Boolean Algebras with Operators

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<ul><li><p>Boolean Algebras with OperatorsAuthor(s): Bjarni Jonnson and Alfred TarskiSource: American Journal of Mathematics, Vol. 74, No. 1 (Jan., 1952), pp. 127-162Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2372074 .Accessed: 09/12/2014 00:52</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms &amp; Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp</p><p> .JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.</p><p> .</p><p>The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.</p><p>http://www.jstor.org </p><p>This content downloaded from 138.251.14.35 on Tue, 9 Dec 2014 00:52:29 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/action/showPublisher?publisherCode=jhuphttp://www.jstor.org/stable/2372074?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>BOOLEAN ALGEBRAS WITH OPERATORS.* </p><p>By BJARNI J0NNSON and ALFRED TARSKI. </p><p>PART II ** </p><p>Section 4. </p><p>Representation Theorems for Relation Algebras. </p><p>Relation algebras are abstract algebraic systems characterized by means of a number of simple postulates which prove to be satisfied if we take the elements of the algebra to be binary relations, and the fundamental operations of this algebra to be set-theoretical operations of addition and multiplicatioli together with relative multiplication and conversion. The relation algebra actually formed by binary relations and the operations just mentioned will be referred to as proper relation algebras. The natural representation problem for relation algebras is the problem whether every relation algebra is isomorphic to a proper relation algebra. It has recently been shown that in general the solution of this problem is negative.14 On the other hand, we shall see in Theorem 4. 22 that every relation algebra has at least a "weak" natural representation in which all the operations except the Boolean multiplication have their natural meaning. In some further theorems, e. g., 4. 29 and 4. 32, we shall obtain a positive solution of the natural representation problem for certain classes of relation algebras which, however, are of a rather special nature. </p><p>When studying abstract relation alegbras it is useful to bear in miind that various notions of the general theory of these algebras take on a familiar meaning and various results can easily be anticipated when applied to proper relation algebras. </p><p>DEFINITION 4. 1. An algebra </p><p>r /A ._ </p></li><li><p>128 BJARNI JONSSON AND ALFRED TARSKI. </p><p>(where [, , and ; are operations on A2 to A, i is an operation on A to A, and 0, 1, and 1' are elements of A) is called a relation algebra if the following conditions are satisfied: </p><p>(i) is a Boolean algebra. </p><p>(ii) (x;y);z =x; (y;z) for any x,y,zeA. (iii) 1';x= x ==x;' for every xeA. (iv) The formulas (x ; y) z = 0, (x-J; z) y = 0, and (z; y-') x = 0 are </p><p>equivalent for any x, y, z e A. </p><p>The operation ; is referred to as relative multiplication, the operation as conversion, and the element 1' as the identity element.'5 </p><p>In view of condition (ii) of this definition we shall in general, when speaking of relation algebras, omit parentheses in expressions like </p><p>(x; y); z and x;(y; z). </p><p>Condition 4. 1 (iv) plays a fundamental role in the theory of relation algebras. It is useful to notice in this connection the following </p><p>THEOREM 4. 2. In Definition 4. 1 condition (iv) can be replaced by the following one: </p><p>(iv') Given any element a ? A, the funzctions f and g defined by the formulas </p><p>f (x) =a;x and g(x) ==a-J;x for every xeA </p><p>are conjugate, and so are the functions f' and g' defined by the formulas </p><p>f'(x) ==x;a and g'(x) =x;aa-' for every xcA. </p><p>Proof. By 1. 11 and 4. 1. </p><p>We shall not develop here either the arithmetic of relation algebras or the proper algebraic theory of these algebras (the study of isomorphisms, homomorphisms, subalgebras, etc.) -except insofar as it is relevant for the main purposes of our discussion. Some arithmetical consequences of 4. 1 are stated in the next theorem. </p><p>THEOREM 4. 3. For any relation algebra </p><p>W ~ </p><p>15 The axiom system 4. 1 (i) - (iv) is equivalent to the one given in J6nsson-Tarski [2]. For the proof of the equivalence of the two systems see Chin-Tarski [1], Theorem 2. 2; for the relation of these systems to the axiom system in Tarski [2] see Chin-Tarski [1], footnote 10. </p><p>This content downloaded from 138.251.14.35 on Tue, 9 Dec 2014 00:52:29 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>BOOLEAN ALGEBRAS WITH OPERATORS. 129 </p><p>we have: </p><p>(i) If x ? A, I is an arbitrary set, and the elements yi e A with i e I are such that 2 y, cA, then </p><p>'Y (x ; y,) F- A, 'Y (y,; x) ? A, iel i 8 </p><p>x;( yi) E (x;yj), antd ( yj);x= (yi;x). iel iel iel iel </p><p>(ii) If x, x, y, y' ? A. x &lt; x', and y y', then x; y &lt; x'; y'. (iii) x; O O O; x for every x e A. </p><p>(iv) x &lt; x; 1 andx &lt; 1;x for every xeA. </p><p>(v) (x + y) -= xU- + y-, (x * y) == x- * y-, and (x; y) = y;- xv for any x,yeA. </p><p>(vi) If x, yeA and x ? y, then x- ?&lt; y'. (vii) x-xJJ - x and x-' x- xkJ for every x ? A. </p><p>(viii) (x; y) * z ? x; xc ;z for any x, y, z ? A. </p><p>(ix) x x; x-; x for every x ? A. </p><p>(x) x-; (x;y)-?y-for any x,yeA. (xi) O0J =O, 1PJ1, and (1')J= 1'. (xii) If x e A is an atom, then x-' is an atom. </p><p>Proof. The proof of parts (i) - (xi) of this theorem can be found elsewhere 16; (xii) obviously follows from (vi), (vii), and (xi). </p><p>THEOREMI 4. 4. Every relation algebra is a normal Boolean algebra with operators. </p><p>Proof. By 2. 13, 4. 1, and 4. 3 (i) (iii) (v) (xi). </p><p>Some further arithmetical notions applying to relation algebras are introduced in the following. </p><p>DEFINITION 4. 5. Let </p><p>% = </p></li><li><p>130 BJARNI JONSSON AND ALFRED TARSKI. </p><p>(iii) a functional element if x-J ; x &lt; 1', (iv) an ideal element if x = 1 ; x ; 1. </p><p>The reasons why we have chosen the terms introduced in 4. 5(ii)(iii)(iv) will become clear in our further discussion (see Theorem 4. 24 and remarks preceding Definition 4. 8). </p><p>THEOREM 4. 6. Given a relation algebra </p><p>we have </p><p>(i WY') =-- O'. </p><p>(ii) 0';0';0';0' 0' ;0'. </p><p>(iii) For every x ? A the following three conditions are equivalent: x is an equivalence element, x ; x = x = xx-', and x-';x ; x. </p><p>(iv) If x, y ? A, x is a functional element, and y ? x, then y is a func- tional element. </p><p>(v) If x, y, z ? A, and x is a futnctional element, then </p><p>x ;(y * z) = (x ;y) * (x ;z) and (y * z) ;x--'= (y ; x) * (z ;x-'). </p><p>(vi) If x e A and x &lt; 1', then x is both an equivalenice element and a functional element. </p><p>(vii) If x, y ? A, x is an atom, and y is a functionial element, then x ; y = 0 or x ; y is an atom. </p><p>(viii) If x, y ? A are idecl elements, then x + y, x y, and x- are ideal elements. </p><p>(ix) If x, y A are ideal elements, then x ;y x y and x- '=x. (x) If x, y, z ? A and x is an ideal element, then </p><p>x (y; z) = (x y) ; (x * z). </p><p>(xi) 0 and 1 are ideal elements. (xii) 1; x ; 1 is an ideal element for every x ? A. (xiii) x c A is an ideal element if, and only if, x and x- are equivalence </p><p>elements. </p><p>Proof. The proof of all parts of this theorem, except (vii), can be found elsewhere.17 To prove (vii), suppose x is an atom and y is a fune- </p><p>17 See Chin-Tarski [1], ? 3. </p><p>This content downloaded from 138.251.14.35 on Tue, 9 Dec 2014 00:52:29 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>BOOLEAN ALGEBRAS WITIH OPERATORS. 131 </p><p>tional eleinent. Consider any element z # 0 such that z &lt; x ; y. Then (x ;y) * z 0 and hence, by 4. 1(iv), (z ;yJ) *x &amp; 0. Since x is an atom, this implies that x &lt; z ; y-J. Hence, by 4. 1 (ii) (iii), 4. 3 (ii), and 4. 5 (iii), </p><p>x ; y ? z ; yJ ; y ? z ; 1' =z, </p><p>and therefore z = x ; y. Since this is true for every z #&amp; 0 with z &lt; x ; y, we obtain the conclusion of (vii) at once. </p><p>We shall now establish some facts belonging to the proper algebraic theory of relation algebras. </p><p>THEOREMi 4. 7. (i) A homomorphic tmage of a relation algebra is agatn a relation algebra. </p><p>(ii) A cardinal product of relation algebras is again a relation algebra. </p><p>Proof. The corresponding theorem for Boolean algebras is well known. Hence and from the form of conditions (ii), (iii), and (iv) of 4. 1 we see directly that 4. 7(ii) holds; to obtain 4. 7(i) we notice in addition that, bv 1. 15 (i) (iii) and 4. 2, condition 4. 1 (iv) can be equivalently replaced by a system of equations. </p><p>By an ideal in the relation algebra </p><p>% </p></li><li><p>132 BJARNI JONSSON AND ALFRED TARSKI. </p><p>to the principal ideal in 5 consisting of all elements x ? I with x C_ a; if a is not in I, the set of all x ? A with x - a is not an ideal in W. Without using the notion of an ideal explicitly, we give a few theorems, 4. 9-4. 14, which are suggested by the above remarks and by the knowledge of analogous results applying to Boolean algebras. </p><p>DEFINITION 4. 8. Let </p><p>% = </p><p>be a relation algebra. Let a be any element in A, and B be the set of all elements x ? A such that x C a. Then the system </p><p>will be denoted by 2(a). </p><p>THEOREM 4. 9. If </p><p>% = </p></li><li><p>BOOLEAN ALGEBRAS WITH OPERATORS. 133 </p><p>be a relation algebra in which 0 # 1. Then the following conditions are eqwivalent: </p><p>(i) i is simple. (ii) I has no ideal elements different from 0 and 1. (iii) For every x eA, x 0 implies 1 ;x; = 1. (iv) For any x,yeA, x; 1 ;y O implies thlat x O or y O.19 </p><p>Proof. By 4. 9, (i) implies (ii). By 4. 3(iv) and 4. 6(xii), (ii) implies (iii). Now assume that (iii) holds. Let 0 be a homomorphism mapping A onto another algebra </p><p>T = </p></li><li><p>134 BJARNI JONSSON AND ALFRED TARSKI. </p><p>be a relation algebra. In order that I be isomorphic to the cardinal product of the algebras , it is necessary and sufficient that there exist elements at E A satisfying the following conditions: </p><p>(i) Eai 1 i e I </p><p>(ii) If i, EI and i -/ j, then ai - aj 0. (iii) E (xi * ai) E A for any elements xi E A correlated with i E I. </p><p>Ie </p><p>(iv) ai is an ideal element for every i E I. </p><p>(v) I (a,) _ Ti for every i E I. </p><p>(Condition (iii) is automatically satisfied in case the set I is finite.) </p><p>Proof. Let = </p></li><li><p>BOOLEAN ALGEBRAS WITH OPERATORS. 135 </p><p>%I(a) -- T and %f(a-) C. </p><p>Proof. By 4. 6 (viii) and 4. 12. </p><p>THEOREM 4. 14. A relation algebra is simple if, and only if, it is indecomposable. </p><p>Proof. By 4. 8, 4. 10, and 4. 13 a relation algebra which is not simple cannot be indecomposable. The converse is known to hold for every algebra. </p><p>THEOREM 4. 15. For every relationt algebra ?f there exist simple relation algebras , correlated with elements i of a set I such that W is isomorphic to a subalgebra of the cardinal product of the algebras , and that, for every i 1, Ti is a homomorphic image of W. </p><p>Proof. If we replace in 4. 15 "relation algebras" by "algebras " and "simple " by "indecomposable," we obtain a statement which holds for arbitrary algebras and is a direct consequence of a result known from the literature.20 Hence, by restricting ourselves to relationi algebras and applying 4. 7(i) and 4. 14, we obtain 4. 15 at once. </p><p>It may be menitioned that relation algebras and specifically simple relation algebras are closely related to cylindric algebras discussed in the preceding section. In fact, if </p><p>A ~</p></li><li><p>136 BJARNI JONSSON AND ALFRED TARSKI. </p><p>DEFINITION 4. 16. A relation algebra </p><p>% </p></li><li><p>BOOLEAN ALGEBRAS WITH OPERATOIRS. 137 </p><p>4. 1 (iii) and 4. 3 (vii) (xi), y = 0. Consequently, by 4. 16, (iv) implies (i). Thus, conditions (i) - (iv) are equivalent. </p><p>From 4. 3 (v) (vii) (xi) it is easily seen that formula x ; 1 1 in 4. 17 (ii) can be replaced by I ; x = 1. </p><p>THEOREM 4. 18. (i) Every relation algebra which is a subalgebra of an integral relation algebra, is itself integral. </p><p>(ii) Every integral relation algebra is simple. </p><p>Proof. (i) immediately follows from 4.16. To obtain (ii) we notice that, by 4. 3 (ii) (iv) and 4. 16, condition 4. 17 (i) directly implies 4. 10 (iv). (Similarly, 4. 17 (ii) directly implies 4. 10 (iii).) </p><p>THEOREM 4. 19. For every relation algebra </p><p>% -- </p></li><li><p>138 BJARNI JONSSON AND ALFRED TARSKI. </p><p>We now turn to the main subject of this section-representation theorems for relation algebras. We begin with the extension theorem: </p><p>THEOREMi 4. 21. For every relation algebra e there is a complete and atomistic relation algebra A which is a perfect extension of S. If e3 is simple, or integral, the same applies to A. </p><p>Proof. By 2. 15 and 4. 4 there is a complete and atomistic Boolean algebra with operators A which is a perfect extension of S. From 2. 11 (i), 2. 18, and 4. 1 it follows that A is a relation algebra. The second part of the theorem results from 2. 11(ii), 4. 10(i) (iii), and 4. 17(i) (ii). </p><p>THEOREM 4. 22. Every relation algebra </p><p>= 'b eU </p><p>This content downloaded from 138.251.14.35 on Tue, 9 Dec 2014 00:52:29 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>BOOLEAN ALGEBRAS WITH OPERATORS. 139 </p><p>Formulas (3)-(5) imply that x y. Thus, by (2), </p><p>(6) F maps A onto A in one-to-one way. </p><p>By (1) and 4. 3(i) (iii) we have </p><p>(7) F(x + y) =JF(x) U F(y) for any x, y 6 A and </p><p>(8) F(0) A. </p><p>Consider any x, y e A. If ? (x ; y), theni, by (1), a : x ;(y ; b) and a, b ? U. Hence A being atomistic, we conclude by 4. 3(i) that there is a c c U such that a x ; c and c &lt; y ; b. Therefore, by (1), ? (X), F(y), and consequently F (x) I F(y). In a similar way, using (1) and 4. 3 (ii), we show that the latter formula implies e F(x ; y). Thus, </p><p>(9) F(x ;y) = F(x) I F(y) for any x,y A. </p><p>Suppose x e A. If ? [F(x)x1, i. e., F (x), then by (1), b &lt; x;a and a, b ? U. Therefore (x; a) b 0 wheice (x'; b) at4=0 by 4. 1 (iv). Consequently, a ? x''; b and e F(x'). Thus </p><p>[F(x) ]-1C F(x') for every x e A. </p><p>Hence the inclusion in the opposite directiol caii easily be derived with the help of 4. 3 (vii), so that finally </p><p>(10) F(x') - [F(x)]-1 for every xeA. </p><p>In view of (6), F has the inverse function F-1 which maps A onto A. If we now define the binary operation ( on elements of A by putting </p><p>R O) S = F(F-1 (R) * F-1 (S) ) for R, S e A, </p><p>we clearly have </p><p>(11) F(x y) F(x)OC) F(y) for any x,yeA. </p><p>By (2) and (6)-(11), the functions F maps A isomorphically onto </p><p>W'-~</p></li><li><p>140 BJARNI JONSSON AND ALFRED TARSKI. </p><p>It should be emphasized that, in genieral, the operation () in the algebra A' of Theorem 4. 22 is not set-theoretical multiplication; i. e., R and S being two relations in A, R () S does not necessarily coincide with the intersection R n S of R and S. However, the meaning of () is unambigu- ously determined by the fact that, according to 4. 1, </p></li><li><p>BOOLEAN ALGEBRAS WITH OPERATORS. 141 </p><p>algebras. Ini fact, W being a proper relation algebra with the universal relation V, let U be the field of V....</p></li></ul>