A N N A L E S S O C IE T A T IS M A T H E M A T IC A E PO LO N A E Series I : C O M M E N TA TIO N E S M A T H E M A T IC A E X X I I I (1983) R O C Z N IK I P O L S K IE G O T O W A B Z Y S T W A M A TEM A TY C ZN EG O
Sé ria I : P R A C E M A T EM A T Y C Z N E X X I I I (1983)
Wieslaw Aleksandee Dudek (Czçstochowa)
Autodistributive w-groups
1. Introduction. In some recent papers several authors have extended the study of ordinary groups and semigroups to the case where the group operation is not a binary, but an w-ary one. The existence of such theories has motivated us to extend the study of ordinary rings to the case where the ring operations are respectively m-ary and n-ary.
The purpose of the present paper is to give a description of n-groups satisfying a natural generalization of the distributive law for operations.
These n-groups are the special case of (m ,n)-rings.
In the sequel we shall use the terminology and notation of [8 ] and [10].
The symbol S? denotes the general algebra (R ) g, />. The algebra (G, />
is denoted by
A general algebra Sê is called an (m, n)-semiring if (B , g} is an m-semi
group and (B , /> is an n-semigroup, and the operation / is distributive
with respect to g, i.e. .
<1) ’ a>?+i) =* g ({f(4 ~ \ Vi, for each i = 1, 2 , . .. ,n .
If (B , g} is a commutative m-group, then an (m, n )-semiring Si. is called an (m, n)-ring. An ordinary ring is a special case of (m, %)-ring, namely for m = n = 2.
We say that St is a commutative (m, n)-semiring if the operation / is commutative. A neutral element of ( B tf ) is called an identity of St. An idempotent of (R , g) is called an additive idempotent:
The concept of an (m, n)-ring was introduced by Cupona [5]. Paper [2 ] is concerned with homomorphism theorems and some ideal-theoretic aspects, (m, n)-quotient rings are studied in [4]. In [4] are also examined cancellative (m, n)-rings and it is proved that an integral domain with a zero may be embedded into a Unique minimal (m, %)-field. We general
ized the theory of divisibility to the' case of (m, w)-rings in [7]. The construc
tion of the covering ring for thé (m, ?i)-ring (see [1] and [3]) is analogous to the construction of the covering group for the а-group (see [19], §3).
2 W. A. D ud ek
In the theory of n-semigroups the following identity (2 ) / ({/ « )}? -.) = / ({/ < < )}?-1)
plays an important role ([11], [13]; see also [16], p. 87). An ^-semigroup with property (2) is called Abelian. "
2. Autodistributiye ^-semigroups. We say that an w-semigroup is autodistributive if / is distributive with respect to itself.
I t is clear that every autodistributive ^-semigroup <(?,/> is an (n, n)- semiring over itself, i.e., <G ; f , f ) is an (n, n)-semiring. I t is easily verified that an idempotent Abelian ^-semigroup is autodistributive. Moreover, a commutative idempotent w-group is a commutative (n, n)-ring and every element of G is an identity in this (n, »)-ring.
t Note that for all natural n there exists an idempotent commutative semigroup (an w-group also). Hence, for every n, there exists an auto
distributive ^-semigroup and an (n, w)-ring such that every element is an identity.
Observe that if the operation / is autodistributive, then the long product
к and the operation g defined as
(3) ~ æn~i> ■ • • > æ2, a?i)
are autodistributive, too. Moreover, < G ;g ,f} is an (n, n)-semiring.
Lemma 1. An n-semigroup (G, дУ (also <0, /)) is autodistributive i f and only i f the algebra <G; g, /) is an (n, n)-semiring with f satisfying (3).
Let now 23 = <B,/> be an autodistribntive ^-semigroup with an idempotent element b. Then, for all оог, x2, . . . , xn e B and г = 1, 2, . .. , щ /(ж*-1, b, х^+1) is also an idempotent. This implies that the set of all idem-
potents in SB forms an ideal.
A zero of ^ is an element в e G such that f(6 , x2) = f( x 2, 0 ,x ”) = . ..
... = f( x”, 6) ~ 6 for all elements x2, x3, . .. , xn e G. G* denotes the set G\ {6}, if в exists, and G otherwise. From the above remarks it follows that if an autodistributive ^-semigroup has only one idempotent, then this element is a zero.
By a zero of an (m, w)-semiring M is meant a zero of <B, />.
Notice that if the condition: xgy if and only if (4) f(4 ~ \ Щ «?+1) = У, <+i)
holds for some i and all ak e 6r, then it defines an equivalence relation in G.
I t is obvious that 0 is autodistributive if and only if this relation is a con
gruence of ^ for all i — 1, 2 , . . . , n.
Autodistributive n-groups 3
We say that the i-cancellative law holds in an w-groupoid <3 if the condition (4) implies x = y for each x , y e G and all ax, a2, . .. , an eG*.
If ^ is i-cancellative for all i = 1, 2, ..., n, then it is called a cancellative n-groupoid. An (m, w)-semiring ^ is cancellative if the ^-semigroup <R , f } is cancellative.
Le m m a 2. I f 3 is an n-semigroup, then the following conditions are equivalent :
(i) & is cancellative,
(ii) 3 is i-cancellative for i — 1 and i = n, (iii) 3 is i-cancellative fo r some i = 2, 3, n — 1.
P roo f. (iii)=>(i). Let 3 be i-cancellative for some 1 < i < n. If (4) holds for j > i, then we have
Ж , / («Г 1, < +I), 4 +I) = f(b\, у, «7+1), 4 +2) , where h — n — 1 - f i — j. Hence
Ж 2, Ж - i,M "), y, «î+i, K+i implies x — y, i.e., 3 is j-cancellative for ail j > i.
If (4) holds for j < i, then putting h = i —j — 1 and proceeding as before, one get x = y.
The remaining part of the proof easily follows (see also [4]).
Theorem 1. i f m is a cancellative {m, n)-semiring, then (i) has not additive idempotents, or
(ii) only zero (if it exists) is an additive idempotent, or (iii) <J2, g} is an idempotent m-semigroup.
( m )
P ro o f. Assume that b = g ( b ) and b is not a zero of Then by dis- tributivity we have
(m) (m)
f(x%,b) = f ( x l , g ( b ) ) =g(f(a% , b),...,f(x% , b)) = f(g (x 2),a%, b)
for all x2, £P3, . . . , xn e R*.
Cancellativity implies Щ ) for each x2 e R*.
Corollary 1. I f & is an autodistributive and cancellative n-semigroup, then
(i) 3 has not an idempotent element, or (ii) only zero (if it exists) is an idempotent, or (iii) every element is an idempotent.
4 W. A* D udek
Following E. L. Post ([19], p. 282), w© define:
(5) (6)
»<*> _ /(a<fc- 1>, V ) for for x :f( x , a<-k-1>(n —2)
fc > 0 , Jfc = 0 , a ) — a for Jc < 0 . The exponential laws given below are easily verified:
(a<r>)<S> __ q(xs{ti lj-bs+r)
f ( a <sP, a<s?y, . .. , a<s«>) = a<si+s2+--+sn+1>.
J)(n —1) + 1
A minimal natural number p such that a iP> — /(p) ( a ) = a is called an n-ary order of a and is denoted by ordn(u).
Theorem 2. Let У be an autodistributive and cancellative n-semigroup.
Then
(i) fo r each x e G there exists a unique y e G such that
(n—i) (i— 1)
f { x , y, x ) = x and y — xin ' fo r all i — 1, 2 , . .. , n, (ii) x — x<n~ly for all x eG ,
(iii) x — (#<n~2>)<1> fo r all x e G ,
(iv) i f ordM(£c) ~ p, then x — (жСр_1>)<1>.
P ro o f, (i) and (ii). We have
and
W (n_1) s , {nK\ w , > ) \ s l {n~ iy ^/(” _1) ( n - 2>x\
a; = /(* X \ x<n~2}) — x<n ly by virtue of cancellativity.
(и - I )
(iii) (x<n 2>)<1;> = f { x <n 2>, . . . , x <n 2>,f ( x <n 3>, x ))
= f ( f ( x <n 2>, . . . , x <n 2>, x <n 3>), x )- 0,
= f{f(X <n~2>, {П°°1)) v • •, f { ^ n~Z>, inx }), f{ x <n~3>, (П®''))-
(iv) Similarly as (iii).
Corollary 2. Every autodistributive and cancellative n-semigroup is regular, i.e., fo r every у eG there exist x2, xz, . . . , xn_x e G such that у
— f(y , x2~x, У)- Moreover, ea'ôh element of this n-semigroup has a finite
■n-ary order and ordn(&) is a divisor o f n —1.
AutodistribuUve n-groups 5
O. W. Kolesnikov proved in [15] that an n-semigroup is inverse if and only if all regular conjugated elements with a given sequence of elements of G are identical ones. Hence each regular and cancellative n-semi
group is inverse. From Theorem 2 follows that each autodistributive and cancellative ^-semigroup is inverse.
Corollary 3. Each autodistributive and cancellative n-semigroup &
is a set-theoretic union o f disjoint cyclic autodistributive n-groups without proper subgroups.
P roo f. Since every element of this ^-semigroup has a finite n-ary order which is a divisor of n — 1, we have G = ( J where G(x) is a cyclic
x eO
n-group generated by an element x e G. Moreover, G(x) has not any proper subgroups. Indeed, if d is the greast common divisor of ordn (x) and k(n —1) + 1, then ordn(a?) = d -o id n{x<k>) (see [19], p. 283). Because every divisor of ordn(&) divides n — 1 therefore GCD{&(n —1) + 1, ord^(&)} = 1, i.e., ordn(a>) — ordn(a?</c>) for every natural k. This implies that G(x) has any proper subgroups and G is a set-theoretic union of disjoint cyclic n-groups.
Theorem 2 (i) implies that, in auto distributive and cancellative n-semigroup, there exists a uniquely defined a unary operation x-+x<n~2>.
An element x<n~2> is a skew element to x.
Theorem 3. I f У is an autodistributive and cancellative n-semigroup, then
(7) M ) = / t â “S *<,*?+1)
for all xlf x21 . .. , xn e G and i — 1 , 2 , . . . , n.
P ro o f. The proof follows from the equalities
( n - l )
f(®i) = fИ x>/( xi ’ xi)>xi+1) = / (/ « )> • ••>/(»?)>/(®î S *<>»?+1)) and Theorem 2 (i).
Corollary 4. Each autodistributive 3-group is Abelian. Conversely, every idempotent 3-group is autodistributive and Abelian.
P ro o f. W. Dornte proved in [6] that f( x ,y ,z ) = f( z ,ÿ ,x ) . Hence f(x, y, z) = f {x, y, z) = f(x, y, z) = f{z, y, X) = f{z, y, X) .
3. Autodistributive n-groups. Let now be an n-group (not necessarily autodistributive). Let x — æ(0) and let ic(s+1) be the skew element to x(*\
where s > 0. In other words, x = х{1\ x = xj2\ x = #(3), etc.
I t is easily verified that the operation x->x is one-to-one if there exists a natural number h such that x = x{k) for all x e G. Conversely, if ^ is a finite n-group and x-^-x is one-to-one, then x = for all x e G
6 W. A. D ud ek
and some 7c. If ^ is also Abelian, then this operation is an automorphism.
Obviously, if ^ is autodistributive then x-^x is one-to-one.
Corollary 5. I f & is an autodistributive n-group, then (8) x<k> — x{n~k~l) and x{k) ~ (ж(Л+1))<1>
is valid for all x e G and h — 0, 1, . .. , n — 1.
P roo f. Applying (7) and Corollary 3, we obtain x = f(x , X, X, .. ., X, x) — f ( X , x,xf-n x)) = 1}
and
x<x> = /(У) — f ( x \ = /(( x \ x, âfn~2)) = x(n~2\ etc.
Analogously as Corollary 5, we prove
Corollary 6. I f an n-group is autodistributive and ordn(a>) == p fo r some element x, then x = x{p\
From Corollaries 1 and 5 we obtain
Corollary 7. I f У is an autodistributive n-group, then it is either an idempotent n-group or, fo r every x e G, we have x Ф x and x — ж<1;>.
Theorem 4. I f Abelian n-group satisfies (7), then it is autodistributive.
P ro o f. In the same manner as Corollary 5 we can prove that condition (7) implies x = ж(п_1) for all elements x. Now, because this n-group is also Abelian,
/(/«)> Vi) = /(/ К ), Л \ Х>»&),•••> f(% n \ ÿn))
= /(/to, y2 ) , • • •, / t o - 1, y2 ), / to , У21 ÿz, • • •, ÿ J)
= /(/to, У2), • • •,/ to -i, 2/2),/toTO_1), 2/2)) = /({/to, 2/2 • Similarly we can prove remaining cases of (2).
Let Z(A) be a center of a binary group 51 — <A, •). It is clear that the operation f(x^) = x1-x2- ... -xn-b is associative if b eZ (A ). Obviously, ( A ,fy is an n-group for every natural n.
Theorem 5. Let SS.be a binary group and let exp (51) be a divisor of n — 1.
I f f ( xi) ~ x1'X2 - ... ‘Xn-b, where b eZ (A ), then an n-group <A,/> satisfies (7). Moreover, <A,/> is autodistributive provided that 5Ï is commutative.
P ro o f. From definition of the skew element we have x = /( x , x) = xn~x ’Х-b -= x-b and x —x-b~l .
!Now, if b eZ (A ), then b~x eZ (A ), too. Hence
/ to ) = / № -b~l = x 1‘x2’ ... -Xi -b~l -xi+l •... -xn -b = / to -1, xit x?+1).
Autodistributive n-group s 7
Therefore (7) holds for all i = 1, 2, ..., я and every x{ e G. The remaining part of the theorem is obvious.
The definition of the skew element implies that x = a?<-1> = xiv~l>- and x — x<n~3>, where ordn(a?) = p . Hence for all 4-groups x = a?<1>, if x has a finite order. Generally:
Theorem 6. For all elements o f an n-group <$, we have:
(i) i f ordn(a?) = p and p s + r = к > 0, where 0 < r < p, then
(ii) ж№+2) = (ж№+1))<-1>,
(iii) i f ordn(a?) = p , then àë(fc+1) = (æ(fc))<p-1> = (х(к~1))<п~3>, /2_<yi)k _x
(iv) âfk) = x<Sk>, where sk = ■---'= — J ? (2 — п)г. ,
^ г=0
P roo f. Using condition (5) and the covering group, we shall prove (i), (ii) and (iii). We can prove condition (iv) by induction.
Corollary 8. An n-ary order o f x is a divisor o f sk i f and only i f x = x(k\
In particular, ordw(co) divides n — 3 i f and only i f x — x.
We can also prove
Corollary 9. I f there exists a natural number к such that x — x(k), then ordn(x) = ordn(x(s)) fo r all s.
P ro o f. Since ordw(àc) divides ordn(ct>) [6], we have
ordn(âë(fc)) < ord^æ** x)) < ... < ordn(æ) < ordre(a?) = OTdn (xfk)).
Thus ordw(a?) = ordre(æ(s)) for all s.
Corollary 10. I f is an autodistributive n-group and p is minimal natural number such that x = з&р\ then ordn(a?) — p.
P roo f. Corollary 6 implies that p < o r d n(ce). On the other hand, x<*> = / №> Г * +,)>
ы *
-up) X )„ ( Я - 1 ) f ( p ) ( X ,
_ ( n - 2 )
X , X ,
_ ( n - 2) _ X . X , X ,
( n - 2)
,, x , X) = X.
Thus ordn(a>) divides p. Finally, ordn(a?) = p.
I t is easily verified that in the free covering group of ^ (the construc
tion—see [17]) ordfc(<») divides {n — l ) 2/(k — l). Generally, the free cover
ing 7c-group of an autodistributive я-group is not autodistributive.
Since an autodistributive я -group ^ is an (n, я)-ring, there exists a free co
vering (m, n)-ring for all m such that я = k(m —1) + 1 .
The problem of existence of autodistributive я-groups is solved by Theorem 7. F or every n there exists a non-reducible autodistributive n-group.
8 W. A. D ud ek .
P roo f. To prove our theorem, let us consider the set G — {|«|:«
= l(m od^)}, where \a\ = {x: x = «(mod w2)} for n ^ 2 . Dôrnte showed in [6] that G with f defined by the formula
/ ( K M «21? •••» K + iD = K I + K I + ••• + K + il
is an (w-f-l)-group. This group is not reducible, i.e., it is not derived from any m-group. Indeed, if [«I is the skew element to |1| in (G, g'), and (G, f } is derived from m-group (G, g}, where n — Tc(m — 1) and 1c > 1, then
(m—2) (m—2)
/(111, I 1 I, l«l) = </w (|l|, I 1 I, U P = HI-
к tim es к tim es
We can rewrite it in the form
|1 | + й (т е -2 )-|1 | + Л-М = I I I,
i.e., 1c(m — 2 + «) =0(m od w 2). This implies that 1c(m — 2-j-a) = rn2 for some r. Multiplying this equation by m — 1 and cancellating by n, we obtain (m — 2 + a) = rn(m — 1 ). Since \a\eG only for a = m + l , we see that s& + l = r1c(m — 1 ). Hence 1 = Jc(rm — r — s), what is impossible for 1c > 1.
This (n -f- l)-group is also autodistributive, because it is commutative and the (w -fl)-ary order of every element in G is a divisor of n.
I t is proved in [12] that the class of all n-groups forms a variety.
(For the defining identities for this class see also [10].) The class of all autodistributive n-groups (n-semigroups) forms a variety, too. Observe that if ^ is an auto distributive n-group and % is an invariant subgroup of
then is an idempotent autodistributive w-group.
4. Reducts and translations. In this section we consider only some binary reducts and some translations of groups. By a binary reduct of Ф with respect to a is meant the algebra reda(^) = (G, o>, where xoy =
(n -3 )
f(x , a , a, y). Clearly, reda(^) is a binary group. If У is an autodistribu
tive %-group, then ord2(a?) divides {n — l )2 for all elements of reda(^).
Th e o b e m 8 . All binary reducts o f an autodistributive n-group are isomorphic.
P ro o f. We shall prove that reda(^) and redc(^) are isomorphic for all a, c eG .
( n - 3)
First we note that mapping h{x) = f(x , a , a, c) is an automorphism of <&. Indeed, frqm axioms of an w-group, h is one-to-one and onto. The distributive law implies that h is a homomorphism, too.
Autodistributive n-groups 9
{ n - 3) _ _ _
Let x A y = f(x , c ,c ,y ) . Since h (a) = c and Ji(a) = c, we have h (x oy ) = h(f(x,( n - 3)a ,
(n 3)
â, У)) h (a), Л(ô), 7i(y))
(n—3)
= / ( Л ( » ) , C , c, (2/)) = Л(а>) A Л(0 ).
Hence reda(^) and redc(^) are isomorphic.
By a basic translation [18] we mean the mapping t(x) = f(a\~l, x, ar-+1).
I t is always invertible. Its inverse is t~l (x) = fib™-1, x, сг2), where (b{, ...
. . . , bn_j, ax, ..., a^-f) and (ai+1, . .. , an, c2, . .. , c{) are (n — l)-adic identities [19]. An elementary translation is a composition of basic translations, and hence it is invertible. If 0 is an group, then the set of all elementary translations forms a transitive binary group.
By a right (left) translation of 0 is meant the mapping <p n(x) — f(x , s2)
S2
(w J x ) — f( s 2, x)). From [15] we know that an n-semigroup ^ is an w-group
s 2
if and only if Ф = {<p^n : s{ e G} and Ф = {ip^: s{ eG} are binary groups.
If s2, ..., s{_x, si+1, . .. , sn are fixed, then Ф is denoted by Фгп. Bemark that
S2
if ^ is an n-group, then the equation s2 = f(t\~1, y, t^+1, s2) has a unique solution for all elements of G. Similarly, the equation sn = f( s n, t f 1, zi ti+1)*
This implies that <p n = cp n for all % = f(2){y,% + i,^ ,ti \ z). Hences0 t„ >
ф\ = Ф1 = ф* №Ш • «,№)
(к—2) (n—k)
<р%т = « , c, z )}
for all sp, tp, e, z eG and i , j , k = 2 , 3 , . . . , n. I t is easily verified that a can- eellative w-semigroup is semi-commutative if and only if for some i all
Фгп are commutative. Therefore, an n-group is Abelian if and only if,
s2
for some fixed i and z, Фгг is commutative (see also [8]).
The remarks above yield to
Corollary 11. I f is an n-group, then Фгп is a transitive group and has card (6r) elements. Moreover, Ф\ ~ Ф{ = Ф for all z,sp eG and i , j
s 2
= 2, 3, . .., n.
Analogous results hold for left translations.
In the same manner as Theorem 2 in [10] we can prove
Theorem 9. An n-semigroup У is an n-group i f and only i f for some i, j = 2 ,3 , n and all a E G there exist b,C EG such that cp^)b (x)
= Va!c(œ) = œ , where y>{£e(x) = / (V * , c, V * , x).
Obviously, if ^ is an autodistributive ^-semigroup (an w-group), then all translations are homomorphisms (automorphisms). If ^ is commuta-
10 W . A. D ud ek
tive, then ^ is autodistributive whenever all right (all elementary) trans
lations are homomorphisms.
5. Final remarks. If ^ is an ^-semigroup, then a non-empty subset B cz G such that B <k} c G \ B <8> is called an (k, s)-mutant in
J . B. Kim [14] proved that аду binary semigroup has no decompo
sitions into a finite number of disjoint (2, l)-mutants. We generalized this result in [9]. This result is not true for n-semigroup where n > 2.
Th e o r e m 10. Bach non-idempotent autodistributive n-group has a decom
position into some number o f disjoint (n — 2 ,n -3)-m utants.
P ro o f. If = {а{}, then B jn~2> = {d j and = (S j. Hence B { are (n — 2 ,n — 3)-mutants.
Added in proof (February, 1983). As it is well known, (see [8], [10], [12]) an
«-group may be defined as a special «-semigroup (G, f ) with a unary operation
— : x-+x, i.e., as a some universal algebra (G ; f , —) of type (« , 1). If (G,f) is an «-group, then this unary operation is uniquely defined- as a solution x of the equation f(x, x, . .. ,æ , x) — x. If (G, f) is an «.-semigroup (but it is not an «-group), then the solution x (if it exists) is not unique. For example, in an «-semigroup (G, / ) with / defined as f ( x ?) = xx every unary operation g satisfies f ( x , x , . . x , g(x)) = x.
An «-semigroup (6r, / ) with a unary operation g satisfying the last equation is called distributive (or weakly distributive) if /(ж|- 1 , g{xi), x™+ l) — gifix™)) for all x lf x 2, . . . , x n e G and i — 1 , 2 , . . . , « . Hence an «-group (G ; f , — ) is distributive if and only if it satisfies (7). Observe that an «-semigroup {G,f) with a unary operation g defined as above is distributive if ( G ; f , g ) is an («, l)-semiring. Moreover, these «- semigroups are special cases of (f/g)-algebras in sense of Hoehnke (comp. J . H. H oe- hnke, On the principle of distributivity, Preprint of the Math. Inst. Hungarian Acad.
Sci., Budapest 1980).
The class of all distributive «-groups forms a variety. The class of all autodistri
butive «-groups is a proper subvariety of a variety of all distributive «-groups. Free distributive (autodistributive) «-groups have «-ary exponent equal to « — 1. There
fore free «-groups in these varieties are set-theoretic unions of disjoint cyclic auto
distributive «-groups without proper subgroups.
All the above results will be proved in my paper On distributive n-groups.
References 1
[1] D. B o c c io n i, Caratterizzazione di una classe di anelli generalizzati, Bend. Sem.
Math. Univ. Padova 35 (1965), 116-127.
[2] G. C rom b ez, On (n,m )-rings, Abhandlungen Math. Sem. Univ. Hamburg 37 (1972), 180-199.
[3] —, The Post coset theorem for (n,m )-rings, Atti. Inst, vento sci. lett. 131 (1972/3), 1-7.
[4] — and J . T im m , On («, m)-quotient rings, Abhandlungen Math. Sem. Univ.
Hamburg 37 (1972), 200-203.
[5] G. C u p on a, On (m, n)-rings, Bull. Soc. Math. Phys. B. S. Macedoine 16 (1965), 5 -1 0 (in Macedonian).
Autodistributive n-groups 11
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