AN IDENTITY FOR /-GROUPS1
J. A. KALMAN
G. Birkhoff [l, p. 109, Theorem 7.8] discovered that2
(1)
\a\Jc-b\Jc\
+ \ar\c-bnc\
= \a-b\
for all a, b and c in any vector lattice. In [2, p. 309] he stated that he
had been unable to generalize the identity (1) to noncommutative
/-groups. However he proved (cf. [3, p. 220, Theorem 8(20)]) that
(2)
| a W c - b \J c | ^ | a - b \ , and dually,
for all a, b and c in any /-group.
We shall prove that the identity (1) actually does hold in all /groups. Clearly, this gives a new proof of (2).
We begin by proving that, for all a, b and c in any /-group,
(3) (ar\b)Uc-c+(ayJb)r\c
= (aUb)nc-c+(al^b)Uc.
Indeed,
(ar\b)\j(b(^c)
w(cMo) = («nj)u
{(«ui)nc}
= «Hi- (an6)n(aWô)nc+(aU6)nc
= ar\b= ar\b-
(af\b)r\c+
(a\Jb)C\c
aC\b+ (ar\b)VJcc+(aUb)nc;
thus
(a n b) yj (b r\ c) \j (c r\ a) = (a r\ b) w c - c + (a yj b) n c.
Dually,
(a U b) i\ (b \J c) H (c VJ a) = (a VJ b) H c - c + (a n b) U c.
Since [3, p. 133] (ar\b)yj(br\c)V)(cr\a)
this proves (3).
We now prove the identity
we have
= (aV)b)r\(bVJc)r\(cKJa),
(1). For all a, b, and c in any /-group,
Presented to the Society, April 23, 1955 under the title Some results in lattice the-
ory; received by the editors October 13, 1955.
1 The result of this paper
University, 1955). The author
versity of New Zealand.
2 We make the convention
over the group operations: for
is taken from the author's doctoral thesis (Harvard
now holds a Research Fund Fellowship from the Unithat the lattice operations in an /-group have priority
example, a —bC\c means a —(b(~\c).
931
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932
| aUc
J. A. KALMAN
- ¿Wc| + \aC\c - bC\c\
= (a\J c)\J (b\J c) - (a\J c) Pi (6U c) + (af\ c) W (b Pi c)
- (a r\ c) r\ (b r\ ¿)
= (a\J b) VJ c - (ar\ b) KJ c + (aVJ b) r\ c - (ar\ b) r\ c
= a\J b - (dU b) P c + e - (aC\b)\J c + («U *) P c
-c+(anô)Uc-anô
= aUS-
{(a P ¿>)U c - c + (a U 6) H c}
+ {(a\J b) n c - c + (ar\ b) \J c} - aPi
= a\JbaC\b, by (3).
Since aVJb—a(~\b = \a —b\, this completes the proof of (1).
References
1. G. Birkhoff, Lattice theory, 1st ed., New York, 1940.
2. -,
Lattice-ordered groups, Ann. of Math. (2) vol. 43 (1942) pp. 298-331.
3. -,
Lattice theory, Amer. Math. Soc. Colloquium
1940; rev. ed., New York, 1948.
Harvard University
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Publications,
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