, we use that of [1]. Definition 1 A Hilbert algebra is a triplet 7t = (H; •, 1), where if is a nonempty set, • is a binary operation and 1 is fixed element of H such that the following axioms hold for each x,y,z G H: (I) X • (tj • X) = 1 (II) (x.(yz))*((x.y)^(x.z)) = l, (III) x • y = 1 and y • x = 1 imply x = y. QI 32 Wieslaw A. DUDEK The following result was proved (cf. for example [4]): Lemma 1 Let H = (i/; •, 1) be a Hilbert algebra and x,y,z G H. Then (1) x • x = 1, (2) 1 • x = x, (.?) x * l = l , (Vj x*(2/*z ) = 3/«(x«2). It is easily checked that in a Hilbert algebra H the relation < defined by x < y <=> x • y = 1 is a partial order on i / with 1 as the largest element. Lemma 2 Let H = (//"; •, 1) be a Hilbert algebra and x,y,z G i / . Tften (5,) (x • y) • y > x, (6) (yz)*(x*z)>x*y, (7) (x • y) • (x • z) > y • z. Proof (5) is a consequence of (4) and (1). (6) is proved in [4]. (7) follows from (6) and (4). • The following concept of ideal of Hilbert algebra was introduced in [3] by I. Chajda and R. Halas. Definition 2 A nonempty subset / of a Hilbert algebra H = (H; •, 1) is called an ideal of H if (0 1 6 / , (ii) x • y G / for all x G /I, y G / , (hi) (j/2 • (y\ • x)) • x G / for all x G #,3/1,3/2 € / . Observe that (i) follows from (ii) and (1). From (ii) it follows also that any ideal is a subalgebra. Moreover, putting in the above definition 3/1 = a and 3/2 = 1, we obtain (a • x) • x G / for all a G / and x e H. But if a < 3/ and a € I, then by (2) we have 3/ — l*3 / = ( a « 3 / ) * 3 / G / , which proves that the following lemma is true. Lemma 3 If I is an ideal of a Hilbert algebra H and a G / , x, 3/ G / / and a < x, £/zen x G / and (a • 3/) • 3/ G / . Definition 3 A nonempty subset Z) of a Hilbert algebra H = (I/; •, 1) is called a deductive system of % if (di) H A (cfe) x € D and x • 3/ G Z) imply y £ D. Directly from the above definition it follows Lemma 4 Le£ D be a deductive system of a Hilbert algebra H. If a G D and a < x, then x G D. On ideals in Hilbert algebras 33 T h e o r e m 1 A nonempty subset A of a Hilbert algebra ri is an ideal if and only if it is a deductive system.. Proof Let A be an ideal. Since 1 E A, then (d\) is satisfied. To prove (d2) assume a £ A and , a • x = a\ E A for some x E IF Then, by Lemma 3, a2 = (a • x) • x E A, which together with (1) and (2) gives x = 1 • x = [((a • x) • x) • ((a • x) • x)] • x = [a2 • (a\ • x)] • x E A. Thus a E A and a • x G .4 imply x E A, which proves (d2). Hence A is a deductive system. Conversely, if A is a deductive system, then 1 £ A and a • (:r • a) = x • (a • a) = a • 1 = 1 E .4, by Lemma 1. This for a E A implies x • a E A for every x £ H. Hence (i) and (ii) are satisfied. We prove (hi). Let alla2 E A. Then a2 • [(a2 • (a\ • x)) • x] = [a2 • (a\ • x)] • (a2 • x) > (a\ • x) • x > a\ by (4), (7) and (5). But, by Lemma 4, for a deductive system A we have a2 • [(a2 • (a\ • x)) • x] > a\ and a i , a 2 E A imply (a2 • (a\ • x)) • x E A, which proves (hi). Thus A is an ideal. D Let R be a binary relation defined on a set H and H = (H; •, 1) be a Hilbert algebra. R is said compatible if (a, b) £ R and (c, d) E i? imply (a *c, &•*/)€ -R. The set [l]tf = { a E H : ( a , l ) E i t } is called a kernel of ,R. Of course, a congruence on 7t is a compatible equivalence relation on if and congruence kernel is its class containing 1. Prom the results obtained in [3] it follows that the kernel of a reflexive and compatible relation defined on a Hilbert algebra is a deductive system. Moreover, if two congruences 0 and $ on a Hilbert algebra have the same kernel, then 0 = $ . This means that the kernel of every congruence on a Hilbert algebra is a deductive system and every congruence is uniquelly determined by its kernel. Hence, as a consequence of our Theorem 1 and the main result from [3], we obtain T h e o r e m 2 Every congruence on a Hilbert algebra is uniquelly determined by a deductive system. Namely, if D is a deductive system of a Hilbert algebra H. = (IF; •, 1). then the relation 0/j defined by (a,b) E QD <=> a9be D and b9ae D is a congruence ofH such that [l]eD = D. 34 Wieslaw A. DUDEK References [1] Busneag, D.: Л note on deductive systems of а Hilbert аlgebrа. Kobe. J. Math. 2 (1985), 29-35. [2] Busneag, D.: Hilbert аlgebrаs of frаctions аnd mаximаl Hilbert аlgebrаs of quotients. Kobe. J. Math. 5 (1988), 161-172. [3] Chajda, L, Halaš, R.: Congruences аnd ideаls in Hilbert аlgebrаs. Kyungpook Math. J. (in press). [4] Diego, A.: Sur les аlgébres de Hilbert. Collection de Logique Math. Ser. A (Ed. Hermann, Paris) 21 (1966), 1-52. [5] Jun , Y. B.: Deductive systems of Hilbert аlgebrаs. Math. Japon . 4 3 (1996), 51-54. 

On ideals in Hilbert algebras

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Institute of Mathematics AS CR, v. v. i.