sets of postulates for boolean groups

Annals of Mathematics

Sets of Postulates for Boolean GroupsAuthor(s): B. A. BernsteinSource: Annals of Mathematics, Second Series, Vol. 40, No. 2 (Apr., 1939), pp. 420-422Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968930 .

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ANNALS OF MATHEMATICS

Vol. 40, No. 2, April, 1939

SETS OF POSTULATES FOR BOOLEAN GROUPS

BY B. A. BERNSTEIN

(Received August 13, 1938; revised January 3, 1939)

1. Object In a Boolean algebra, the operation o given by ao b = ab' + a'b is an operation

with respect to which the elements of the algebra form a group.2 The group is an (additive) abelian group in which every element is its own inverse. An abelian group in which every element is its own inverse I call a Boolean group. The object of my paper is to give a number of sets of postulates for (non-trivial) Boolean groups in terms free from the usual symbolism of Boolean algebra.

2. List of postulates The postulates constituting the various sets are taken from the following list

of conditions on an undefined class K and a binary operation o. In Postulates 2-2i"y 3, 7_7v, 8, 8' there is to be supplied the restriction: whenever the elements involved and their indicated combinations are in K. The list follows.

0. K has at least two distinct elements. 1: ao b is in K whenever a, b are in K. 2. ao (bo c) = (ao b)o c. 2'. ao (bo c) = bo (co a). 2". ao (bo c) = co (bo a). 2"'. (aob)oc = (aoc)ob. 2iv. ao(boc) = (aoc)ob. 3. aob = boa. 4. There is an element 0 in K such that ao 0 = a for every element a in K. 5. If the element 0 of Postulate 4 exists, and is unique, then for every element a

in K there is an element a' in K such that aoa' = 0. 6. If the elements 0, a' of Postulates 4, 5 exist, and are unique, then a' = a. 7. a= (bob)oa. 7'. a= (boa)ob. 7". a = (ao b)o b. 7"'. a = bo(boa). 7iv a = bo(aob). 7v a = ao(bob).

I Presented to the American Mathematical Society, April 9, 1938. 2 See my two papers: (I) "Operations with respect to which the elements of a Boolean

algebra form a group," Trans. Amer. Math Soc., vol. 26 (1924), pp. 171-175, and vol. 27 (1925) p. 600; (II) "A set of postulates for Boolean algebra involving the operation of com- plete disjunction," these Annals, vol. 37 (1936), pp. 317-325.

420



SETS OF POSTULATES FOR BOOLEAN GROUPS 421

8. If ao b = c then b = coa. 8'. If aob = cthenb = aoc. 9. For any two elements a, b in K there is an element x in K such that ao x = b.

3. The postulate sets

The various sets of postulates are given in the table below. For each set is furnished a proof of the mutual independence of the postulates of the set. The independence examples are taken from the arithmetic systems i-xii following.

i. K= 0; aob = 0.

ii. K = 0,1; aob = a + b + 0/(ab + b + 1) mod. 2.

iii. K = 0,1,2; aob = a2(b2 + 2b) + a(2b2 + 2b + 1) + b mod. 3.

iv. K = 0,1; aob = a.

v. K= 0,1; aob = 0.

vi. K= 0,1; aob = ab+a+bmod.2.

vii. K= 0,1,2; aob = a+bmod.3.

viii. K = 0,1; aob=0 *. 0.

ix. K = 0,1; aob = b.

x. K =0,1,2; aob =2(a + b) mod. 3.

xi. K=0,1,2; aob=a+2bmod.3. xii. K=0,1; aob=a *. b.

The table of postulate sets follows. In this table the independence system for the ith postulate in any set occupies the ith place among the independence systems for that set. Thus, for Set II, whose postulates are 0, 1, 2', 7, the independence systems are respectively i, viii, ix, v.

TABLE OF POSTULATE SETS

Set Postulates Independence Systems

I 0, 1, 2, 3, 4, 5, 6 i, ii, iiiiv, v, vivii II 0, 1, 2', 7 i, viii, ix, v III 0,1, 2', 7' i, viii, x, v IV 0, 1, 2', 7" i, viii, iv, v V 0, 1, 2', 7"' i, viii, ix, v VI 0, 1, 2', 7iv i, viii, x, v VII 0, 1, 2', 7v I, viii, iv, v VIII 0, 1, 2, 7iv i, viii, x, v IX 0, 1, 2", 7 i, viii, ix, v X 0, 1, 2"', 7 i, viii, ix, v XI 0, 1, 2iv, 7 i, viii, ix, iv



422 B. A. BERNSTEIN

XII 0, 1, 2, 8 i, viii, x, ix XIII 0, 1, 2', 8 i, viii, x, vii XIV 0, 2, 8, 9 i, x, vii, viii XV 0, 2', 8, 9 i, x, vii, viii XVI O 2iv 8, 9 i, x, vii, viii XVII 0, 1, 2, 3, 7"' i, viii, x, ix, v XVIII 0, 2, 3, 9, 8' i, x, ix, viii, vii XIX 0, 1, 2", 7"', 3 i, Viii, x, v, xi XX 0, 2", 9, 3, 8' i, x, xii, xi, vii

4. Remarks on the postulate sets All the sets in the above table, except Set I, are free from unconditioned exist-

ence postulates, other than Postulate 0, which merely rules out trivial systems. Set I gives a Boolean group directly in accordance with the definition of a

Boolean group. It contains both the ordinary associative law and the commutative law. Postulates 0-5 of this set, taken by themselves, form a set of postulates for additive abelian groups.

Sets II-VII are briefer than Set I. In each of these sets the cyclic associative law is used instead of the ordinary associative law; and each set contains the closure condition and a (direct) cancellation law, but no commutative law.

Sets VIII-XI are variations of Sets II-VII. In these sets, other associative laws than the cyclic associative law are used.

Sets XII and XIII have a transposition law, instead of a cancellation law. Sets XIV-XVI contain no closure postulates. Set XVII is a set, briefer than Set I, in which the ordinary associative law

and the commutative law both appear. Set XVIII, like Set I, "embeds" a set of postulates for additive abelian groups

(Postulates 0, 2, 3, 9); but it is briefer than Set I. Set XIX embeds a set of postulates for subtractive abelian groups (Postulates

0, 1, 2",Y 7"'i).3 Set XX embeds both a set of postulates for additive abelian groups (Postu-

lates 0, 2", 9, 3) and a set of postulates for subtractive abelian groups (Postu- lates 0, 2", 9, 8').

3 See my "Postulates for abelian groups and fields in terms of non-associative operations," Trans. Amer. Math. Soc., vol. 43 (1938), pp. 1-6.

THE UNIVERSITY OF CALIFORNIA.



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