Michailov, M. M.

Pinus, Alexander G.

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

Acta Universitatis Palackianae Olomucensis. Facultas RerumNaturalium. MathematicaAlexander G. Pinus; M. M. MichailovOn the approximation of the Boolean powers by Cartesian powersActa Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Vol. 33 (1994), No.1, 105--108Persistent URL: http://dml.cz/dmlcz/120303Terms of use:© Palacký University Olomouc, Faculty of Science, 1994Institute 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.czACTA UNIVERSITATIS PALACKIANAE OLOMUCENSIS FACULTAS RERUM NATURALIUM 1994 Matheraat ica XXXIII Vo l . 114 ON THE APPROXIMATION OF THE BOOLEAN POWERS BY CARTESIAN POWERS A. G. PINUS AND M. M. MICHAILOV (Received December 12, 1993) Abstract On the base of the Malcev — Cleave theorem on quasi-universal for-mulas it is proved, that a number of properties of congruences, tolerances, quasiorders can be transported from finite cartesian powers of algebraic systems on any boolean powers. Key words: boolean power, congruence property, second order for-mula MS Classification: 08A05, 08A30 The concept of boolean power (the restricted boolean power) is introduced in the paper by Foster [1] and seemed to be useful enough in studying of some problems in the universal algebra and model theory. The survey of results, which are connected with the construction of boolean power, can be found in [2], [3]. This note is devoted to some observation', which are connected with the question of approximation of boolean power properties by cartesian power properties of algebraic systems. Recal that if 21 is some algebraic system, 03 is a boolean algebra, then the boolean power 21 ̂  is the subsystem of direct power 2193 with the underlying set C(03*,2l). Here 03* is the Stone space of the boolean algebra 03, and C(03*,2l) is a set of continuous maps of the space 03* into the system 21, equiped by the discrete topology. As the space 21 is discrete and the space 03* is compact, then the elements from C(03*,21) are exactly functions / G 21 . More precisely, for any finite subset {a\,.. ., an} C 21 and any finite subset { 6 1 , . . . , bn} C 03 which is the partition of unity of the boolean algebra 03 the equality f~1(ai) == b; holds if elements b2- are identified with the corresponding clopen subsets of the space 03*. If is clear that if 0$ is a finite boolean algebra with n atoms, then 105 21® ~ 21n. A natural question arises, how far the properties of the cartesian powers 21n influent the properties of arbitrary boolean powers 21 . The concept of congruence-boolean power 21 (0) , where 0 is a congruence of an algebraic system 21 , was introduced in [4]. Here 21 (0) is a subsystem of the system 21 with the underlying set { /€2 t® | ( / ( * ) , / ( j ) ) G 0 for any i,j £%*}. By analogy, 2in(0) is a subsystem of the system 2in with the underlying set {/ G 2 t" | (f(i),f(j)) G 6 for any i,j G n}, moreover, if 03 is a finite boolean algebra with n atoms, then 21 (0) ~ 2l n (0) . In the papers by A. I. Malcev [5] and L P . Cleave [6] there was introduced the notion of quasi-universal formula. A formula of the second order predicate calculus in prenex normal form is said to be object-universal if there is no occurences of the quantifier 3 referring to an object variable. A formula of the second order predicate calculus is quasi-universal if it is obtained from object -universal formulas without free object variables, by first combining them using connectives only, and then using the universal quantifier V to bind every predicate-variable. A family 21z- (i £ I) of subsystems of an algebraic system 21 is said to form a local covering of 21, if for any a £ 21 there exists i £ I such that a £ 21z-, and if for each pair 212-,21y (z,j £ I) there is a third 21& (k £ I) containing both of them. In [5], [6] (see also [7]) it is proved the following theorem. Theorem 1 If a quasi-universal formula is true on subsystems 212- (i £ I) locally covering an algebraic system 21, then <p is also true on 21. Proposit ion 1 For every quasi-universal formula <p, and each algebraic system 21. if for any n £ w 2ln \= p, then for any boolean algebra 03 21® f= p. Analogously, if 0 is a congruence on 21 and 2tn(0) |= p for any n £ u, then 21 (0) |= <p for every boolean algebra 03. Proo f It is sufficient to show that for any boolean algebra S there exists a local covering of the system 21® by its subsystems which are isomorphic to the systems 2in (n Ew) . For any partition {bi, . . . . bn} of the unit of the boolean algebra 03 we put 2l< B(b1 , . . . ,bn) = {f£21® | for some au...,an £21, f~l(ai)2h, for i < n} . Obviously, the family of subsystems 21® (bi,. . ., bn), where {bi,. . ., bn} is any partition of the unit of the algebra 03 is a local covering of the system 21®. Moreover, 21® (b ! , . . . ,b n ) ^21n . Thus the proposition follows from the result of A. I. Malcev and I. P. Cleave. The analogy can be proved for congruence-boolean powers. 106 Let Con'2l be the lattice of congruences of an algebraic system 21 which is equipped by the partial operation o — the product of relations. By analogy the lattices Tol'21, Qord'21 of tolerances and quasiorders on system 21 are defined, which are also enriched by the partial operation o (on the Tol2i and Qord2l see, as example, [8], [9]). Corollary 1 If cp is a umversal formula of the signature (V,A,o) and if for some algebraic system 21 of a finite signature Con/(2ln) |= cp (Tol/(2ln) |= cp, Qord /(2ln) h (p) for any n £ u, then Con/(2l<B) |= (p (Tol/(2l<B) |= cp, Qord'(2l ) \= cp) for any boolean algebra 03. Proo f The proposition implies that it is sufficient to show that for any uni-versal formula cp of the signature (V,A,o) there exists a quasi-universal for-mula a^p {Pep) Jip) such that for any algebraic system 21 of a finite signature Con'(21) f= <p (Tol/(2l) |= <p, Qord/(2l) f= cp) iff 21 \= a^ (21 |= /^ ,2 l (= T v ) . Therefore, let us note that the property of a binary relation 0 C 21 to be a congruence (tolerance, quasiorder) on the algebraic system 21 of some fixed finite signature is expressible in an object — universal formula, as well as in relations 6 i A 02 = 03 , 0 i V 02 = 03 , 0 i o 0 2 = 03 for congruences (tolerances, quasiorders) are expressable. In particular, from Corollary 1 it follows that if for some algebraic system 21 and any n £ LO the algebraic system 2in is congruence-modular, congruence-distributive, (congruence-permutable, congruence-n-permutable, have Fraser-Horn property) then for any boolean algebra 03 the system 21 also has the corresponding property. An analogous statement remains true if we replace congruence-properties by corresponding tolerance and quasi-order-properties. Here the Fraser-Horn property for congruences (tolerances, quasiorders) on 21 means, that for any systems 2li,2l2 such that 21 = 211 x 2I2 and 0 £ Con 21 there exist 0 i £ Con2ii, 02 £ Con2l2 such that 0 = 0 i x 0 2 (identifying 21 and 2li x 212). In the same way it can be shown that if for any n £ UJ the system 2ln is a-congruence regular (tolerance-trivial, a-trivial), then for any boolean algebra 03 the system 21 is a-congruence regular (tolerant-trivial, g-trivial), where a is some constant of an algebraic system 21. The variety DJl has the Fraser-Horn congruence-property if any OTl-algebra has this property. It is known that any congruence-distributive variety, any variety of rings with a unit and so on, has the Fraser-Horn congruence-property (on the varieties with Fraser-Horn congruence-property, see e.g. [10]). Corollary 2 Let cp be a universal Horn formula, let 21 be an algebraic system from a variety with the Fraser-Horn congruence-property and let Con(2l) |= cp, then Con(2t ) |= cp for any boolean algebra 03. Proo f This statement follows from Corollary 1 and the fact that the Fraser-Horn congruence-property implies that Con(21n) = (Cori2l)n. 107 References [1] Foster, A. L.: Functюnal completeness in the small. Algebraic structure theorems and identities. Math. Ann., v. 143, N 1, 1961, 29-53. [2] Burris, S.: Boolean powers. Alg. Univ., v. 5, N 3, 1975, 341-360. [3] Pinus, A. G.: Boolean constructions in universal algebra. Kluwer Academ. Publish., Dordrecht-Boston-London, 1993. [4] Pinus, A. G.: On the covers in epimorphism skeletons of varгeties of algebras. Algebra and logic, v. 27, N 3, 1988, 316-326, (in Russian). [5] Malcev, A. I.: The model correspondences. Izvestia AN SSSR, ser. math., v. 23, N 3, 1959, 313-336, (in Russian). [6] Cleave, I. P.: Local propertгes of systems. J. London Math. Soc. (1), v. 44, N 1, 1969, 121-130, Addendum., J. London. Math. Soc. (2), v. 1, N 2, 1969, p. 384. [7] Kargapolov, M. I., Merzljakov, Ju. I.: Fundaments ofthe theory of groups. Nauka, Moscow, 1972, (in Russian). [8] Chajda, I.: Algebraic thevry of tolerance relations. UP Olomouc, Mono-graphy series., 1991. [9] Pinus, A. G., Chajda, I.: Quasiorders on the universal algebras. Algebra and logic, v. 32, N 3, 1993, (in Russian). [10] Fraser, G. A., Horn, A.: Congruence relations in direct products. Proc. Amer. Math. Soc, v. 26, N 2, 1970, 390-394. Author's address: Department of Mathematics University of Novosibirsk Novosibirsk Russia 108 

English

On the approximation of the Boolean powers by Cartesian powers

Czech digital mathematics library

http://dml.cz/bitstream/handle/10338.dmlcz/120303/ActaOlom_33-1994-1_11.pdf

On the approximation of the Boolean powers by Cartesian powers

Abstract

Similar works

Full text

Available Versions

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

Czech digital mathematics library