ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I V (1970) ROCZNIKI POLSKIEGO T O W A R ZYSTW A MATEMATYCZNEGO
Séria I : PEACE MATE M A TY CZNE X I Y (1970)
E. M arczewski (Wroclaw)
A remark on independence in abstract algebras
This note is a small complement to my paper [1].
A subset I of the carrier of an algebra is called independent, if for every finite sequence ax, . .. , am (m > 0) of distinct elements of I and for every pair of m-ary algebraic operations / x and f 2, if
(1) / i K > ••• ? ^m)
. / 2(^l > • •• J am) ? then
(2) < II
It is essential in this definition that we consider all pairs of m-ary algebraic operations. W e cannot assume that they depend on every of their variables. In a semilattice ($ ; + ) , for instance, for every m there is only one such m-ary operation (namely x x- { - x m), and consequently, for pairs of such operations the implication (1) => (2) always occurs, although there exist dependent sets (e.g. {ax, a2, ax-\-af$).
Nevertheless, it is possible to describe the independence merely by means of operations which depend on all their variables. Such a formulation, which is the aim of this note, is needed in some research on independence (see e.g. [2]).
In spite of the ordinary definition of independence, quoted above, this new description requires an explicit use of nullary operations. Such an operation (i.e. a constant) will be treated as an operation dependent on all its variables.
I shall prove that in every algebra, a subset D of the carrier is depend
ent, if and only if there are: a sequence ax, ..., am (m > 0) of distinct ele
ments of D, a sequence am+1, . .., am+n (n > 0) of distinct elements of D, an m-ary algebraic operation f x and an n-ary algebraic operation f 2 both dependent on all their variables and such that
( ^ ) f l 1 > • • • > ®m) f 2 i.^m+1 f • • ‘ f ®m + n )
and either
(bx) f x and / 2 are distinct and sequences ax, . . . , a m and am+1, . . . , a TO+n
are identical {i.e. m = n and a = am+j for m)
16 E. Marczewski
or
(b2) the sets {ax, . . . , a m} and {am+l1 . . . , am+n} are distinct.
N e c e s s i t y . D being dependent, there are a sequence of distinct elements bx, ..., bkeD and two distinct fc-ary algebraic operations gx, g2 (k > 1) such that
9i(bi, ••.,&*) = gzibu
By omitting the variables on which gx or g2 does not depend we obtain an equality of the form
fl (bri ? • • • J ЬГи) — fz{bsX1 * * ' 5 >
where w > 0, v > 0 and where f x and / 2 are algebraic and depend on all their variables.
Since gx and g2 are distinct, if the sequences
(3) rx, . . . , r u and 8l f ...,8 v
are identical, then f x and / 2 are also distinct and we obtain (a) and (bx) . If sequences (3) are distinct, then the sets {ôri, . .. , } and
} are also distinct and we have (a) and (b2).
S u f f i c i e n c y . If (a) and' (bx), then D is dependent by definition.
Let us assume (a) and (b2) , where f x and / 2 depend on all their varia
bles. W e can assume e.g. that
(4) m > l and ax4{am+x, . .., am+n}.
We arrange now elements
in such a sequence
ax, . .. , am, am^_x, . .. , am4rn
Ьц • • • ? bm, . .. , br
without repetitions that aj — bj for j < m. We have of course m < r and am+j — bk. for 1 < j < n (where kx, ..., kn form a sequence of distinct numbers with 1 < Ц < r).
We put now
gx{xx, . . . , x r) = / x(a?x,
g2{ccx, . . . , xr) = f 2(xkl, . . . , xkn) .
If n = 0, then f 2 is a nullary algebraic operation, i.e. an algebraic
constant and the definition of g2 obviously remains valid. In any case
it follows from (4) that kj Ф 1 for 1 < j < n and hence g2 does not depend
Independence in abstract algebras
17
on the variable x x. On the other hand, f x depends on x x and hence gx depends on x x too. Consequently, gx
фgz and simidtaneonsly
9l{bl1 •••? br) ~ •••» ^m) ~ •••? ^m)
= fzi.^m+1 J • • * 5 Q'm+n) ~ fzi^kx1 * * • ) Ь]сп) == • • • j ^r) • Thus В is dependent by definition and the proof is completed.
References
[1] E. M a r c z e w s k i, Independence and JiomomorpMsms in abstract algebras, Fund.
Math. 50 (1961), pp. 45-61
[2] J. P lo n k a , On sums of direct systems of Boolean algebras, Coll. Math. 20 (1969), pp. 209-214.
2 — Prace matematyczne X IV
